Abstract
We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible \(*\)-structures, encompassing the quasitriangular case.
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1 Introduction
The gauge group of a principal bundle can be given as bundle automorphisms (diffeomorphisms of the total space onto itself which respect the group action) covering the identity map on the base space. Elements of the gauge group act by pullback on the space of connection one-forms on the bundle, thus playing a central role for the definition of the moduli space of connections. In the dual algebraic language a principal bundle is given as an algebra extension \(B\subseteq A \) which is H-Hopf–Galois, for H a Hopf algebra. A gauge transformation would then be given as an H-equivariant algebra morphism of A onto itself which restricts to the identity on B. This dual definition works well for the case of commutative algebras, and also for Hopf–Galois extensions with H coquasitriangular and B commutative [4]. It is however too restrictive in general, due to the scarcity of morphisms for a generic noncommutative algebra. Finding a good notion of bundle automorphisms and gauge transformations for noncommutative principal bundles is still an open problem.
In our previous paper [5] we looked at the problem from the infinitesimal view-point by considering algebra derivations, rather than algebra morphisms. We considered derivations of algebras which are commutative up to a braiding. The derivations are required to form a braided Lie algebra as well as a module over the algebra. Compatibility of these two structures is better understood in the context of triangular braidings.
We then studied the case of H-Hopf–Galois extensions which are equivariant under a triangular Hopf algebra \((K,{\textsf {R} })\). Infinitesimal gauge transformations are now given by H-comodule maps that are vertical braided derivations, with the brading implemented by the triangular structure \({\textsf {R} }\) of the symmetry K. These maps form a braided Lie algebra \(\textrm{aut}^{\textsf {R} }_B(A)\) and lead to a braided Hopf algebra \({\mathcal {U}}(\textrm{aut}^{\textsf {R} }_B(A))\) of infinitesimal gauge transformations. The construction is shown to be compatible with the theory of Drinfeld twists, and thus suitable for the study of noncommutative principal bundles that are obtained via twist deformation (quantization) of classical ones. We refer to [5] for details and for a discussion of different approaches and of the literature on the subject.
In the present paper we complement the general theory developed in [5], and briefly reviewed in § 2, with a systematic analysis of \(*\)-structures on braided Hopf algebras associated with quasitriangular Hopf algebras. This is done in §3 where we also study their compatibility with actions on \(*\)-algebras. In the triangular case we further consider braided Lie \(*\)-algebras and their representations on \(*\)-algebras.
We then illustrate the general theory with the computation of the braided Lie \(*\)-algebras of infinitesimal gauge transformations of two important examples of noncommutative principal bundles. These are given by two Hopf–Galois extensions of the algebra \({\mathcal {O}}(S^4_\theta )\) of the noncommutative 4-sphere \(S^4_\theta \) of [8] associated to an abelian twist. Additional examples obtained from cotriangular quantum groups, and from abelian as well as Jordanian twists are in [5, §7.1, §8.1]. In §4.1 we determine the braided Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta )}({\mathcal {O}}(S^7_\theta ))\) of infinitesimal gauge transformations of the \({\mathcal {O}}(SU(2))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(S^7_\theta )\) of [12]. This bundle can also be obtained as a deformation by a twist on \({\mathcal {O}}({\mathbb {T}}^2)\) of the Hopf–Galois extension \({\mathcal {O}}(S^4) \subset {\mathcal {O}}(S^7)\) of the classical SU(2)-Hopf bundle [3]. This allows for the construction of \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ) )\) from its classical counterpart, following the general theory. The explicit description of the classical gauge Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\) relies on the Spin(5) equivariance of the principal bundle \(S^7\rightarrow S^4\). This equivariance also implies that, as linear space, \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7)) \) splits as a direct sum over a class of representations of the Lie \(*\)-algebra so(5), of vertical \({\mathcal {O}}(SU(2))\)-equivariant derivations, this is done in § 4.1.4. Following a similar procedure, in § 4.2 we compute the braided Lie \(*\)-algebra of infinitesimal gauge transformations of the \({\mathcal {O}}(SO_\theta (4,{\mathbb {R}}))\)-Hopf–Galois extension \({\mathcal {O}}(S^{4}_\theta ) \subset {\mathcal {O}}(SO_\theta (5,{\mathbb {R}}))\) of the quantum homogeneous space \({\mathcal {O}}(S^{4}_\theta )\) [3, 14].
The braided Lie algebras of gauge transformations of these Hopf–Galois extensions of \({\mathcal {O}}(S^4_\theta )\) are re-obtained in [6] via an intrinsic construction, which does not use the twist procedure. They are further studied there in the context of Atiyah sequences of braided Lie algebras, generalising the Atiyah sequence of a principal bundle.
2 Braided Lie algebras of gauge transformations
The main objects investigated in this paper are K-equivariant Hopf–Galois extensions, for \((K, {\textsf {R} })\) a triangular Hopf algebra, and their braided Lie algebras of gauge symmetries. We briefly recall from [5] the main notions and results that are needed.
We work in the category of \(\mathbb {k}\)-modules for \(\mathbb {k}\) a commutative field, or the ring of formal power series in an indeterminate and coefficients in a field. All algebras are assumed to be unital and associative; morphisms of algebras preserve the unit. Dually for coalgebras. We use standard terminologies and notations in Hopf algebra theory. For H a bialgebra we also call H-equivariant a map of H-modules or H-comodules.
Recall that an algebra A is a right H-comodule algebra for a Hopf algebra H if it carries a right coaction \(\delta : A \rightarrow A \otimes H\) which is a morphism of algebras. As usual we write \(\delta (a)= {a}_{\scriptscriptstyle {(0)}} \otimes {a}_{\scriptscriptstyle {(1)}}\) in Sweedler notation with an implicit sum. Then the subspace of coinvariants \(B:= A^{coH}=\big \{b\in A ~|~ \delta (b) = b \otimes 1_H \big \}\) is a subalgebra of A. The algebra extension \(B\subseteq A\) is called an H-Hopf–Galois extension if the canonical map
is bijective. There may be additional requirements, such as faithful flatness of A as a right B-module, to be mentioned when needed.
In the present paper we deal with H-Hopf–Galois extensions which are K-equivariant for a Hopf algebra K. That is A carries also a left action \(\vartriangleright \,: K\otimes A \rightarrow A\) that commutes with the right H-coaction, \( \delta \,\circ \vartriangleright = (\vartriangleright \otimes \textrm{id}) \circ (\textrm{id}\otimes \delta )\) (the coaction \(\delta \) is a K-module map where H has trivial K-action). On elements \(k\in K, a\in A\),
We further assume the Hopf algebra K to be quasitriangular. Recall that a bialgebra (or Hopf algebra) K is quasitriangular if there exists an invertible element \({\textsf {R} }\in K \otimes K\) (the universal R-matrix of K) with respect to which the coproduct \(\Delta \) of K is quasi-cocommutative
for each \(k \in K\), with \(\Delta ^{cop}:= \tau \circ \Delta \), \(\tau \) the flip map, and \({{\overline{{\textsf {R} }}}}\in K \otimes K\) the inverse of \({\textsf {R} }\), \({\textsf {R} }{{\overline{{\textsf {R} }}}}= {{\overline{{\textsf {R} }}}}{\textsf {R} }= 1 \otimes 1 \). Moreover \({\textsf {R} }\) is required to satisfy,
We write \({\textsf {R} }:={\textsf {R} }^\alpha \otimes {\textsf {R} }_\alpha \) with an implicit sum. Then \({\textsf {R} }_{12}={\textsf {R} }^\alpha \otimes {\textsf {R} }_\alpha \otimes 1\), and similarly for \({\textsf {R} }_{23}\) and \({\textsf {R} }_{13}\). From conditions (2.3) and (2.4) it follows that \({\textsf {R} }\) satisfies the quantum Yang–Baxter equation \( {\textsf {R} }_{12} {\textsf {R} }_{13} {\textsf {R} }_{23}= {\textsf {R} }_{23} {\textsf {R} }_{13}{\textsf {R} }_{12} \). The \({\textsf {R} }\)-matrix of a quasitriangular bialgebra \((K,{\textsf {R} })\) is unital: \((\varepsilon \otimes \textrm{id}) {\textsf {R} }= 1 = (\textrm{id}\otimes \varepsilon ) {\textsf {R} }\), with \(\varepsilon \) the counit of K. When K is a Hopf algebra, quasitriangularity implies that its antipode S is invertible and satisfies
The Hopf algebra K is said to be triangular when \({{\overline{{\textsf {R} }}}}= {{\textsf {R} }}_{21},\) with \({{\textsf {R} }}_{21}= \tau ({\textsf {R} })={\textsf {R} }_\alpha \otimes {\textsf {R} }^\alpha \).
2.1 Braided Hopf algebras
We recall that a braided bialgebra associated with a quasitriangular Hopf algebra \(( K,{\textsf {R} })\) is a K-module \((L,\vartriangleright _L)\) which is both a K-module algebra \((L, m_L, \eta _L, \vartriangleright _L)\) and a K-module coalgebra \((L, \Delta _L,\varepsilon _L, \vartriangleright _L)\) and is a bialgebra in the braided monoidal category of K-modules. That is, \(\varepsilon _L: L \rightarrow \mathbb {k}\) and \(\Delta _L: L \rightarrow L \boxtimes L \) are algebra maps with respect to the product in L and the product
in
\(L\otimes L\) defined by
for \( x, y, x', y' \in L\) and
the braiding. We denote
. It is a K-module algebra with action
Such an L is a braided Hopf algebra if there is a K-module map \(S_L: L\rightarrow L \), the braided antipode, which satisfies
It turns out that \(S_L\) is a braided algebra map:
and a braided coalgebra map:
(For \(\Psi _{\textsf {R} }= \tau \) the flip map, the previous conditions state that \(S_L\) is an antialgebra and anticoalgebra map.)
Due to the quasi-cocommutativity property (2.3), the action in (2.8) commutes with the braiding: \(\vartriangleright _{L \boxtimes L} \circ \,\Psi _{\textsf {R} }=\Psi _{\textsf {R} }\circ \vartriangleright _{L \boxtimes L}\). More generally, given two K-modules V, W and braiding \( \Psi _{\textsf {R} }:V\otimes W\rightarrow W\otimes V\), \(\Psi _{\textsf {R} }(v\otimes w)={\textsf {R} }_\alpha \vartriangleright _W w\otimes {\textsf {R} }^\alpha \vartriangleright _V v\), the actions of the coalgebra K on their tensor products satisfy \( k \vartriangleright _{W \otimes V} \circ \Psi _{\textsf {R} }=\Psi _{\textsf {R} }\circ k \vartriangleright _{V\otimes W} \) for \(k \in K\).
2.2 Braided Lie algebras of derivations
We study derivations of quasi-commutative algebras. As mentioned in the introduction, compatibility of the braided Lie algebra and the module structures works well in the context of triangular braidings.
Here we take \((K,{\textsf {R} })\) to be triangular, an assumption which is enough for the purposes of the present paper. A braided Lie algebra associated with a triangular Hopf algebra \((K, {\textsf {R} })\), or simply a K-braided Lie algebra, is a K-module \({\mathfrak {g}}\) with a bilinear map
that satisfies the following conditions:
-
(i)
K-equivariance: for \(\Delta (k) = {k}_{\scriptscriptstyle {(1)}} \otimes {k}_{\scriptscriptstyle {(2)}}\) the coproduct of K,
$$\begin{aligned} k\vartriangleright [u,v]=[{k}_{\scriptscriptstyle {(1)}}\vartriangleright u,{k}_{\scriptscriptstyle {(2)}}\vartriangleright v] \end{aligned}$$ -
(ii)
braided antisymmetry:
$$\begin{aligned}{}[u,v]=-[{\textsf {R} }_\alpha \vartriangleright v, {\textsf {R} }^\alpha \vartriangleright u ], \end{aligned}$$ -
(iii)
braided Jacobi identity:
$$\begin{aligned}{}[u,[v,w]]=[[u,v],w]+[{\textsf {R} }_\alpha \vartriangleright v, [{\textsf {R} }^\alpha \vartriangleright u,w] ], \end{aligned}$$
for all \(u,v,w \in {\mathfrak {g}}\), \(k\in K\).
As shown in [5, §5.1], the universal enveloping algebra \({\mathcal {U}}({\mathfrak {g}})\) of a braided Lie algebra \({\mathfrak {g}}\) associated with \((K,{\textsf {R} })\) is a braided Hopf algebra. The coproduct of \({\mathcal {U}}({\mathfrak {g}})\) is determined requiring the elements of \({\mathfrak {g}}\) to be primitive, \(\Delta (u)=u\boxtimes 1+1\boxtimes u\), for all \(u\in {\mathfrak {g}}\).
Any K-module algebra A is a K-braided Lie algebra with bracket given by the braided commutator
(See [5, Lemma 5.2].) In particular, if A is a K-module algebra, then also the K-module algebra \(({\textrm{Hom}}(A,A), \vartriangleright _{{\textrm{Hom}}(A,A)})\) of linear maps from A to A with action
is a braided Lie algebra with the braided commutator; here S is the antipode of K. Elements \(\psi \) in \({\textrm{Hom}}(A,A)\) which satisfy
for any \(a,a'\) in A are called braided derivations. We denote \( {\textrm{Der}}^{\textsf {R} }(A) \) the \(\mathbb {k}\)-module of braided derivations of A (to lighten notation we often drop the subscript \({\textsf {R} }\)). It is a K-submodule of \({\textrm{Hom}}(A,A)\), with action given by the restriction of \(\vartriangleright _{{\textrm{Hom}}(A,A)}\)
and moreover, see [5, Prop. 5.7], a braided Lie subalgebra of \({\textrm{Hom}}(A,A)\) with
When the K-module algebra A is quasi-commutative, that is when
for all \( a,a' \in A\), the braided Lie algebra \(\textrm{Der}{(A)}\) with
for \(\psi \in {\textrm{Hom}}(A,A)\), \(a,a'\in A\), is also a left A-submodule of \({\textrm{Hom}}(A,A)\). The Lie bracket of \(\textrm{Der}{(A)}\) satisfies ( [5, Prop. 5.8])
for all \(a,a'\in A, \psi ,\psi '\in \textrm{Der}{(A)}\).
2.3 Infinitesimal gauge transformations
Let now \(B=A^{co H} \subseteq A \) be a K-equivariant Hopf–Galois extension, for \((K,{\textsf {R} })\) a triangular Hopf algebra. Inside the braided Lie algebra \(\textrm{Der}{(A)}\) we consider the subspace of braided derivations that are H-comodule maps,
and then those derivations that are vertical,
The linear spaces \(\textrm{Der}^{\textsf {R} }_{{{\mathcal {M}}^H}}{(A)}\) and \(\textrm{aut}^{\textsf {R} }_B(A)\) are K-braided Lie subalgebras of \(\textrm{Der}{(A)}\), [5, Prop. 7.2]. Elements of \(\textrm{aut}^{\textsf {R} }_B(A)\) are regarded as infinitesimal gauge transformations of the K-equivariant Hopf–Galois extension \(B=A^{co H} \subseteq A\), [5, Def. 7.1]. There is the corresponding braided Hopf algebra \({\mathcal {U}}(\textrm{aut}^{\textsf {R} }_B(A))\) of gauge transformations.
2.4 Twisting of braided Lie algebras
Important examples of noncommutative principal bundles come from twisting classical structures. Aiming at studying their braided Lie algebras of infinitesimal gauge transformations, we need to first consider twist deformations of braided Lie algebras.
We recall some basic results of the theory of Drinfeld twists [9].
Let K be a bialgebra (or Hopf algebra). A twist for K is an invertible element \(\textsf {F} \in K \otimes K\) which is unital, \( (\varepsilon \otimes \textrm{id}) (\textsf {F} )= 1= (\textrm{id}\otimes \varepsilon )(\textsf {F} ) \), and satisfies the twist condition
For \(\textsf {F} \) and its inverse \({\overline{\textsf {F} }}\) we write \(\textsf {F} =\textsf {F} ^{ \alpha } \otimes \textsf {F} _\alpha \) and \({\overline{\textsf {F} }}=: {\overline{\textsf {F} }}^{\alpha } \otimes {\overline{\textsf {F} }}_{\alpha }\), with an implicit sum. The R-matrix \({\textsf {R} }\) of a quasitriangular bialgebra K is a twist for K.
When K has a twist it can be endowed with a second bialgebra structure which is obtained by deforming its coproduct and leaving its counit and multiplication unchanged. Moreover if K is triangular, or more in general quasitriangular, so is the new bialgebra:
Proposition 2.1
Let \(\textsf {F} =\textsf {F} ^{ \alpha } \otimes \textsf {F} _\alpha \) be a twist on a bialgebra \((K, m, \eta , \Delta , \varepsilon )\). Then the algebra \((K, m, \eta )\) with coproduct
and counit \(\varepsilon \) is a bialgebra. If in addition K is a Hopf algebra, then the twisted bialgebra \({K_\textsf {F} }:=(K, m, \eta , \Delta _\textsf {F} , \varepsilon )\) is a Hopf algebra with antipode \(S_\textsf {F} (k):=\textsf {u} _\textsf {F} S(k) \bar{{\textsf {u} }}_\textsf {F} , \) where \(\textsf {u} _\textsf {F} \) is the invertible element \(\textsf {u} _\textsf {F} :=\textsf {F} ^{\alpha } S(\textsf {F} _\alpha )\) with \(\bar{{\textsf {u} }}_\textsf {F} = S({\overline{\textsf {F} }}^{\alpha }){\overline{\textsf {F} }}_{\alpha }\) its inverse.
Finally, if \((K, {\textsf {R} })\) is a quasitriangular bialgebra (a Hopf algebra), such is the twisted bialgebra (Hopf algebra) \({K_\textsf {F} }\) with R-matrix
and inverse \({{{\overline{{\textsf {R} }}}}_\textsf {F} }:=\textsf {F} \, {{\overline{{\textsf {R} }}}}\, {\overline{\textsf {F} }}_{21} = \textsf {F} ^\alpha {{\overline{{\textsf {R} }}}}^\beta {\overline{\textsf {F} }}_{\gamma } \otimes \textsf {F} _\alpha {{\overline{{\textsf {R} }}}}_\beta {\overline{\textsf {F} }}^{\gamma }\). If \((K, {\textsf {R} })\) is triangular, so is \(({K_\textsf {F} },{{\textsf {R} }_\textsf {F} })\): \({{{\textsf {R} }_\textsf {F} }}_{\,21} = \textsf {F} \, {\textsf {R} }_{21} \, {\overline{\textsf {F} }}_{21}=\textsf {F} \, {{\overline{{\textsf {R} }}}}\, {\overline{\textsf {F} }}_{21}={{{\overline{{\textsf {R} }}}}_\textsf {F} }\).
Any K-module V with left action \(\vartriangleright _V: K \otimes V \rightarrow V\), is also a \({K_\textsf {F} }\)-module with the same linear map \(\vartriangleright _V\), now thought as a map \(\vartriangleright _V: {K_\textsf {F} }\otimes V \rightarrow V \). When thinking of V as a \({K_\textsf {F} }\)-module we denote it by \(V_\textsf {F} \), with action \(\vartriangleright _{V_\textsf {F} }\). Moreover, any K-module morphism \(\psi : V\rightarrow W\) can be thought of as a morphism \(\psi _\textsf {F} : V_\textsf {F} \rightarrow W_\textsf {F} \).
If A is a K-module algebra, with multiplication \(m_A\) and unit \(\eta _A\), in order for the action \(\vartriangleright _{A_\textsf {F} }\) to be an algebra map one has to endow the \({K_\textsf {F} }\)-module \(A_\textsf {F} \) with a new algebra structure: the unit is unchanged, while the product is deformed to
For any K-module algebra map \(\psi : A\rightarrow A^\prime \), the \({K_\textsf {F} }\)-module map \(\psi : A_\textsf {F} \rightarrow A^\prime _\textsf {F} \) is an algebra map for the deformed products.
If C is a K-module coalgebra, the \({K_\textsf {F} }\)-module \(C_\textsf {F} \) is a \(K_\textsf {F} \)-module coalgebra with counit \(\varepsilon _\textsf {F} =\varepsilon \) as linear map and coproduct
The twist \(L_\textsf {F} \) of a braided Hopf algebra L is obtained twisting L as a K-module algebra and as a K-module coalgebra, cf. [5, Prop. 4.11].
We next recall that the action of a braided Hopf algebra (or just bialgebra) L on a K-module algebra A is a K-equivariant action \(\blacktriangleright _A:L\otimes A\rightarrow A\) which satisfies
for all \(a,a'\in A\). When twisting this leads to an action
When \({\mathfrak {g}}\) is a braided Lie algebra associated with a triangular Hopf algebra \((K,{\textsf {R} })\), and \(\textsf {F} \) is a twist for K, the \(K_\textsf {F} \)-module \({\mathfrak {g}}_\textsf {F} \) inherits from \({\mathfrak {g}}\) a twisted bracket ([5, Prop. 5.14]):
Proposition 2.2
The \(K_\textsf {F} \)-module \({\mathfrak {g}}_\textsf {F} \) with bilinear map
is a braided Lie algebra associated with \((K_\textsf {F} , {\textsf {R} }_\textsf {F} )\).
As a particular case of the above, consider the braided Lie algebra \(\big (\textrm{Der}{(A)}, [\,, \, ] \big )\), for A a K-module algebra. It consists of the K-module of braided derivations of A to itself, with action \(\vartriangleright _{{\textrm{Hom}}(A,A)}\) as in (2.15), and bracket the braided commutator (2.16). It is a braided Lie algebra associated with the triangular Hopf algebra \((K, {\textsf {R} })\).
On the one hand, we obtain the braided Lie algebra \((\textrm{Der}{(A)}_\textsf {F} ), [\,,\, ]_\textsf {F} )\) associated with the triangular Hopf algebra \((K_\textsf {F} , {\textsf {R} }_\textsf {F} )\). The \(K_\textsf {F} \)-action \(\vartriangleright _{\textrm{Der}{(A)}_\textsf {F} }\) coincides with \(\vartriangleright _{\textrm{Der}{(A)}}\) as linear map. The Lie bracket is given by the braided commutator
with the composition (in fact in \(({\textrm{Hom}}(A,A), \circ )\)) that is changed as in (2.25):
On the other hand, there is the braided Lie algebra \(\textrm{Der}{(A_\textsf {F} )}\) of the \(K_\textsf {F} \)-module \(A_\textsf {F} \) associated with \((K_\textsf {F} , {\textsf {R} }_\textsf {F} )\). We use the notation \(\Delta _\textsf {F} (k)=: {k}_{\scriptscriptstyle {[1]}} \otimes {k}_{\scriptscriptstyle {[2]}}\) for the coproduct in \({K_\textsf {F} }\) to distinguish it from the original one \(\Delta (k)= {k}_{\scriptscriptstyle {(1)}} \otimes {k}_{\scriptscriptstyle {(2)}}\) in K. The \(K_\textsf {F} \)-action is
with bracket
These two braided Lie algebras are isomorphic [5, Thm. 5.19]:
Theorem 2.3
The braided Lie algebras \((\textrm{Der}{(A)}_\textsf {F} , [\,,\, ]_\textsf {F} )\) and \((\textrm{Der}{(A_\textsf {F} )}, [\,,\, ]_{{{\textsf {R} }_\textsf {F} }})\) are isomorphic via the map
which satisfies \({\mathcal {D}}\big ([\psi ,\lambda ]_\textsf {F} \big ) = [{\mathcal {D}}(\psi ), {\mathcal {D}}( \lambda ) ]_{{{\textsf {R} }_\textsf {F} }}\), for all \(\psi ,\lambda \in \textrm{Der}{(A)}_\textsf {F} \). It has inverse
This isomorphism extends as algebra map to the universal enveloping algebras
resulting into a braided Hopf algebra isomorphism. We further have the braided Hopf algebras isomorphisms (see [5, Prop. 5.18]) \({\mathcal {U}}(\textrm{Der}{(A)})_\textsf {F} \simeq {\mathcal {U}}(\textrm{Der}{(A)}_\textsf {F} )\simeq {\mathcal {U}}(\textrm{Der}{(A_\textsf {F} )})\).
Remark 2.4
As shown in [5], the isomorphism \({\mathcal {D}}:\textrm{Der}{(A)}_\textsf {F} \rightarrow \textrm{Der}{(A_\textsf {F} )}\) is the restriction of a more general isomorphism \({\mathcal {D}}: (({\textrm{Hom}}(A,A)_\textsf {F} , \circ _\textsf {F} ), [\,,\, ]_\textsf {F} )\rightarrow (({\textrm{Hom}}(A_\textsf {F} ,A_\textsf {F} ), \circ ),[\,, \,]_{{{\textsf {R} }_\textsf {F} }})\). This result indeed holds in more generality for A just a K-module and not necessarily a K-module algebra.
As mentioned, when A is quasi-commutative the K-braided Lie algebra \(\textrm{Der}{(A)}\) has an A-module structure defined in (2.18) that is compatible with the Lie bracket of \(\textrm{Der}{(A)}\).
The \(K_\textsf {F} \)-braided Lie algebra \(\textrm{Der}{(A)}_\textsf {F} \) has \(A_\textsf {F} \)-module structure
for all \(\psi \in \textrm{Der}{(A)}_\textsf {F} \) and \(a\in A_\textsf {F} \). The compatibility of the braided bracket with this module structure then, for all \(\psi ,\psi '\in \textrm{Der}{(A)}_\textsf {F} \), \(a,a'\in A_\textsf {F} \), reads
Here an element in A is thought as a linear map \(A\rightarrow A\) given by left multiplication. Then \([\psi ,a]_\textsf {F} = [{\overline{\textsf {F} }}^\alpha \vartriangleright _{\textrm{Der}{(A)}} \psi ,{\overline{\textsf {F} }}_\alpha \vartriangleright _A a]= ({\overline{\textsf {F} }}^\alpha \vartriangleright _{\textrm{Der}{(A)}} \psi )({\overline{\textsf {F} }}_\alpha \vartriangleright _A a)\).
Also the \(K_\textsf {F} \)-braided Lie algebra \(\textrm{Der}{(A_\textsf {F} )}\) has compatible \(A_\textsf {F} \)-module structure. With the product \({\scriptstyle {\,\bullet }}_{\scriptscriptstyle {\textsf {F} \,}}\) in (2.25) this is given as in (2.18) by
for any \(a,a'\in A_\textsf {F} ,\psi \in \textrm{Der}{(A_\textsf {F} )}\).
The isomorphism \({\mathcal {D}}:\textrm{Der}{(A)}_\textsf {F} \rightarrow \textrm{Der}{(A_\textsf {F} )}\) respects the \(A_\textsf {F} \)-module structures:
Corollary 2.5
If the K-module algebra A is quasi-commutative, the braided Lie algebra isomorphism \({\mathcal {D}}: (\textrm{Der}{(A)}_\textsf {F} , [\,,\, ]_\textsf {F} )\rightarrow (\textrm{Der}{(A_\textsf {F} )}, [\,,\, ]_{{{\textsf {R} }_\textsf {F} }})\) of Theorem 2.3 is also an isomorphism of the \(A_\textsf {F} \)-modules \(\textrm{Der}{(A)}_\textsf {F} \) and \(\textrm{Der}{(A_\textsf {F} )}\):
for \(a\in A_\textsf {F} \) and \(\psi \in \textrm{Der}{(A)}_\textsf {F} \).
Next, let \(B \subseteq A\) be a K-equivariant Hopf–Galois extension. We use the above isomorphisms for the K-braided Lie algebra of derivations \(\textrm{Der}{(A)}\) and its braided subalgebras \(\textrm{Der}^{\textsf {R} }_{{{\mathcal {M}}^H}}{(A)}\) and \(\textrm{aut}^{\textsf {R} }_B(A)\) defined in (2.20) and in (2.21).
The K-braided Lie algebras \((\textrm{aut}^{\textsf {R} }_B(A), [~,~])\subseteq (\textrm{Der}^{\textsf {R} }_{{{\mathcal {M}}^H}}{(A)}, [~,~])\) are twisted to the \(K_\textsf {F} \)-braided Lie algebras \((\textrm{aut}^{\textsf {R} }_B(A)^{}_\textsf {F} ,[~,~]_\textsf {F} )\subseteq (\textrm{Der}^{\textsf {R} }_{{{\mathcal {M}}^H}}{(A)}_\textsf {F} , [~,~]_\textsf {F} )\) with bracket \([~,~]_\textsf {F} \). These are braided Lie subalgebras of \((\textrm{Der}{(A)}_\textsf {F} , [~,~]_\textsf {F} )\). We can equivalently consider the \(K_\textsf {F} \)-braided Lie algebras \((\textrm{aut}^{{\textsf {R} }_\textsf {F} }_{B_\textsf {F} }(A_\textsf {F} ),[~,~]_{{\textsf {R} }_\textsf {F} })\subseteq (\textrm{Der}^{{\textsf {R} }_\textsf {F} }_{{{\mathcal {M}}^{H_\textsf {F} }}}{(A_\textsf {F} )}, [~,~]_{{\textsf {R} }_\textsf {F} })\) that are braided Lie subalgebras of \((\textrm{Der}{(A_\textsf {F} )}, [~,~]_{{\textsf {R} }_\textsf {F} })\). These are isomorphic, [5, Prop. 8.1]:
Proposition 2.6
The isomorphism \({\mathcal {D}}: (\textrm{Der}{(A)}_\textsf {F} , [~,~]_\textsf {F} ){\, \rightarrow \,} (\textrm{Der}{(A_\textsf {F} )}, [~,~]_{{\textsf {R} }_\textsf {F} })\) of braided Lie algebras in Theorem 2.3 restricts to isomorphisms \({\mathcal {D}}: \textrm{Der}^{\textsf {R} }_{{{\mathcal {M}}^H}}{(A)}_\textsf {F} {\, \rightarrow \,} \textrm{Der}^{{\textsf {R} }_\textsf {F} }_{{{\mathcal {M}}^{H_\textsf {F} }}}{(A_\textsf {F} )}\) and \({\mathcal {D}}: \textrm{aut}^{\textsf {R} }_B(A)^{}_\textsf {F} {\, \rightarrow \,}\textrm{aut}^{{\textsf {R} }_\textsf {F} }_{B_\textsf {F} }(A_\textsf {F} )\) of \((K_\textsf {F} ,{\textsf {R} }_\textsf {F} )\)-braided Lie algebras.
In § 4 we work out the braided Lie algebra of equivariant derivations and of infinitesimal gauge transformations for two important examples of principal bundles over the noncommutative 4-sphere \(S^4_\theta \) of [8]. We use the general theory developed in this section to obtain the braided Lie algebra of equivariant derivations and of infinitesimal gauge transformations of these noncommutative bundles from their classical counterparts.
These noncommutative principal bundles are \(*\)-algebra extensions (which can be completed to \(C^*\)-algebras). The \(*\)-structures canonically lift to the Lie algebras of braided derivations and of gauge transformations. Before considering these examples, in the next section we proceed with a systematic analysis of \(*\)-structures for braided Hopf and Lie algebras. The results of § 4 are however presented in a self contained way so that § 3 might be skipped in a first reading.
3 Braided Hopf and Lie \(*\)-algebras
In this section the ground field is \(\mathbb {k}={\mathbb {C}}\). We present a study of compatibility conditions for defining \(*\)-structures on Hopf algebras and their representations. The study of braided Hopf \(*\)-algebra actions on braided \(*\)-algebras associated with quasitriangular Hopf algebras is new to the best of our knowledge.
A \(*\)-structure on a vector space V is an antilinear involution \(*:V\rightarrow V\), \(v \mapsto v^*\), on an algebra A one also requires \(*:A\rightarrow A\) to be antimultiplicative. A \(*\)-structure on a Hopf algebra K is a \(*\)-structure on the algebra K that satisfies \(\Delta (k^*)=\Delta (k)^{*\otimes *}\) for all \(k\in K\); it then follows that \(\varepsilon (k^*)=\overline{\varepsilon (k)}\) and \((S\circ *)^2=\textrm{id}\). In particular S is invertible, with \(S^{-1}= * \circ S\circ *\). If V is a K-module with a \(*\)-structure, one requires the compatibility condition
for all \(k\in K, v \in V\). This condition is well defined: it is equivalent to require that \(\succ _V: K\otimes V\rightarrow V\), defined by
is an action of the Hopf algebra K on V that coincides with the starting one \( \vartriangleright _V\). Indeed
The condition (3.1) is also required for A a K-module \(*\)-algebra. In this case (3.1) is also well defined with respect to the multiplication in A. Indeed, \(k \vartriangleright _L 1_A = \varepsilon (k)1_A\) and
for all \(k,h\in K, \,a, b\in A\). Here and in the following to lighten the notations we frequently omit the subscript on the actions.
Example 3.1
Any Hopf \(*\)-algebra K with adjoint action \(K\otimes K\rightarrow K\), \(k\,\vartriangleright \, k'={k}_{\scriptscriptstyle {(1)}}k'S({k}_{\scriptscriptstyle {(2)}})\) is a K-module \(*\)-algebra. This motivates the definition (3.1) following the conventions in [10] rather than those in [13].
Example 3.2
Condition (3.1) is dual to that for a comodule \(*\)-algebra. Given a \(*\)-algebra A which is a right comodule algebra for a Hopf \(*\)-algebra \({\mathcal {U}}\), with coaction \(\delta : A \rightarrow A \otimes {\mathcal {U}}\), \(a \mapsto {a}_{\scriptscriptstyle {(0)}} \otimes {a}_{\scriptscriptstyle {(1)}}\), one requires \(\delta (a^*)= ({a}_{\scriptscriptstyle {(0)}})^* \otimes ({a}_{\scriptscriptstyle {(1)}})^*\). If K and \({\mathcal {U}}\) are dually paired Hopf \(*\)-algebras, one has \(\langle k, u^* \rangle = \overline{\langle S^{-1}(k^*), u \rangle }\). Then A is a module \(*\)-algebra with K-action \(k \vartriangleright a = {a}_{\scriptscriptstyle {(0)}} \langle k, {a}_{\scriptscriptstyle {(1)}} \rangle \) satisfying (3.1).
In the present paper we deal with braided Hopf and Lie algebras associated with a (quasi)triangular Hopf algebra \((K, {\textsf {R} })\).
When K is quasitriangular we require its R-matrix to be antireal, that is, \({\textsf {R} }^{*\otimes *}={{\overline{{\textsf {R} }}}}\). When K is triangular this condition coincides with the reality condition \({\textsf {R} }^{*\otimes *}={\textsf {R} }_{21}\).
3.1 Braided Hopf \(*\)-algebras
In this section we take \((K, {\textsf {R} })\) quasitriangular with \({\textsf {R} }\) antireal. We use the braiding \(\Psi _{\textsf {R} }:L\boxtimes L\rightarrow L\boxtimes L, \Psi _{\textsf {R} }(x\boxtimes y)={\textsf {R} }_\alpha \vartriangleright _L y\boxtimes {\textsf {R} }^\alpha \vartriangleright _L x\), in (2.7) to induce the \(*\)-structure from a K-module \(*\)-algebra L to the K-module algebra \(L\boxtimes L\) defined in §2.1 (and to more general tensor products of K-module \(*\)-algebras).
Lemma 3.3
Given a K-module \(*\)-algebra L, the K-module algebra \(L \boxtimes L\) with product 
in (2.6) and involution \((x \boxtimes y)^* = \Psi _{\textsf {R} }(y^*\boxtimes x^*)\), that is,
for \(x,y\in L\), is a K-module \(*\)-algebra.
Proof
The matrix \({\textsf {R} }\) being antireal implies that (3.5) is an involution. It is antimultiplicative: using (3.1) for \(\vartriangleright _L\) together with antireality and properties (2.5) of the \({\textsf {R} }\)-matrix we compute (omitting the subscript on the actions)
Finally, we show the compatibility condition (3.1) between the \(*\)-structures. Recalling that \(L \boxtimes L\) has action (2.8), we compute
where for the third last equality we used the quasi-cocommutative condition (2.3). \(\square \)
Considering the last and third expression in (3.6) one has
and, recalling the action (3.2), this reads \(k\succ _{L\boxtimes L} (x\boxtimes y)={k}_{\scriptscriptstyle {(1)}}\succ _L x\,\boxtimes \, {k}_{\scriptscriptstyle {(2)}}\succ _L y\). This proves that \(\succ _{L \boxtimes L}\) is an action of the coalgebra K on \(L\boxtimes L\).
Lemma 3.4
The action \(\succ : K\otimes L\rightarrow L\) commutes with the braiding isomorphism \(\Psi _{\textsf {R} }:L\boxtimes L\rightarrow L\boxtimes L\), \(\Psi _{\textsf {R} }(x\boxtimes y)={\textsf {R} }_\alpha \vartriangleright y\boxtimes {\textsf {R} }^\alpha \vartriangleright x\) of the original K-action \(\vartriangleright \),
for all \(k\in K, x, y\in L\).
Proof
We first compute, using the antireality of the R-matrix,
for all \(x,y\in {\mathfrak {g}}\). This and the \(*\)-structure of \(L \boxtimes L\) in (3.5) prove the lemma:
\(\square \)
Definition 3.5
A \(*\)-structure on a braided Hopf algebra L associated with \((K,{\textsf {R} })\) is a \(*\)-structure on the K-module algebra L such that the braided coproduct \(\Delta _L: L\rightarrow L \boxtimes L\) is a \(*\)-algebra map, \(\Delta _L(x^*) = (\Delta _L (x))^*\) with \(*\)-structure on \(L \boxtimes L\) in (3.5).
The \(*\)-algebra map condition for \(\Delta _L\) is equivalent to \(\Delta _L = *\circ \Delta _L\circ *\). This is well defined since \(*\circ \Delta _L\circ *: L\rightarrow L \boxtimes L\) is a coassociative K-module map and an algebra map. The K-equivariance follows from that of \(\Delta _L\) and the compatibility (3.1). The \(*\)-algebra map property is straighforward and coassociativity is verified by direct computation.
The (braided) antipode \(S_L\) of a braided Hopf \(*\)-algebra satisfies \(S_L \circ * \circ S_L \circ * =\textrm{id}\) and so is invertible. Using (2.11), \(\Delta _L \circ S_L = (S_L \otimes S_L ) \circ \Psi _{\textsf {R} }\circ \Delta _L\), one gets:
with \({\overline{\Psi }}_{\textsf {R} }\) the inverse of \(\Psi _{\textsf {R} }\): \( {\overline{\Psi }}_{\textsf {R} }(x\otimes y)={{\overline{{\textsf {R} }}}}^\alpha \vartriangleright _L y\otimes {{\overline{{\textsf {R} }}}}_\alpha \vartriangleright _L x\), for \(x,y \in L\). Then, using (3.5), one gets
for each \(x \in L\), together with
A braided Hopf \(*\)-algebra L acts on a K-module \(*\)-algebra A, with action \(\blacktriangleright _A:L\otimes A\rightarrow A\) satisfying (2.27), if the \(*\)-structure of A satisfies the compatibility condition
that generalizes condition (3.1).
Proposition 3.6
The compatibility condition (3.11) is well-defined.
Proof
The compatibility condition is equivalent to require that
is an action of the braided Hopf algebra L on the K-module algebra A, which coincides with the starting action \(\blacktriangleright _A\). We need to show that the map \(\succ \) defines a K-equivariant action that satisfies (2.27). To lighten the notation we omit the subscript on the actions. Firstly, \(\succ \) is an action. The condition \(x \succ 1_A = \varepsilon _L(x) 1_A\) follows from the unitality of \({\textsf {R} }\). Observing that the compatibility of the inverse antipode with the multiplication uses the inverse braiding, \( S^{-1}_L (xy) = ( {{\overline{{\textsf {R} }}}}^\alpha \vartriangleright _L S^{-1}_L( y))({{\overline{{\textsf {R} }}}}_\alpha \vartriangleright _L S^{-1}_L( x)) \), cf. (2.10), we compute
For the last equality we used the analogous of property (2.4) for \({{\overline{{\textsf {R} }}}}\). On the other hand
having used property (2.4) again. The two expressions then coincide due to the quantum Yang–Baxter equation \( {{\overline{{\textsf {R} }}}}_{12} {{\overline{{\textsf {R} }}}}_{13} {{\overline{{\textsf {R} }}}}_{23}= {{\overline{{\textsf {R} }}}}_{23} {{\overline{{\textsf {R} }}}}_{13}{{\overline{{\textsf {R} }}}}_{12} \). Next we show K-equivariance:
where for the fourth equality we used the co-commutativity (2.3) of the coproduct of K and for the sixth equality the K-equivariance of the braided antipode S of L.
We are left to show that (2.27) holds: using the same property for the action \(\blacktriangleright \) and property (3.10) for the inverse of the braided antipode, we compute
On the other hand, also using the antireality of \({\textsf {R} }\),
and the two expressions coincide due to K-equivariance of the braided antipode. \(\square \)
3.2 Braided Lie \(*\)-algebras
In this section \((K,{\textsf {R} })\) is triangular, with \({\textsf {R} }^{*\otimes *}={\textsf {R} }_{21}={{\overline{{\textsf {R} }}}}\).
Definition 3.7
A \(*\)-structure on a K-braided Lie algebra \({\mathfrak {g}}\) is an antilinear involution \(*:{\mathfrak {g}}\rightarrow {\mathfrak {g}}\) which satisfies (3.1) and in addition \(([u,v])^*=[v^*,u^*]\) for all \(u,v\in {\mathfrak {g}}\).
The compatibility condition (3.1) is well-defined because it is equivalent to require that the K-action \(\succ _{\mathfrak {g}}:K\otimes {\mathfrak {g}}\rightarrow {\mathfrak {g}}\) in (3.2), with \((A, \cdot )\) replaced by \(({\mathfrak {g}}, [~,~])\), is an action of K on the braided Lie algebra \({\mathfrak {g}}\), which coincides with the starting action \(\vartriangleright _{\mathfrak {g}}\). The proof that \(\succ _{\mathfrak {g}}: K\otimes {\mathfrak {g}}\rightarrow {\mathfrak {g}}\) is an action on the K-module \({\mathfrak {g}}\) is as in (3.1). The K-equivariance property of the bracket, \(k\succ _{\mathfrak {g}}[u,v]=[{k}_{\scriptscriptstyle {(1)}}\succ _{\mathfrak {g}}u,{k}_{\scriptscriptstyle {(2)}}\succ _{\mathfrak {g}}v]\) for all \(k\in K\), that is, \(k\succ _{\mathfrak {g}}\circ \, [~,~]=[~,~]\circ k\succ _{\mathfrak {g}}\) as maps \({\mathfrak {g}}\otimes {\mathfrak {g}}\rightarrow {\mathfrak {g}}\), is as in (3.4).
The compatibility of the action \(\succ _{\mathfrak {g}}\) with the braided antisymmetry \([~,~]=-[~,~]\circ \Psi _{\textsf {R} }\) is due to K-equivariance of the braiding \(\Psi _{\textsf {R} }\), see (3.7). Similarly, the compatibility of the action \(\succ _{\mathfrak {g}}\) with the braided Jacobi identity (which is an equality between maps obtained from the bracket and the braiding) is due to K-equivariance of all the maps involved.
In the present paper the main example of K-braided Lie \(*\)-algebra is that of braided derivations \(\textrm{Der}{(A)}\) of a K-module \(*\)-algebra A. Its \(*\)-structure is defined by
for all \(\psi \in \textrm{Der}{(A)}\), \(a\in A\). It lifts as an antilinear and antimultiplicative map to a \(*\)-structure on the universal enveloping algebra \(L={\mathcal {U}}(\textrm{Der}{(A)})\). This \(*\)-structure is compatible with the braided action \(\blacktriangleright _A: {\mathcal {U}}(\textrm{Der}{(A)})\otimes A\rightarrow A\) defined by \(\psi \blacktriangleright _A a=\psi (a)\) for all \(\psi \in \textrm{Der}{(A)}\subseteq {\mathcal {U}}(\textrm{Der}{(A)}) \), \(a\in A\). Since \(\textrm{Der}{(A)}\) is the K-submodule of primitive elements, \(S^{-1}(\psi )=-\psi \) and (3.12) also reads \( \psi ^*(a) = \left( ( {{\overline{{\textsf {R} }}}}^\alpha \vartriangleright _L S^{-1}(\psi ) ) ({{\overline{{\textsf {R} }}}}_\alpha \vartriangleright _A a^*) \right) ^* \). This implies the compatibility:
This is the unique \(*\)-structure compatible with \(\blacktriangleright _A\), indeed (3.11) is equivalent to (3.13).
If the K-module \(*\)-algebra A is quasi-commutative, see (2.17), the K-braided Lie \(*\)-algebra \(\textrm{Der}{(A)}\) is also a left A-module with action \(\cdot : A\otimes \textrm{Der}{(A)}\rightarrow A\) defined in (2.18). This is compatible with the \(*\)-structure: on \(A\otimes \textrm{Der}{(A)}\) we have the \(*\)-structure \((a\otimes \psi )^*=({\textsf {R} }_\alpha \vartriangleright a^*)\otimes ({\textsf {R} }^\alpha \vartriangleright \psi ^*)\) (cf. Lemma 3.3), then
for all \(a\in A\), \(\psi \in \textrm{Der}{(A)}\), that is, \(*\circ \cdot =\cdot \circ *\) as maps \(A\otimes \textrm{Der}{(A)}\rightarrow \textrm{Der}{(A)}\).
3.3 Twists and \(*\)-structures
3.3.1 Twist of Hopf \(*\)-algebras and their representations
A twist \(\textsf {F} \) of a Hopf \(*\)-algebra K is a twist of the Hopf algebra K that satisfies
Then \(K_\textsf {F} \) is a Hopf \(*\)-algebra with
where \(\textsf {u}=\textsf {F} ^\alpha S(\textsf {F} _\alpha )\) and \({{\bar{\textsf {u}}}}=S({{\bar{\textsf {F} }}}^\alpha ){{\bar{\textsf {F} }}}_\alpha \) is its inverse. From (3.14) we have \(\textsf {u}^*=\textsf {u}\), which implies that \({*_\textsf {F} }\) is involutive. The twist condition (2.22) implies the identity \(\textsf {F} \Delta (\textsf {u})=\textsf {u}S({\textsf {F} _{\,\beta }}_{\scriptscriptstyle {(2)}})\otimes \textsf {F} ^\beta S({\textsf {F} _{\,\beta }}_{\scriptscriptstyle {(1)}})\) or equivalently
Compatibility with the coproduct then follows: \(\Delta _\textsf {F} (k^{*_\textsf {F} })=\Delta _\textsf {F} (k)^{*_\textsf {F} \otimes *_\textsf {F} }\) for all \(k\in K\).
If A is a K-module \(*\)-algebra, \(A_\textsf {F} \) is a \(K_\textsf {F} \)-module \(*\)-algebra with \(*:A_\textsf {F} \rightarrow A_\textsf {F} \) that is the same as the initial \(*\) as antilinear map. Indeed this is antimultiplicative:
where we used (3.1) and (3.14). Moreover it is compatible with the \(K_\textsf {F} \)-action,
where we used \((S_\textsf {F} (k))^{*_\textsf {F} }=(\textsf {u}S(k) {{\bar{\textsf {u}}}})^{*_\textsf {F} }=(S(k))^*\) which holds since \(\textsf {u}^*=\textsf {u}\).
3.3.2 Twist of quasitriangular and braided Hopf \(*\)-algebras, and of their representations
If \((K, {\textsf {R} }, *)\) is a quasitriangular Hopf \(*\)-algebra with \({\textsf {R} }\) antireal, so is \((K_\textsf {F} , {{\textsf {R} }_\textsf {F} },*_\textsf {F} )\) with \({{\textsf {R} }_\textsf {F} }\) antireal. From (3.15), (3.16) and the equivalent expression \(\textsf {F} _{21}^{*\otimes *} ({\bar{u}} \otimes {\bar{u}})= \Delta ^{cop} ({\bar{u}}) {\overline{\textsf {F} }}_{21}\), we compute
where we used the quasi-cocommutativity property (2.3).
For a K-module algebra L one has the \(K_\textsf {F} \)-module algebra isomorphism
(leading to the monoidal equivalence of the categories of K-module algebras and \(K_\textsf {F} \)-module algebras). When L is a K-module \(*\)-algebra so is \(L\boxtimes L\) while \(L_\textsf {F} \), \((L\boxtimes L)_\textsf {F} \) and \(L_\textsf {F} \boxtimes _\textsf {F} L_\textsf {F} \) are \(K_\textsf {F} \)-module \(*\)-algebras. This latter with \(*\)-structure (cf. Lemma 3.3)
Lemma 3.8
Let \((K,{\textsf {R} }, *)\) be a quasitriangular Hopf \(*\)-algebra, with twist \(\textsf {F} \) and L a K-module \(*\)-algebra. The isomorphism \(\varphi : L_\textsf {F} \boxtimes _\textsf {F} L_\textsf {F} \rightarrow (L\boxtimes L)_\textsf {F} \) in (3.19) is a \(K_\textsf {F} \)-module \(*\)-algebra isomorphism.
Proof
We show \(\varphi ^{-1}\circ *\circ \varphi =*_\textsf {F} \). Using the compatibility condition (3.1), (3.14) and \({\textsf {R} }_\textsf {F} =\textsf {F} _{21}{\textsf {R} }{\overline{\textsf {F} }}\) we have, for all \(x,y\in L\),
\(\square \)
Theorem 3.9
\((L_\textsf {F} ,{\textsf {R} }_\textsf {F} , *)\) is a \(K_\textsf {F} \)-braided Hopf \(*\)-algebra with \(*\)-structure that, as an antilinear involution, is the same as that of \((L,{\textsf {R} }, *)\).
Proof
We already know that \((L_\textsf {F} ,{\textsf {R} }_\textsf {F} )\) is a \(K_\textsf {F} \)-braided Hopf algebra, we prove the compatibility with \(*\). From (3.18) we see that \((L_\textsf {F} ,*)\) is a \(K_\textsf {F} \)-module \(*\)-algebra. Moreover the braided coproduct \(\Delta _{L_\textsf {F} }\) is a \(*\)-algebra map: \(\Delta _{L_\textsf {F} }(x^*)={(\Delta _{L_\textsf {F} }(x))}^{*_\textsf {F} }\), for all \(x\in L_\textsf {F} \). Indeed from (2.26) we have \(\Delta _{L_\textsf {F} }\circ *=\varphi ^{-1}\circ \Delta _L\circ *= \varphi ^{-1}\circ *\circ \Delta _L=\varphi ^{-1}\circ * \circ \varphi \circ \Delta _{L_\textsf {F} }=*_\textsf {F} \circ \Delta _{L_\textsf {F} }\). Thus \((L_\textsf {F} ,{\textsf {R} }_\textsf {F} , *)\) is a \(K_\textsf {F} \)-braided Hopf \(*\)-algebra according to Definition 3.5. \(\square \)
If a K-braided Hopf \(*\)-algebra L acts on a K-module \(*\)-algebra A then the twisted \(K_\textsf {F} \)-braided Hopf \(*\)-algebra \(L_\textsf {F} \) acts on the twisted \(K_\textsf {F} \)-module \(*\)-algebra \(A_\textsf {F} \) with action
The compatibility of the \(*\)-structure of \(A_\textsf {F} \) with the braided action \(\blacktriangleright _{A_\textsf {F} }\) of \(L_\textsf {F} \) reads \((x\blacktriangleright _{A_\textsf {F} } a)^{*}=({{{\overline{{\textsf {R} }}}}_\textsf {F} }^\beta \vartriangleright _{L_\textsf {F} } S^{-1}_{L_\textsf {F} }(x^*))\blacktriangleright _{A_\textsf {F} }({{{\overline{{\textsf {R} }}}}_\textsf {F} }_\beta \vartriangleright _{A_\textsf {F} } a) \), cf. (3.11). This follows from \(S^{-1}_L((k\vartriangleright x)^*)=S^{-1}_L(S^{-1}(k^*)\vartriangleright x^*)= S^{-1}(k^*)\vartriangleright S^{-1}_L(x^*)\), for \(k\in K\), \(x\in L\), owing to the K-equivariance of the braided antipode \(S_L\).
3.3.3 Twist of braided Lie \(*\)-algebras of derivations and gauge transformations
Let \((K,{\textsf {R} })\) be triangular. If \({\mathfrak {g}}\) is a K-module Lie \(*\)-algebra then \({\mathfrak {g}}_\textsf {F} \) is a \({K_\textsf {F} }\)-module Lie \(*\)-algebra with the initial involution of \({\mathfrak {g}}\). The property \(([u,v]_\textsf {F} )^*=[v^*,u^*]_\textsf {F} \) for all \(u,v\in {\mathfrak {g}}_\textsf {F} \) is proven along the same lines of those in (3.17).
We now consider the K-braided Lie \(*\)-algebra \({\mathfrak {g}}=\textrm{Der}{(A)}\) of derivations of the K-module \(*\)-algebra A, with \(*\)-structure defined in (3.12) and its universal enveloping \(*\)-algebra \(L={\mathcal {U}}(\textrm{Der}{(A)})\). Similarly we have the \(K_\textsf {F} \)-braided Lie \(*\)-algebra \(\textrm{Der}{(A_\textsf {F} )}\), of derivations of the \(K_\textsf {F} \)-module \(*\)-algebra \(A_\textsf {F} \), with \(*\)-structure \({{{\tilde{*}}}}:=*^{}_{\textrm{Der}{(A_\textsf {F} )}}\), defined as in (3.12),
for all \(\psi \in \textrm{Der}{(A_\textsf {F} )}\), \(a\in A_\textsf {F} \). The associated universal enveloping \(*\)-algebra is \(\,{\mathcal {U}}(\textrm{Der}{(A_\textsf {F} )})\).
Proposition 3.10
The isomorphism \({\mathcal {D}}: \textrm{Der}{(A)}_\textsf {F} \rightarrow \textrm{Der}{(A_\textsf {F} )}\) of \(K_\textsf {F} \)-braided Lie algebras of Theorem 2.3 is a \(K_\textsf {F} \)-braided Lie \(*\)-algebra isomorphism. It lifts to the isomorphism \({\mathcal {D}}: {\mathcal {U}}(\textrm{Der}{(A)}_\textsf {F} ) \rightarrow {\mathcal {U}}(\textrm{Der}{(A_\textsf {F} )})\) of \(K_\textsf {F} \)-braided Hopf \(*\)-algebras.
Proof
We have to show \({\mathcal {D}}(\psi ^*)(a)={{\mathcal {D}}(\psi )}^{{{{\tilde{*}}}}}(a)\) for all \(\psi \in \textrm{Der}{(A)}_\textsf {F} \), \(a\in A_\textsf {F} \). In analogy with (3.17) we have \( ({\overline{\textsf {F} }}^\alpha \vartriangleright a)^*\otimes ({\overline{\textsf {F} }}_\alpha \vartriangleright \psi )^* = ({\overline{\textsf {F} }}_\alpha \vartriangleright a^* ) \otimes ({\overline{\textsf {F} }}^\alpha \vartriangleright \psi ^*)\) and therefore,
where we used (3.12) with \(S(\psi )=-\psi \) (or (3.13)) and K-equivariance of the braided antipode \(S_L\). On the other hand, by definition of the \(*\)-structure on \(\textrm{Der}{(A_\textsf {F} )}\) we have
The proof then follows from \({\textsf {R} }_\textsf {F} =\textsf {F} _{21}{\textsf {R} }\textsf {F} ^{-1}\).
The isomorphism \({\mathcal {D}}: {\mathcal {U}}(\textrm{Der}{(A)}_\textsf {F} ) \rightarrow {\mathcal {U}}(\textrm{Der}{(A_\textsf {F} )})\) of \(K_\textsf {F} \)-braided Hopf algebras commutes with the \(*\)-structures when restricted to the primitive elements \(\textrm{Der}{(A)}_\textsf {F} \) and, since these are the generators, on all elements of \({\mathcal {U}}(\textrm{Der}{(A)}_\textsf {F} )\). The proof is by induction. If \(x,y\in {\mathcal {U}}(\textrm{Der}{(A)}_\textsf {F} )\) satisfy \({\mathcal {D}}(x^*)={\mathcal {D}}(x)^{{{\tilde{*}}}}\), \({\mathcal {D}}(y^*)={\mathcal {D}}(y)^{{{\tilde{*}}}}\), then so does their product, \({\mathcal {D}}((x\cdot _\textsf {F} y)^*)={\mathcal {D}}(y^*\cdot _\textsf {F} x^*)={\mathcal {D}}(y^*){\scriptstyle {\,\bullet }}_{\scriptscriptstyle {\textsf {F} \,}}{\mathcal {D}}(x^*)= {\mathcal {D}}(y)^{{{\tilde{*}}}}{\scriptstyle {\,\bullet }}_{\scriptscriptstyle {\textsf {F} \,}}{\mathcal {D}}(x)^{{{\tilde{*}}}}=({\mathcal {D}}(x){\scriptstyle {\,\bullet }}_{\scriptscriptstyle {\textsf {F} \,}}{\mathcal {D}}(y))^{{{\tilde{*}}}}=({\mathcal {D}}(x\cdot _\textsf {F} y))^{{{\tilde{*}}}}\). \(\square \)
Finally, for A a quasi-commutative K-module \(*\)-algebra, \(\textrm{Der}{(A)}\) is a K-braided Lie and A-module \(*\)-algebra. The twisted algebra \(A_\textsf {F} \) is a \(K_\textsf {F} \)-braided quasi-commutative \(*\)-algebra and \(\textrm{Der}{(A_\textsf {F} )}\) a \(K_\textsf {F} \)-braided Lie and \(A_\textsf {F} \)-module \(*\)-algebra. In particular,
From Theorem 2.3, Corollary 2.5 and Proposition 3.10, \({\mathcal {D}}: \textrm{Der}{(A)}_\textsf {F} \rightarrow \textrm{Der}{(A_\textsf {F} )}\) is an isomorphism of \(K_\textsf {F} \)-braided Lie and \(A_\textsf {F} \)-module \(*\)-algebras. As in Theorem 2.6, for a Hopf–Galois extension this isomorphism restricts to equivariant derivations and to infinitesimal gauge transformations:
Corollary 3.11
For \(B=A^{co H}\subseteq A\) a \((K, {\textsf {R} })\)-equivariant Hopf–Galois extension with A a quasi-commutative H-comodule \(*\)-algebra, the isomorphism \({\mathcal {D}}: \textrm{Der}{(A)}_\textsf {F} {\, \rightarrow \,}\textrm{Der}{(A_\textsf {F} )}\) of \((K_\textsf {F} ,r_\textsf {F} )\)-braided Lie and \(A_\textsf {F} \)-module \(*\)-algebras restricts to isomorphisms
of \((K_\textsf {F} ,{\textsf {R} }_\textsf {F} )\)-braided Lie and \(B_\textsf {F} \)-module \(*\)-algebras.
4 Principal bundles over \(S^4_\theta \) and their gauge transformations
In this section we consider the twist deformation of the Hopf SU(2)-bundle over the 4-sphere \(S^4_\theta \), and then of the SO(4)-bundle over \(S^4_\theta \), seen as a homogeneous space.
4.1 The instanton bundle
The \(H={\mathcal {O}}(SU(2))\) Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(S^7_\theta )\) of [12] can be obtained as a deformation by a twist on \(K={\mathcal {O}}({\mathbb {T}}^2)\) of the Hopf–Galois extension \({\mathcal {O}}(S^4) \subset {\mathcal {O}}(S^7)\) of the classical SU(2) Hopf bundle, [3]. We use that twist deformation in the framework of the theory developed in § 2.4, to obtain the braided Lie algebras \(\textrm{Der}_{{{\mathcal {M}}^H}}{({\mathcal {O}}(S^7_\theta ))}\) and \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ) )\) from their classical counterparts \(\textrm{Der}_{{{\mathcal {M}}^H}}{({\mathcal {O}}(S^7)})\) and \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7) )\).
4.1.1 The classical Hopf bundle
Let us start with the Hopf–Galois extension \(B \subset A\) of the classical SU(2)-Hopf bundle \(\pi : S^7 \rightarrow S^4\). The algebra \(A:= {\mathcal {O}}(S^7)\) is the commutative \(*\)-algebra of coordinate functions on the 7-sphere \(S^7\) with generators \(\{ z_a, z_a^*\), \(a=1,\dots , 4\}\), satisfying the sphere relation \(\sum z^*_a z_a=1\). It carries a right coation of the Hopf algebra \({\mathcal {O}}(SU(2))\) of coordinate functions on SU(2). This is the \(*\)-algebra generated by commuting elements \(\{w_j, w_j^*\), \(j=1,2\}\), with \(\sum w^*_j w_j=1\), and standard Hopf algebra structure induced from the group structure of SU(2). The right coaction \(\delta : {\mathcal {O}}(S^7) \longrightarrow {\mathcal {O}}(S^7) \otimes {\mathcal {O}}(SU(2))\) is defined on the algebra generators as
Here \(\overset{.}{\otimes }\) denotes the composition of the tensor product \(\otimes \) with matrix multiplication. As usual the coaction is extended to the whole \({\mathcal {O}}(S^7)\) as a \(*\)-algebra morphism.
The \(*\)-subalgebra \(B={\mathcal {O}}(S^7)^{co{\mathcal {O}}(SU(2))}\) of coinvariant elements for the coaction is identified with the \(*\)-algebra \({\mathcal {O}}(S^4)\) of coordinate functions on the 4-sphere \(S^4\). As the algebraic counterpart of the principality of the Hopf bundle \(\pi :S^7 \rightarrow S^4\), one has that the algebra \({\mathcal {O}}(S^7)\) is a (not trivial) faithfully flat Hopf–Galois extension of \({\mathcal {O}}(S^4)\).
A set of generators for the algebra B is given by the elements
and their \(*\)-conjugated \(\alpha ^*, \beta ^*\), with \(x^*=x\). From the 7-sphere relation \(\sum z_\mu ^* z_\mu =1\), it follows that they satisfy a 4-sphere relation \(\alpha ^* \alpha + \beta ^* \beta + x^2=1\).
For future use we also note these generators satisfy the relations
together with their \(*\)-conjugated.
4.1.2 The equivariant derivations
Since the sphere \(S^7\) and \(S^4\) are the homogeneous spaces \(S^7=Spin(5)/SU(2)\) and \(S^4=Spin(5)/Spin(4)\simeq Spin(5)/SU(2)\times SU(2)\), the Hopf fibration \(S^7\rightarrow S^4\) is a Spin(5)-equivariant SU(2)-principal bundle. Then, the right-invariant vector fields \(X \in so(5)\simeq spin(5)\) on Spin(5) project to the right cosets \(S^7\) and \(S^4\) and generate the \({\mathcal {O}}(S^7)\)-module of vector fields on \(S^7\) and the \({\mathcal {O}}(S^4)\)-module of those on \(S^4\). A convenient generating set for the \({\mathcal {O}}(S^7)\)-module is given by the following right SU(2)-invariant vector fields on \(S^7\) (cf. [11]):
Here the partial derivatives \(\partial _a, \partial _a^*\), are defined by \(\partial _a(z_c) = \delta _{a c}\) and \(\partial _a(z_c^*)=0\) and similarly for \(\partial _a^*\), \(a, c=1,2,3,4\). The vector fields above are chosen so that their commutators close the Lie \(*\)-algebra so(5) in the form
The elements \(H_1,H_2\) are the generators of the Cartan subalgebra, and \(E_\textsf {r} \) is labelled by
one of the eight roots. Also, \(N_\textsf {rs}=0\) if \(\textsf {r+s}\) is not a root and \(N_\textsf {rs} \in \{1,-1\}\) otherwise. The \(*\)-structure is given by
The \(*\)-structure on vector fields X is defined by \(X^*(f)= (S(X)(f^*))^* = -(X(f^*))^*\) for any function f, and one accordingly checks that for the vector fields in (4.4) and (4.5), \(E_{-\textsf {r} } (z_a) =-(E_{\textsf {r} }(z^*_a))^*\) and \(H_j (z_a)=-(H_j(z^*_a))^*\).
The vector fields (4.4) and (4.5), being invariant under the action of SU(2), projects to a generating set for the \({\mathcal {O}}(S^4)\)-module of vector fields on \(S^4\). Explicitly one finds,
using analogous partial derivatives on \({\mathcal {O}}(S^4)\). Indeed the \({\mathcal {O}}(S^4)\)-module of vector fields on \(S^4\) can be generated by the five elements \(H_\mu = \partial _{\mu ^*} - x_\mu D\), for \(D = \sum \nolimits _\mu x_\mu \partial _{\mu }\) the Liouville vector field. The five weights \(\mu \) are those of the representation [5] of so(5) with
and sphere relation \(\sum _\mu x_\mu ^* x_\mu =1\). The commutators \([H_\mu , H_\nu ]\) give the generators in (4.8).
Dually, the vector fields (4.4) and (4.5) are \(H={\mathcal {O}}(SU(2))\)-equivariant derivations and generate the \({\mathcal {O}}(S^4)\)-module of such derivations
The general H-equivariant derivation is then of the form
for generic elements \(b_j, b_\textsf {r} \in {\mathcal {O}}(S^4)\). These derivations are real, that is \(X^*= X\), if and only if \(b_j^*= b_j\) and \(b_\textsf {r} ^*=b_{-\textsf {r} }\). On the generators of \({\mathcal {O}}(S^7)\) the derivation X is given as
where \(\textsf {M} \) is the \(4 \times 4\) matrix with entries in \( {\mathcal {O}}(S^4)\)
The derivation (4.11) restricts to
with
4.1.3 The Lie \(*\)-algebra of gauge transformations
We next look for infinitesimal gauge transformations, that is H-equivariant derivations X as in (4.11) which are vertical: \(X^\pi (b)=0\), for \(b \in {\mathcal {O}}(S^4)\). These are the kernel of the matrix \(\textsf {M} ^\pi \) in (4.14). Their collection \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\) is clearly an \({\mathcal {O}}(S^4)\)-module. It is also a Lie algebra with Lie bracket \([bX,b'X']=bb'[X,X']\) for any \(b,b' \in {\mathcal {O}}(S^4)\) and \(X,X' \in \textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\).
The Spin(5) equivariance of the principal bundle \(S^7\rightarrow S^4\) implies that the Lie algebra \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\) can be organised using the representation theory of the Lie algebra so(5). Indeed, the Spin(5) action on \(S^7\) lifts to \(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))\) via the adjoint action, \(Ad_gX=L_g\circ X\circ L_g^{-1}\), where, as usual, \(L_g(a)(p)=a(g^{-1}p)\) for \(g\in Spin(5)\), \(p\in S^7\) and \(a\in {\mathcal {O}}(S^7)\). Since the Spin(5)-action closes on the subalgebra \({\mathcal {O}}(S^4) \subseteq {\mathcal {O}}(S^7)\), it also closes on the Lie subalgebra \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7) )\) of vertical derivations, indeed \(Ad_gX(b)=L_g( X(L_g^{-1}(b)))=0\) for all \(g\in Spin(5)\), \(b\in {\mathcal {O}}(S^4)\). Infinitesimally, \([T, X](b) = 0\) for all \(T\in so(5)\), \(b\in {\mathcal {O}}(S^4)\).
It follows that \( \textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7)) =\oplus _\pi V_\pi \) as linear space, with the sum over a class of representations \(V_\pi \) of so(5) of vertical \({\mathcal {O}}(SU(2))\)-equivariant derivations. This decomposition will be worked out in details in § 4.1.4.
Proposition 4.1
The Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\) of infinitesimal gauge transformations of the \({\mathcal {O}}(SU(2))\)-Hopf–Galois extension \({\mathcal {O}}(S^4 ) \subset {\mathcal {O}}(S^7 )\) is generated, as an \({\mathcal {O}}(S^4 )\)-module, by the elements
The \(*\)-structure is given by
Proof
An H-equivariant real derivation \(X= b_1 H_1 + b_2 H_2 + \sum \nolimits _\textsf {r} b_\textsf {r} E_\textsf {r} \) vanishes on \({\mathcal {O}}(S^4 )\) if \(\textsf {M} ^\pi \begin{pmatrix} \alpha&\alpha ^*&\beta&\beta ^*&x \end{pmatrix}^t = 0\), for the associated matrix \(\textsf {M} ^\pi \) in (4.14). This reads
At the algebraic level of the present paper, it is enough to look for solutions with entries of the matrix \(\textsf {M} ^\pi \) that are linear in the \({\mathcal {O}}(S^4)\) generators. An explicit computation leads to the derivations
The derivations in (4.15) are obtained as the linear combinations
Each vertical derivation, \( X= b_1 H_1 + b_2 H_2 + \sum \nolimits _\textsf {r} b_\textsf {r} E_\textsf {r} \), with \(b_j, b_\textsf {r} \in {\mathcal {O}}(S^4)\) which satisfy (4.17) is expressed as combination of the vertical derivations \(K_j, W_\textsf {r} \) in (4.15) as
with coefficients \(c_1,c_2, c_\textsf {r} \in {\mathcal {O}}(S^4)\) given by
The proof uses the equation (4.17) for the kernel of \(\textsf {M} ^\pi \). Indeed, from (4.15) one computes:
where the last equality follows from equations (4.17) for the coefficients \(b_j\), \(b_\textsf {r} \).
The generators in (4.15) satisfy \(K_j (f^*)=-(K_j(f))^*\) and \(W_\textsf {r} (f^*) =-(W_{-\textsf {r} }(f))^*\) for \(f\in {\mathcal {O}}(S^7)\), from which one gets the \(*\)-structure in (4.16). This also follows from \(H_j^*=H_j\) and \(E_\textsf {r} ^*=E_{-\textsf {r} }\), in (4.7) using \((bX)^*= b^* X^*\), for \(b \in {\mathcal {O}}(S^4)\) and X a derivation. \(\square \)
The action of the vertical derivations \(K_j, W_\textsf {r} \) on the algebra generators \(z_a\) of \({\mathcal {O}}(S^7)\) is listed in Table 2 in Appendix 1.
Proposition 4.2
The generators in (4.15) transform under the adjoint representation of so(5) with highest weight vector \(W_{11}\):
with \(N_\textsf {rs}\) the structure constants of so(5) as before, with \(N_\textsf {rs}=0\) if \(\textsf {r+s}\) is not a root.
Proof
By direct computation. \(\square \)
Remark 4.3
The generators in (4.15) are not independent over the algebra \({\mathcal {O}}(S^4)\). Indeed one finds they satisfy the relations:
These relations have a deep geometrical meaning. They are the vanishing ‘vertical’ components of five vector fields which are horizontal for a canonical connection on the principal bundle [6]. These horizontal vector fields carry the five dimensional representation of so(5) the smallest not trivial vector representation of so(5) with highest weight vector of weight (1, 0). On the other hand, any d-dimensional representation of so(5) as vertical vector fields on \(S^7\) vanishes when \(d<10\). Indeed, the only vertical equivariant derivation which is linear in the generators of \({\mathcal {O}}(S^4)\) and with weight (1, 0) is \(X_{10} = \alpha ^* H_2 + \beta ^* E_{-11} - \beta E_{-1-1}\). This generator is annihilated by \(E_{1,1}\), \(E_{1,0}\) and \(E_{1,-1}\), but it is not by \(E_{0,1}\). Since \( E_{0,1} \vartriangleright X_{10}\) has weight (1, 1) which is not present in the five-dimensional representation, we conclude that the minimal space of (linear in the generators of \({\mathcal {O}}(S^4)\)) derivations is ten-dimensional.
4.1.4 A representation theoretical decomposition of \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\)
The result of multiplying the generators of \({\mathcal {O}}(S^4)\) with the ten vector fields in (4.15) can be organised using the representation theory of so(5) (cf. [1, 2]). An irreducible representation of so(5) is characterised by two non negative integers (s, n) and we denote it [d(s, n)]. It has highest weight vector of weight \(\tfrac{s}{2}(1,1)+ n(1,0)\) and is of dimension \(d(s,n)= \tfrac{1}{6} (1 + s)(1+n)(2+s+n)(3+s+2n)\). The generic vector field in the \({\mathcal {O}}(S^4)\)-module \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\) is a combination of the vector fields in (4.15) with coefficients given by polynomials in the generators of \({\mathcal {O}}(S^4)\). The algebra \({\mathcal {O}}(S^4)\) decomposes in the sum of irreducible representations of so(5) (spherical harmonics on \(S^4\)) as
with [d(0, n)] the representation of highest weight vector \(\alpha ^{n}\) of weight (n, 0) consisting of polynomials of homogeneous degree n in the generators of \({\mathcal {O}}(S^4)\) (see Appendix 1).
The 50 vector fields obtained by multiplying the vector fields in (4.15) with the generators of \({\mathcal {O}}(S^4)\) can be arranged according to the representations \([35] \oplus [10] \oplus [5]\). The highest weight vectors for these three representations are worked out to be given respectively by:
with the label denoting the value of the corresponding weight.
Lemma 4.4
When represented as vector fields on the bundle, the representation [5] generated by the vector \(X_{10}\) above vanishes. Also, \(Y_{11} = - \sqrt{2} W_{11}\) so that the representation [10] generated by \(Y_{11}\) is the one in Proposition 4.2. The vector \(Z_{21}\) makes up the representation [35], none of whose vectors do vanish.
Proof
The action of so(5) on the vector \(X_{10}\) yields the additional four vectors
These five derivations vanish on \({\mathcal {O}}(S^7)\); they are in fact the vanishing combinations of derivations in (4.20). (The relations (4.20) have then a representation-theoretical meaning.) Using the 4-sphere relation and the relations in (4.3) one shows that
Then the vector \(Y_{11}\) generates the starting representation [10] in (4.15) as stated. \(\square \)
By construction \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\) is closed under commutator. It turns out that the commutators of the derivations in (4.15) can be expressed again in terms of the derivations in (4.15) with coefficients which are linear in the generators of \({\mathcal {O}}(S^4)\) (all commutators are listed in Appendix 1. More specifically we have the following:
Lemma 4.5
The commutators of the derivations in (4.15) can be organised according to the representation \([35] \oplus [10]\) of so(5) already found and generated by \(\alpha W_{11}\) and \(W_{11}\).
Proof
There are 45 commutators. The non vanishing commutator with highest weight is \([W_{11}, W_{10}]\) with weight (2, 1). A direct computation shows that
and the corresponding representation is the [35] found in the previous lemma.
The commutator \([W_{11}, W_{1-1}]\), the only one of weight (2, 0), belongs to the representation [35]. The latter comprises also two combinations of the three vectors
of weight (1, 1). On the other hand, their combination
is annihilated by all positive element of so(5) and generates a copy of the representation [10]. In fact this is just the starting representation in (4.15) of Proposition 4.2. An explicit computation gives
so that using the relation (4.24) one obtains
Thus the representation [10] generated by \(T_{11}\) is the one in (4.15) as stated. \(\square \)
Proposition 4.6
The Lie algebra \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\) decomposes as
Here [d(2, n)] is the representation of so(5) as derivations on \({\mathcal {O}}(S^7)\) of highest weight vector \(\alpha ^n W_{11}\) of weight \((n+1,1)\) and dimension \( d(2,n)=\tfrac{1}{2}(n+1)(n+4)(2n+5) \).
Proof
From the splitting (4.21) of \({\mathcal {O}}(S^4)\) (and Appendix 1), we need to consider the \( 10 \cdot d(0,n)\) vector fields obtained by multiplying the 10 vector fields in (4.15) with the polynomials of degree n in the generators of \({\mathcal {O}}(S^4)\) that are in the representation [d(0, n)] of highest weight vector \(\alpha ^n\). Of these, \(\alpha ^n W_{11} = \alpha ^{n-1} Z_{21}\) is a highest weight vector and generates the representation [d(2, n)]. For the remaining vectors fields, \(\alpha ^{n-1} Y_{11}\) is a highest weight vector generating the representation \([d(2,n-1)]\). There is then the highest weight vector \(\alpha ^{n-1} X_{10}\) generating the representation [d(0, n)]. And finally there is the highest weight vector \(\alpha ^{n-3} \rho ^{2}Z_{21} = \alpha ^{n-3} Z_{21}\) of the representation \([d(2,n-2)]\) (here \(\rho ^2:=\alpha \alpha ^*+\beta \beta ^*+x^2=1\)). There is no room for additional vectors since a direct computation shows that \(d(2,n) + d(2,n-1) + d(0,n) + d(2,n-2) = 10 \cdot d(0,n)\). Since \( Z_{21}=\alpha W_{11}\), \(Y_{11} =- \sqrt{2} W_{11}\) and \(X_{10}=0\), the only representation which has not yet appeared in lower degree is [d(2, n)], the one of highest weight \(\alpha ^n W_{11}\). \(\square \)
4.1.5 Braided derivations and infinitesimal gauge transformations
The right invariant vector fields \(H_1\) and \(H_2\) of Spin(5) are the vector fields of a maximal torus \({\mathbb {T}}^2 \subset Spin(5)\). They define the universal enveloping algebra K of the abelian Lie algebra \([H_1,H_2]=0\). Their action (4.4) on \({\mathcal {O}}(S^7)\) commutes with the \({\mathcal {O}}(SU(2))\) right coaction on \({\mathcal {O}}(S^7)\). To the torus 2-cocycle of [3, Ex. 3.21] there corresponds then a twist
with universal R-matrix \({\textsf {R} }_\textsf {F} ={\overline{\textsf {F} }}^2\). In fact these elements belong to a topological completion of the algebraic tensor product \(K\otimes K\). This fact does not play a role here since we diagonalise \(\textsf {F} \) (we systematically use it on eigen-functions of the generators \(H_1, H_2\)).
The twist \(\textsf {F} \) in (4.26) hence leads to the \({\mathcal {O}}(SU(2))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_{\theta }) ={\mathcal {O}}(S^7_\theta )^{co{\mathcal {O}}(SU(2))} \subset {\mathcal {O}}(S^7_\theta )\) introduced in [12]. It satisfies equation (3.14) so that it is compatible with the \(*\)-structure. To conform with the literature, in the following we use the subscript \(\theta \) instead of \(\textsf {F} \) for twisted algebras and their multiplications, as well as for module structures. The \(*\)-algebra \({\mathcal {O}}(S^7_\theta )\) is generated by coordinates \(z_a, z_a^*\), \(a=1,2,3,4\). Their commutation relations are obtained from (2.25) given that for the action of \(H_1\) and \(H_2\) the \(z_a\) have eigenvalues \(\frac{1}{2}(1, -1), \, \frac{1}{2}(-1, 1), \, \frac{1}{2}(-1, -1), \, \frac{1}{2}(1, 1)\), for \(a=1,2,3,4\). The only nontrivial relations among the \(z_a\) are:
Those with the \(z_a^*\) are obtained using that they have eigenvalues opposite to the eigenvalues of the \(z_a\). These coordinates satisfy the relation \(z_1{\scriptstyle {\bullet }_\theta }z^*_1+z_2{\scriptstyle {\bullet }_\theta }z^*_2+z_3{\scriptstyle {\bullet }_\theta }z^*_3+z_4{\scriptstyle {\bullet }_\theta }z^*_4=1\). The \(*\)-subalgebra \({\mathcal {O}}(S^4_{\theta })\) of \({\mathcal {O}}(SU(2))\)-coinvariants is generated by
The only nontrivial commutation relations are
and their complex conjugates. They can be obtained from the twisted multiplication rule (2.25) by using that \(\upalpha , \upbeta , {\textsf {x} }\) are eigen-functions of \(H_1\) and \(H_2\), with eigenvalues (1, 0), (0, 1) and (0, 0) respectively. They satisfy the relations \(\upalpha {\scriptstyle {\bullet }_\theta }\upalpha ^*+\upbeta {\scriptstyle {\bullet }_\theta }\upbeta ^*+{\textsf {x} }{\scriptstyle {\bullet }_\theta }{\textsf {x} }=1\). From these one then establishes:
Remark 4.7
The relations (4.29) are the analogues of the classical ones (4.3). However, in passing from the algebra \({\mathcal {O}}(S^4)\) to the algebra \({\mathcal {O}}(S^4_\theta ) \) we rescaled by a phase the classical elements \(\alpha , \beta \). In the vector space \({\mathcal {O}}(S^4_\theta )={\mathcal {O}}(S^4) \), one has \({\textsf {x} }=x\) and
Since the Lie algebra so(5) is a braided Lie algebra associated with K with trivial R-matrix \({\textsf {R} }=1 \otimes 1\), we can twist it to the braided Lie algebra \(so_\theta (5)\) associated with \((K_\textsf {F} , {\textsf {R} }_\textsf {F} = {\overline{\textsf {F} }}^2)\). It has Lie brackets (see Proposition 2.2)
with \({\textsf {r} } \wedge {\textsf {s} }:= r_1 s_2 - r_2 s_1\). Here, as for the so(5)-commutators in (4.6), \(N_\textsf {rs}=0\) if \(\textsf {r+s}\) is not a root.
Similarly, the \({\mathcal {O}}(S^4)\)-module and Lie \(*\)-algebra \( \mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))\) is deformed to the \({\mathcal {O}}(S^4_\theta )\)-module and braided Lie \(*\)-algebra \((\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))_\textsf {F} , [~,~]_\textsf {F} , \cdot _\textsf {F} , *)\) associated with \((K_\textsf {F} , {\textsf {R} }_\textsf {F} )\). The module \((\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))_\textsf {F} \) is generated by derivations \(H_j\) and \(E_\textsf {r} \) with module structure in (2.34):
for all \(a\in {\mathcal {O}}(S^7_\theta )\) and \( a_\textsf {s} \in {\mathcal {O}}(S^7_\theta ) \) eigen-functions of \(H_j\) with eigenvalues \({s}_j\) (being \(E_\textsf {r} \) eigenvectors of \(H_j\)). The Lie brackets are determined by those of \(so_\theta (5)\) in (4.30) using equation (2.35) for the module structure. The \(*\)-structure is the same as that of \(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))\), as stated in the first sentence of § 3.3.3.
The Lie \(*\)-algebra of infinitesimal gauge transformations \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\) is generated, as an \({\mathcal {O}}(S^4)\)-module, by the operators in (4.15). Its twist deformation is the \({\mathcal {O}}(S^4_\theta )\)-module and braided Lie \(*\)-algebra \(\big (\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7) )_\textsf {F} , [~,~]_\textsf {F} ,\cdot _\textsf {F} , *\big )\) associated with \((K_\textsf {F} , {\textsf {R} }_\textsf {F} = {\overline{\textsf {F} }}^2)\). It has braided Lie bracket determined on generators:
in parallel with the result in (4.30). On generic elements \(X,X'\) in the linear span of the generators in (4.15) and \(b,b'\in {\mathcal {O}}(S^4)\), the equation (2.35) gives
From Corollary 3.11 we have \(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7_\theta )) = {\mathcal {D}}((\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7))_\textsf {F} )\) with \({\mathcal {D}}\) an isomorphism of \(K_\textsf {F} \)-braided Lie and \(A_\textsf {F} \)-module \(*\)-algebras. In particular \({\mathcal {D}}(H_j)^{{{\tilde{*}}}}={\mathcal {D}}(H_j^*)={\mathcal {D}}(H_j)\) and \({\mathcal {D}}({E}_{\textsf {r} })^{{{\tilde{*}}}}={\mathcal {D}}({E}_{\textsf {r} }^*)={\mathcal {D}}({E}_{-\textsf {r} })\), \(j=1,2\), \(\textsf {r} \in \Gamma \) (cf. (4.7)). Thus, recalling (4.30):
Proposition 4.8
The braided Lie \(*\)-algebra \(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(S^7_\theta ))\) of equivariant derivations of the \({\mathcal {O}}(SU(2))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(S^7_\theta )\) is generated, as an \({\mathcal {O}}(S^4_\theta )\)-module by elements
with bracket closing the braided Lie algebra \(so_\theta (5)\):
(and \(N_\textsf {rs}=0\) if \(\textsf {r+s}\) is not a root). The \({\mathcal {O}}(S^4_\theta )\)-module structure is in (2.36) (with \({\scriptstyle {\,\bullet }}_{\scriptscriptstyle {\textsf {F} \,}}={\scriptstyle {\bullet }_\theta })\). The \(*\)-structure on generators is given by \({\widetilde{H^{}}_j}^{\,{{{\tilde{*}}}}}={\widetilde{H}}_j\) and \({{\widetilde{E^{}}_\textsf {r} }}^{\,{{{\tilde{*}}}}}= {\widetilde{E}}_{-\textsf {r} }\).
For \(a_\textsf {s} \in {\mathcal {O}}(S^7_\theta )\) an eigen-function of \(H_j\) of eigenvalue \({s}_j\), the derivation \({\widetilde{E}}_\textsf {r} \) acts as
On the product of two such eigen-functions \(a_\textsf {s} , {a}_\textsf {m} \in {\mathcal {O}}(S^7_\theta )\), we can explicitly see that \({\widetilde{E}}_\textsf {r} \) acts as a braided derivation, with respect to the braiding \({{\textsf {R} }_\textsf {F} }= \textsf {F} _{21} \, {\overline{\textsf {F} }}= {\overline{\textsf {F} }}^2\):
Using these results for the subalgebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ))={\mathcal {D}}(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7) )_\textsf {F} )\) of vertical derivations, we have the following characterization of \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta )}({\mathcal {O}}(S^7_\theta ))\).
Proposition 4.9
The braided Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ) )\) of infinitesimal gauge transformations of the \({\mathcal {O}}(SU(2))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(S^7_\theta )\) is generated, as an \({\mathcal {O}}(S^4_\theta )\)-module, by the elements
with bracket given in Table 1. The braided Lie bracket of generic elements \({\widetilde{X}},{\widetilde{X}}'\) in \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta )}({\mathcal {O}}(S^7_\theta ) )\) and \(b,b'\in {\mathcal {O}}(S^4_\theta )\) is then given by
The \(*\)-structure on generators is given by \({\widetilde{K^{}}_j}^{\,{{{\tilde{*}}}}}={\widetilde{K}}_j\) and \({\widetilde{W^{}}_\textsf {r} }^{\,{{{\tilde{*}}}}}= {\widetilde{W}}_{-\textsf {r} }\). It is extended on the whole \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ) )\) via (3.21).
Proof
For all \({\widetilde{X}},{\widetilde{X}}' \in \textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ) )\) we have \([{\widetilde{X}}, {\widetilde{X}}']_{{\textsf {R} }_\textsf {F} }={\mathcal {D}}([X,X']_\textsf {F} )\) from Proposition 2.6 with the bracket on the right hand side given in (4.31). Using the classical Lie brackets listed in Table 3 of Appendix 1, we can compute the brackets of the generators of the braided gauge Lie algebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ))\). For instance, for \([\widetilde{W}_{-1-1},{{\widetilde{W}}}_{01}]_{{\textsf {R} }_\textsf {F} } \) we first compute
Here we used (4.31) to relate the brackets \([~,~]_\textsf {F} \) and [ , ], the module structure of \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7) )_\textsf {F} \) in (2.34) and that the coordinates of the sphere \(S^4\) are eigen-functions of \(H_1\) and \(H_2\). Next, applying the algebra map \({\mathcal {D}}\) leads to
Here to pass from the generators of \({\mathcal {O}}(S^4)\) to those of \({\mathcal {O}}(S^4_\theta )\) we used (see Remark 4.7):
Half of the brackets among the generators (4.37) of the braided Lie algebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(S^7_\theta ))\) are listed in Table 1. The remaining ones can be obtained similarly, or more directly using the compatibility of the \(*\)-structure with the braided Lie algebra and \({\mathcal {O}}(S^4)\)-module structures as in (3.21),
Recalling that \({\mathcal {D}}\) is a \(*\)-isomorphism we have
Therefore,
with
For instance,
while
\(\square \)
The action of any element \({\widetilde{W}}_\textsf {r} \) on an algebra element \(a_\textsf {s} \in {\mathcal {O}}(S^7_\theta )\) is as in (4.35),
with a braided derivation property as in (4.36),
4.2 The orthogonal bundle on the homogeneous space \(S^{4}_\theta \)
The 4-sphere of the previous example is the prototype of more general noncommutative \(\theta \)-spheres \(S^{2n}_\theta \). These are quantum homogeneous spaces of quantum groups \(SO_\theta (2n+1,{\mathbb {R}})\) [7, 14]. The Hopf–Galois extension \({\mathcal {O}}(S^{2n}_\theta )={\mathcal {O}}(SO_\theta (2n+1,{\mathbb {R}}))^{co {\mathcal {O}}(SO_\theta (2n,{\mathbb {R}}))} \subset {\mathcal {O}}(SO_\theta (2n+1,{\mathbb {R}}))\) was obtained in [3] from the extension of the classical SO(2n)-bundle \(SO(2n+1) \rightarrow S^{2n}\) via a twist deformation process for quantum homogeneous spaces.
As in the previous section, the modules of infinitesimal gauge transformations of these noncommutative Hopf–Galois extensions are obtained by deforming those of the corresponding classical bundles. We study here the case \(n=2\) of the Hopf–Galois extension \({\mathcal {O}}(S^{4}_\theta )={\mathcal {O}}(SO_\theta (5,{\mathbb {R}}))^{co {\mathcal {O}}(SO_\theta (4,{\mathbb {R}}))} \subset {\mathcal {O}}(SO_\theta (5,{\mathbb {R}}))\). We address the general case in [6].
Let \({\mathcal {O}}(M(4, {\mathbb {R}}))\) be the commutative complex \(*\)-algebra with generators \(m_{JL}\), and capital indices J, L running from 1 to 4. It has the standard bialgebra structures
in matrix notation, where \(\overset{.}{\otimes }\) denotes the combination of tensor product and matrix multiplication. The Hopf algebra \({\mathcal {O}}(SO(4, {\mathbb {R}})) \) of coordinate functions on \(SO(4, {\mathbb {R}})\) is the quotient of \({\mathcal {O}}(M(4, {\mathbb {R}}))\) by the bialgebra ideal
and the further assumption \(\det (M)=1\). Indeed, this is a \(*\)-ideal for the \(*\)-structure \(*(M)=QMQ^t\) in \({\mathcal {O}}(M(4, {\mathbb {R}}))\). If we introduce on the set of indices \(\{1, \dots , 4\}\) the involution defined by \(1'=3\) and \(2'=4\), the \(*\)-structure can be given as
The \(*\)-bialgebra \({\mathcal {O}}(SO(4, {\mathbb {R}}))\) is a Hopf \(*\)-algebra with antipode \(S(M):= Q M^tQ^{-1}\).
Similarly, one has the algebra \({\mathcal {O}}(M(5, {\mathbb {R}}))\), the commutative \(*\)-bialgebra with generators \(n_{JL}\) (capital indices J, L now run from 1 to 5). The coproduct and counit are
The algebra of coordinate functions on \(SO(5, {\mathbb {R}})\) is the quotient of \({\mathcal {O}}(M(5, {\mathbb {R}}))\) by the bialgebra \(*\)-ideal
and the additional requirement that \(\det (N)=1\). The \(*\)-structure is now
with \(5'=5\). Then \({\mathcal {O}}(SO(5, {\mathbb {R}}))\) is a Hopf \(*\)-algebra with antipode \(S(N):= Q N^tQ^{-1}\).
We shall select the last column of N by writing \(n_{j5} = u_j\), for \(j=1, \cdots , 4\) and \(n_{55} = x\).
The surjective Hopf \(*\)-algebra morphism
induces a right coaction of \({\mathcal {O}}(SO(4, {\mathbb {R}}))\) on \({\mathcal {O}}(SO(5, {\mathbb {R}}))\):
The \(*\)-subalgebra of \( {\mathcal {O}}(SO(5, {\mathbb {R}}))\) made of coinvariant elements is isomorphic to the \(*\)-algebra of coordinate functions \({\mathcal {O}}(S^{4})\) on the 4-sphere \(S^{4} \). It is indeed generated by the elements \(u_i, u_i^*\) and x in the last column of the defining matrix \(N=(n_{JK})\) of \({\mathcal {O}}(SO(5, {\mathbb {R}}))\), which satisfy the sphere equation \( (S(N)N)_{55}= 2 (u_1^* u_1 + u_2^* u_2) +x^2=1\). The algebra extension \({\mathcal {O}}(S^{4})={\mathcal {O}}(SO(5,{\mathbb {R}}))^{co {\mathcal {O}}(SO(4,{\mathbb {R}}))} \subset {\mathcal {O}}(SO(5,{\mathbb {R}}))\) is Hopf–Galois.
The coordinate algebra of this orthogonal 4-sphere is isomorphic to the one of the 4-sphere of the previous section (the base space algebra of the SU(2)-fibration) with generators in (4.2) via the identification
We now determine infinitesimal gauge transformations of the Hopf–Galois extension \({\mathcal {O}}(S^{4}) \subset {\mathcal {O}}(SO(5,{\mathbb {R}}))\) using, as done for the previous example, the representation theory of so(5) as vector fields on the bundle. Like it was the case for the instanton bundle, the \({\mathcal {O}}(S^{4})\)-module of infinitesimal gauge transformations is generated by the ten generators of so(5). The crucial difference (see Proposition 4.13) is that in the present example, vector fields of given degree n in the generators of \({\mathcal {O}}(S^{4})\) split in the sum of two irreducible representations with distinct highest weight vectors with same weight.
We then start with the \({\mathcal {O}}(S^4)\)-module of equivariant derivations for \(H:={\mathcal {O}}(SO(4,{\mathbb {R}}))\).
Let \(\partial _{\scriptscriptstyle {IJ}}\) denote the derivation on \({\mathcal {O}}(M(5, {\mathbb {R}}))\) given on the generators by \(\partial _{\scriptscriptstyle {IJ}}(n_{KL}) = \delta _{IK} \delta _{JL}\), for \(I,J,K,L=1,\dots , 5\). Then, recalling the coaction (4.49), \(\mathrm{{Der}}_{{\mathcal {M}}^H} ({\mathcal {O}}(SO(5, {\mathbb {R}})))\) is generated by the derivations
with summation on \(K=1, \dots , 5\) understood. They satisfy \( H_j^*=H_j\) and \(E_\textsf {r} ^*=E_{-\textsf {r} }\). These ten generators close the Lie \(*\)-algebra of so(5) in (4.6), from which the labels used. It is important to notice that due to the equivariance for the right coaction of \({\mathcal {O}}(SO(4,{\mathbb {R}}))\), when applied to a generator \(n_{JK}\), these derivations do not move the second index. This fact will play a role later on. Being equivariant, they restrict to derivations on the \(*\)-subalgebra \({\mathcal {O}}(S^4)\) of coinvariants:
using partial derivations for the generators \(n_{j5}\) of \({\mathcal {O}}(S^4)\). With the isomorphism in (4.50) these derivations coincide with the ones in (4.8).
The generic equivariant derivation is of the form \( X= b_1 H_1 + b_2 H_2 + \sum \nolimits _\textsf {r} b_\textsf {r} E_\textsf {r} \), with \(b_j, b_\textsf {r} \in {\mathcal {O}}(S^4)\) and \(H_j, E_{\textsf {r} }\) in (4.52). The condition for X to be vertical only uses its restriction to \({\mathcal {O}}(S^4)\), that is the derivations in (4.53) and thus coincides with the conditions (4.17) under the isomorphism (4.50). Then, in parallel with Proposition 4.1 for the generators (4.15), and noticing that its proof only uses the algebra structure of \({\mathcal {O}}(S^4)\), we have the following result (we are dropping an overall factor of 2).
Proposition 4.10
The Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(SO(5, {\mathbb {R}})))\) of infinitesimal gauge transformations of the \({\mathcal {O}}(SO(4, {\mathbb {R}}))\)-Hopf–Galois extension \({\mathcal {O}}(S^4 ) \subset {\mathcal {O}}(SO(5, {\mathbb {R}}))\) is generated, as an \({\mathcal {O}}(S^4 )\)-module, by the derivations
with \(K_j^*=K_j\) and \(W^*_\textsf {r} = W_{-\textsf {r} }\). They are eigen-operators for \(H_1\) and \(H_2\) and transform under the adjoint representation [10] of so(5) (that is (4.19) hold), with \(W_{11}\) the highest weight vector.
In particular we have that \(H_j \vartriangleright K_l = 0\) and \(H_j \vartriangleright W_\textsf {r} = r_j W_\textsf {r} \), and this induces a (left, see later) \({\mathbb {Z}}^2\)-grading on the derivations. They satisfy analogue relation of those in (4.20):
Remark 4.11
The Lie bracket of the generators \(H_j\) and \(W_\textsf {r} \) are those in Table 3 of Appendix 1 (with the identification (4.50) and up to a rescaling). While the Lie \(*\)-algebras \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(S^7))\) and \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(SO(5, {\mathbb {R}})))\) have the same Lie bracket on generators, they differ as \({\mathcal {O}}(S^4)\)-modules, and hence as Lie \(*\)-algebras, since in the orthogonal case there is no analogue of the constraint (4.24) occurring in the instanton case.
As in the case of the instanton bundle (cf. Lemma 4.4), the fifty vector fields obtained by multiplying the vector fields in (4.54) with the generators of \({\mathcal {O}}(S^4)\) can be arranged according to the representations \([35] \oplus [10] \oplus [5]\) of so(5). The highest weight vectors for these three representations are given respectively by:
with the label denoting the value of the corresponding weight. These are the analogous of the vector fields found in (4.22) for the SU(2) Hopf bundle.
When represented as vector fields on the bundle, the vector \(X_{10}\) is zero (because of relations analogous to (4.23)) and the representation [5] it generates vanishes. On the other hand, the vector field \( Y_{11}\) is no longer proportional to \(W_{11}\), as it was for the Hopf bundle case due to the constraint (4.24). An explicit computation, using also the condition \((N^\dag N)_{5K} = \delta _{5K}\) yields:
The vector field \({Y}_{11}\) generates a different copy of the ten-dimensional representation of so(5) that we denote by \(\widehat{[10]}\) to distinguish it from the ten-dimensional representation [10] of highest weight vector \(W_{11}\). Notice that while [10] consists of vector fields which are combinations of those in (4.54) with coefficients of degree zero in the generators of \({\mathcal {O}}(S^4)\), elements of \(\widehat{[10]}\) are combinations with coefficients of degree one.
Next, in parallel with Lemma 4.5, the representation \([35] \oplus \widehat{[10]}\) of so(5) just found are the ones that occur in the decomposition of the commutators.
Lemma 4.12
The commutators of the derivations in (4.54) can be organised according to the representations \([35] \oplus \widehat{[10]}\) of so(5) of highest weight vectors \(\alpha W_{11}\) and \(Y_{11}\) respectively.
Proof
There are 45 non vanishing commutators. The non vanishing commutator with highest weight is \([W_{11}, W_{10}]\) with weight (2, 1). A direct computation shows that
and the corresponding representation is the [35] found above.
The other highest weight vector, of weight (1, 1), is
Thus the representation generated by \(T _{11}\) is the ten dimensional \(\widehat{[10]}\). \(\square \)
By using the decomposition of \({\mathcal {O}}(S^4)\) in (4.21), in parallel to Proposition 4.6, we have:
Proposition 4.13
There is a decomposition
Here [d(2, n)], respectively \(\widehat{[d(2,n-1)]}\), is the representation of so(5) with highest weight vector \(\alpha ^n W_{11}\) of weight \((n+1,1)\), respectively \(\alpha ^{n-1} Y_{11}\) of weight (n, 1); they consist of derivations on \({\mathcal {O}}(SO(5, {\mathbb {R}}))\) which are combinations of the derivations in (4.54) with polynomials coefficients of degree n in the generators of \({\mathcal {O}}(S^4)\).
4.2.1 Braided derivations and infinitesimal gauge transformations
Let us now pass to the twisted Hopf–Galois extension
We briefly recall its construction from twist deformation (see [3, §4.1] for details). Consider the 2-cocycle \(\gamma : {\mathcal {O}}({\mathbb {T}}^2)\otimes {\mathcal {O}}({\mathbb {T}}^2)\rightarrow {\mathbb {C}}\) on \({\mathcal {O}}({\mathbb {T}}^2) \subset {\mathcal {O}}(SO(4, {\mathbb {R}}))\), given on the generators by \(\gamma (t_1\otimes t_2)=e^{-\pi i \theta }\), \(\gamma (t_2\otimes t_1)=e^{ \pi i \theta }\) and \({\gamma (t_1\otimes t_1)=\gamma (t_2\otimes t_2)=1}\). Notice that here \(\gamma = \sigma ^2\), where \(\sigma \) is the cocycle used in §4.1.5. We use it to deform the Hopf \(*\)-algebra \({\mathcal {O}}(SO(4, {\mathbb {R}}))\) into the noncommutative Hopf \(*\)-algebra \({\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))\). This latter has same coalgebra structure as the original one but twisted algebra multiplication,
where \(T:= \textrm{diag}(t_1, t_2, t_1^*, t_2^*)\). (Here again, to conform with the literature we use the subscript \(\theta \) instead of \(\gamma \) for twisted algebras and their multiplications.) We set \(\lambda \)
so that \(\lambda _{I J}= \exp (-2i\pi \theta _{I J})\). Since \({{\bar{\gamma }}}(T_J\otimes T_L)=\gamma (T_L\otimes T_J)\), and \(\gamma (T_L\otimes T^*_J)={{\bar{\gamma }}}(T_L\otimes T_J)\) we have \(\lambda _{JI}={\lambda _{IJ}}^{-1}=\lambda _{IJ'}\). It follows that the generators in \({\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))\) satisfy the commutation relations
The twisted antipode turns out to be equal to the starting one, \(S_\theta (m_{IJ}) =S(m_{IJ}).\) The relations (4.43) become
together with \({\det }_{\theta }(M)=1\). The \(*\)-structure is as in (4.44).
Next, using the projection \(\pi \) in (4.48) we lift the 2-cocycle from \({\mathcal {O}}(SO(4, {\mathbb {R}}))\) to \({\mathcal {O}}(SO(5, {\mathbb {R}}))\) (or equivalently we consider the same torus \({\mathbb {T}}^2\) embedded in SO(5)). The resulting Hopf \(*\)-algebra is denoted by \({\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) and has generators \(n_{IJ}\) with relations
where now \(T:= \textrm{diag}(t_1, t_2, t_1^*, t_2^*,1),\) and orthogonality conditions \(N^t {\scriptstyle {\bullet }_\theta }Q {\scriptstyle {\bullet }_\theta }N =Q\) and \(N {\scriptstyle {\bullet }_\theta }Q {\scriptstyle {\bullet }_\theta }N^t=Q\), with \({\det }_{\theta }(N)=1\). The \(*\)-structure is as in (4.47).
The quantum homogeneous space \({\mathcal {O}}(S^{4})\) is deformed into the quantum homogeneous space \({\mathcal {O}}(S_\theta ^{4})\subset {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\), consisting of coinvariants of \({\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) under the \({\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))\)-coaction. This noncommutative \(*\)-subalgebra \({\mathcal {O}}(S_\theta ^{4})\) is generated by five elements \(\{u_I=n_{I5}\}_{I=1,\ldots , 5}=\{u_i, u_{i'}=u^*_i, x\}_{i=1,2}\) with commutation relations, obtained from (4.59),
The orthogonality conditions imply the sphere relation 2 \(\sum \nolimits _{i=1}^{2} u_i^* {\scriptstyle {\bullet }_\theta }u_i +x^2=1\). From the general theory in [3], the algebra extension \({\mathcal {O}}(S_\theta ^{4})\subset {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) of the quantum homogeneous space \({\mathcal {O}}(S_\theta ^{4})= {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))^{co{\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))}\) is still Hopf–Galois.
When considering the braided Lie \(*\)-algebra of infinitesimal gauge transformations of this Hopf–Galois extension, it is useful to think of the latter as the result of a double deformation done with commuting left coaction of \({\mathcal {O}}({\mathbb {T}}^2)\) and right coaction of \({\mathcal {O}}({\mathbb {T}}^2) \subset {\mathcal {O}}(SO(4, {\mathbb {R}}))\). This second \({\mathcal {O}}({\mathbb {T}}^2)\) disappears when considering \({\mathcal {O}}(SO(4, {\mathbb {R}}))\) equivariant quantities. This is the case for the algebra \({\mathcal {O}}(S^{4})\) of coinvariant elements. It is also the case for the equivariant derivations in Proposition 4.10 and it is in this sense that those derivations can be thought of as having trivial right \({\mathbb {Z}}^2\)-grading (they do not move the second index in a generator \(n_{JK}\) as already mentioned).
Thus for the braided Lie \(*\)-algebra of infinitesimal gauge transformations of the Hopf–Galois extension \({\mathcal {O}}(S_\theta ^{4})= {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))^{co{\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))}\) we just need to consider the left torus action and the construction goes exactly as for the SU(2) instanton case of the previous section. In particular we can repeat the construction in §4.1.5 verbatim by considering the maximal torus \({\mathbb {T}}^2 \subset SO(5)\), generated by the right invariant vector fields \(H_1\) and \(H_2\) of SO(5), and use the twist
of K, where K is generated by the right invariant vector fields \(H_1\) and \(H_2\). Hence K is the universal enveloping algebra of the Cartan subalgebra of \(so(5)^{op}\), the Lie algebra so(5) being that of left invariant vector fields on SO(5), cf. [5, §7.1]. The braided Lie \(*\)-algebras of braided derivations and of infinitesimal gauge transformations of the Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) are the twist (left) deformations of \( \mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(SO(5, {\mathbb {R}})))\) and of \(\textrm{aut}_{{\mathcal {O}}(S^4) }({\mathcal {O}}(SO(5, {\mathbb {R}})))\) respectively (with the right torus action playing no role).
As \({\mathcal {O}}(S^4)\)-modules, the Lie algebra \(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(SO(5, {\mathbb {R}})))\), \(H={\mathcal {O}}(SO(4, {\mathbb {R}}))\), is generated by the operators in (4.52) while the Lie algebra of infinitesimal gauge transformations \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(SO(5, {\mathbb {R}})))\) is generated by the operators in (4.54).
From Corollary 3.11\(\mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))) = {\mathcal {D}}( \mathrm{{Der}}_{{\mathcal {M}}^H}({\mathcal {O}}(SO (5, {\mathbb {R}}) )_\textsf {F} )\), where now \(H=O(SO_\theta (4, {\mathbb {R}}))\), and \({\mathcal {D}}\) is an isomorphism of \({\mathcal {O}}(SO_\theta (5,{\mathbb {R}})))\)-braided Lie and \({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})))\)-module \(*\)-algebras. In parallel with Proposition 4.8, we then have:
Proposition 4.14
The braided Lie \(*\)-algebra \(\mathrm{{Der}}_{{\mathcal {M}}^H}(SO_\theta (5, {\mathbb {R}}))\) of equivariant derivations of the \({\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) is generated, as an \({\mathcal {O}}(S^4_\theta )\)-module, by elements
As in (4.34) the bracket closes the braided Lie algebra \(so_\theta (5)\),
with \(N_\textsf {rs}=0\) if \(\textsf {r+s}\) is not a root. The \(*\)-structure is \({\widetilde{H^{}}_j}^{\,{{{\tilde{*}}}}}={\widetilde{H}}_j \) and \(\widetilde{E^{}}^{\,{{{\tilde{*}}}}}_\textsf {r} = {\widetilde{E}}_{-\textsf {r} }.\)
Corollary 3.11 also yields \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})) ) ={\mathcal {D}}(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(SO (5, {\mathbb {R}}))_\theta )\). From Remark 4.11 the brackets among the generators \({\mathcal {D}}(K_j)\) and \({\mathcal {D}}(W_\textsf {r} )\) of \(\textrm{aut}_{{\mathcal {O}}(S_\theta ^4) }({\mathcal {O}}(S_\theta ^7))\) and among those of \(\textrm{aut}_{{\mathcal {O}}(S_\theta ^4) }({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})))\) have the same expression. This follows comparing the twist expressions (4.26) and (4.61) that define the isomorphisms \({\mathcal {D}}\). Nevertheless the braided Lie \(*\)-algebras \(\textrm{aut}_{{\mathcal {O}}(S_\theta ^4) }({\mathcal {O}}(S_\theta ^7))\) and of \(\textrm{aut}_{{\mathcal {O}}(S_\theta ^4) }({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})))\) are different because of the different \({\mathcal {O}}(S_\theta ^4)\)-module structures. Thus as in Proposition 4.9:
Proposition 4.15
The braided Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})) )\) of infinitesimal gauge transformations of the \({\mathcal {O}}(SO_\theta (4, {\mathbb {R}}))\)-Hopf–Galois extension \({\mathcal {O}}(S^4_\theta ) \subset {\mathcal {O}}(SO_\theta (5, {\mathbb {R}}))\) is generated, as an \({\mathcal {O}}(S^4_\theta )\)-module, by the elements
with brackets in Table 1 with the identifications \(\,\varphi _{\scriptscriptstyle {\upalpha }}\upalpha \rightarrow \sqrt{2} u_1, \,\varphi _{\scriptscriptstyle {\upbeta }}\upbeta \rightarrow \sqrt{2} u_2\), \(x\rightarrow x\) following from (4.39) and (4.50) (and up to a rescaling). The braided Lie bracket of generic elements in \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }({\mathcal {O}}(SO_\theta (5, {\mathbb {R}})) )\) is given by
for \(b,b'\in {\mathcal {O}}(S^4_\theta )\) and \({\widetilde{X}},{\widetilde{X}}'\) in the linear span of the generators in (4.63). The \(*\)-structure on the generators \({\widetilde{K^{}}_j}^{\,{{{\tilde{*}}}}}={\widetilde{K}}_j\), \({\widetilde{W^{}}_\textsf {r} }^{\,{{{\tilde{*}}}}}= {\widetilde{W}}_{-\textsf {r} }\) is extended to the whole \(\textrm{aut}_{{\mathcal {O}}(S^4_\theta ) }(SO_\theta (5, {\mathbb {R}}) )\) via the \({\mathcal {O}}(S_\theta ^4)\)-module compatibility (3.21), \((b{\scriptstyle {\bullet }_\theta }{{\widetilde{X}}})^{{{\tilde{*}}}}= ({{\textsf {R} }_\textsf {F} }_\gamma \vartriangleright b^*) {\scriptstyle {\bullet }_\theta }({{\textsf {R} }_\textsf {F} }^\gamma \vartriangleright {{\widetilde{X}}}^{{{\tilde{*}}}}) \). It is compatible with the bracket, \( [{{{{\widetilde{X}}}}'}{}^{{{\tilde{*}}}}, {\widetilde{X}}^{{{\tilde{*}}}}]_{{\textsf {R} }_\textsf {F} }=([{{\widetilde{X}}}, {{\widetilde{X}}}']_{{\textsf {R} }_\textsf {F} })^{{{\tilde{*}}}}\).
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Acknowledgements
This research has a financial support from Università del Piemonte Orientale. PA acknowledges partial support from INFN, CSN4, Iniziativa Specifica GSS, INFN-ICTP (giornate uomo) and from INdAM-GNFM. PA acknowledges hospitality from Università di Trieste and ICTP. GL acknowledges partial support from INFN, Iniziativa Specifica GAST and INFN Torino DGR4. CP acknowledges support from Università di Trieste (assegni Dipartimenti di Eccellenza, legge n. 232 del 2016). GL and CP acknowledge partial support from the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA–INdAM) and hospitality from Università del Piemonte Orientale and INFN Torino.
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Appendices
Appendix A. Decomposition of \({\mathcal {O}}(S^4)\)
We study how \({\mathcal {O}}(S^4)\) decomposes in the sum of irreducible representations [d(s, n)] of so(5). In the algebra \({\mathcal {O}}({\mathbb {R}}^5)\), both \(\alpha \) and \(\rho ^2=\alpha \alpha ^*+\beta \beta ^*+x^2\) are annihilated by all raising operators \(W_\textsf {r} \) (the ones for positive roots), and thus their powers and products. They are of weight (1, 0) and (0, 0) respectively.
Let \(V^{\scriptscriptstyle {(r)}}\) be the \(\left( {\begin{array}{c}4+r\\ r\end{array}}\right) \)-dimensional vector space of monomials of degree r in the indeterminates \(\alpha ,\alpha ^*,\beta ,\beta ^*,x\). The vectors \(\alpha ^{r-2k}\rho ^{2k}\) are highest weight vectors of \(V^{\scriptscriptstyle {(r)}}\) and
where \([d(0,r-2k)]\) is the irreducible representation with highest weight vector \(\alpha ^{r-2k}\rho ^{2k}\) of weight \((r-2k,0)\), and of dimension \(d(0,r-2k)=\tfrac{1}{6} (1+r-2k)(2+r-2k)(3+2r-4k)\). Indeed, for \(r=2\,m\),
which coincides with the dimension \( \left( {\begin{array}{c}4+2m\\ 2m\end{array}}\right) \) of \(V^{\scriptscriptstyle {(2m)}}\). Similar computations go for r odd.
For \(\alpha ,\alpha ^*,\beta ,\beta ^*,x\) coordinate functions on \({\mathcal {O}}(S^4)\), \(\rho ^2=1\) and with fixed r, all representations [d] in the decomposition (4.65) already appeared in \(V^{\scriptscriptstyle {(r')}} \) for some \(r' <r\), but for [d(0, r)]. We hence conclude that
where [d(0, n)] has highest weight vector \(\alpha ^{n}\) of weight (n, 0).
Appendix B. Matrix representation of the braided Lie algebra \(so_\theta (5)\)
We give a matrix representation of the braided Lie algebra \(so_\theta (5)\) as defined in (4.34).
Consider weights \(\mu , \nu = (0,0), (\pm 1,0), (0,\pm 1)\) with order
Using this order for an identification between weights and row/column indices, define matrices \({\mathcal {E}}_{\mu \nu }\) of components
The product of two such matrices is found to be
The minus signs in the exponents are due to \({\mathcal {E}}_{\mu \nu }\) having weight \(\mu -\nu \) (cf. (4.57)).
A direct computation shows the following:
Lemma 1
The matrices
give a matrix representation of the algebra \(so_\theta (5)\) (see (4.34)) with the identification \(\textsf {K} _\textsf {r} \leftrightarrow {\widetilde{E}}_\textsf {r} \) and setting \([\textsf {K} _\textsf {r} , \textsf {K} _\textsf {s} ]_{{\textsf {R} }_\textsf {F} }:= \textsf {K} _\textsf {r} \textsf {K} _\textsf {s} - \lambda ^{2 \textsf {r} \wedge \textsf {s} } \textsf {K} _\textsf {s} \textsf {K} _\textsf {r} \) for the braided commutator of matrices.
The matrices \(\textsf {K} _\textsf {r} \) in the lemma are of the form
The braided commutator \([\textsf {K} _{\mu , \nu }, \textsf {K} _{\tau , \sigma }]_{{\textsf {R} }_\textsf {F} } =\textsf {K} _{\mu , \nu } \textsf {K} _{\tau , \sigma } - \lambda ^{2 (\mu + \nu ) \wedge (\tau + \sigma ))}\textsf {K} _{\tau , \sigma } \textsf {K} _{\mu , \nu } \) is found to be
In the classical limit, \(\lambda =1\), the matrices \({\mathcal {E}}_{\mu \nu }\) reduce to the usual elementary matrices and those in Lemma 1 give the defining matrix representation of the Lie algebra so(5).
The Lie \(*\)-algebra \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\)
In Table 2 we list the action of the generators (4.15) of the Lie algebra of infinitesimal gauge transformations on the generators of the algebra \({\mathcal {O}}(S^7)\).
We list in Table 3 half of the brackets of the generators (4.15) of the Lie algebra of infinitesimal gauge transformations \(\textrm{aut}_{{\mathcal {O}}(S^4)}({\mathcal {O}}(S^7))\), obtained by direct computation. The remaining brackets are obtained using the \(*\)-structure:
with \(K_j^*=K_j\) and \(W^*_\textsf {r} = W_{-\textsf {r} }\) (see (4.16)) and \((bX)^*= b^* X^*\) for each \(b \in {\mathcal {O}}(S^4)\) and X a derivation. For example, one computes
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Aschieri, P., Landi, G. & Pagani, C. Braided Hopf Algebras and Gauge Transformations II: \(*\)-Structures and Examples. Math Phys Anal Geom 26, 13 (2023). https://doi.org/10.1007/s11040-023-09454-9
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DOI: https://doi.org/10.1007/s11040-023-09454-9