Abstract
Let H be a quasi-Hopf algebra, \({}_H^H{{{\mathcal {M}}}}_H^H\) the category of two-sided two-cosided Hopf modules over H and \({}_H^H{\mathcal YD}\) the category of left Yetter–Drinfeld modules over H. We show that \({}_H^H{{{\mathcal {M}}}}_H^H\) admits a braided monoidal structure for which the strong monoidal equivalence \({}_H^H{\mathcal M}_H^H\cong {}_H^H{\mathcal YD}\) established by the structure theorem for quasi-Hopf bimodules becomes braided monoidal. Using this braided monoidal equivalence, we prove that Hopf algebras within \({}_H^H{{{\mathcal {M}}}}_H^H\) can be characterized as quasi-Hopf algebras with a projection or as biproduct quasi-Hopf algebras in the sense of Bulacu and Nauwelaerts (J Pure Appl Algebra 174:1–42, 2002) . A particular class of such (braided, quasi-) Hopf algebras is obtained from a tensor product Hopf algebra type construction. Our arguments rely on general categorical facts.
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1 Introduction
The so-called quantum shuffle Hopf algebras are cotensor Hopf algebras of a Hopf bimodule M over a Hopf algebra H. Their importance resides on the fact that all quantized enveloping algebras associated with finite-dimensional simple Lie algebras or with affine Kac–Moody Lie algebras are of this type; see [25]. As the cotensor defines a monoidal structure on the category of Hopf H-bimodules isomorphic to the one determined by the tensor product over H, it follows that quantum shuffle algebras can be as well introduced as tensor Hopf algebras within the braided category of Hopf H-bimodules (also known under the name of two-sided two-cosided Hopf modules).
The structure of a Hopf algebra H with a projection \(\pi : B\rightarrow H\) is due to Radford [24]. Up to an isomorphism, B is a biproduct Hopf algebra \(A\times H\) between a left H-module algebra and left H-comodule coalgebra A and H, satisfying appropriate compatibility relations. Majid [19] observed that all these conditions are equivalent to the fact that A is a Hopf algebra within \({}_H^H{\mathcal YD}\), the braided monoidal category of left Yetter–Drinfeld modules over H. A second characterization of Hopf algebras with a projection is due to Bespalov and Drabant [1], where Hopf algebras with a projection are identified with Hopf algebras within \({}_H^H{{{\mathcal {M}}}}_H^H\), the braided monoidal category of two-sided two-cosided Hopf modules over H introduced by Woronowicz in [28]. The connection with the Hopf algebras in \({}_H^H{\mathcal YD}\) becomes clear now, since \({}_H^H{{{\mathcal {M}}}}_H^H\) and \({}_H^H{\mathcal YD}\) are braided monoidally equivalent. The latest result was proved by Schauenburg in [26]; see also [25]. We should mention that in all this theory a key role is played by the structure theorem for two-sided two-cosided Hopf modules. Furthermore, by moving backwards, these equivalences associate to any vector space (viewed in a canonical way as Yetter–Drinfeld module) a two-sided two-cosided Hopf module, and then a quantum shuffle Hopf algebra.
The purpose of this note is to construct quasi-quantum shuffle groups, i.e., tensor Hopf algebras within categories of quasi-Hopf bimodules. This is possible because many of the above-mentioned results have already been generalized to the quasi-Hopf case. For instance, a structure theorem for quasi-Hopf (bi)comodule algebras was given in [9, 23]. It is not possible to prove a similar structure theorem for quasi-Hopf module coalgebras, since H is not, in general, a module coalgebra over itself. Instead, it is more natural to try describing the bimodule coalgebras C over a quasi-Hopf algebra H, as H is a bimodule coalgebra over itself in a canonical way. We did this in [2, Theorem 5.6] where we proved that, up to an isomorphism, C is a smash product coalgebra between a coalgebra in \({}_H^H{\mathcal YD}\) and H. Note that all the mentioned structure theorems actually characterize the (co)algebras within some monoidal categories of quasi-Hopf (bi)modules. Furthermore, the involved structures are a smash product algebra and a smash product coalgebra, as they were defined in [2, 7]; they are required to define a Hopf like object, and this leads naturally to the biproduct quasi-Hopf algebra construction from [5], as well to the structure of a quasi-Hopf algebra with a projection and its relation to the Hopf algebras in \({}_H^H{\mathcal YD}\). Although a quasi-Hopf algebra cannot be regarded as a braided Hopf algebra, we were able to adapt the categorical techniques used in [1] to the setting provided by quasi-Hopf algebras. Otherwise stated, we could produce structure theorems for the bialgebras and Hopf algebras in \({}_H^H{{{\mathcal {M}}}}_H^H\), similar to the ones in [1]. The choice of the category \({}_H^H{\mathcal M}_H^H\) rather than \({}_H{{{\mathcal {M}}}}_H^H\) is imposed by the fact that the former is braided, while the latter is not, and so we can consider bialgebras and Hopf algebras only within \({}_H^H{\mathcal M}_H^H\). Finally, \({}_H^H{{{\mathcal {M}}}}_H^H\) with \(\otimes _H\) is strict monoidal and although is isomorphic to the monoidal structure given by the cotensor product, the latter is not strict; this led us to work with tensor braided Hopf algebras instead of cotensor ones.
The paper is organized as follows. In Sect. 2, we briefly recall the definition of a quasi-Hopf algebra, the language of braided monoidal categories and braided monoidally equivalences, and the monoidally equivalence between \({}_H^H{{{\mathcal {M}}}}_H^H\) and \({}_H^H{\mathcal YD}\). Using a general categorical result, in Sect. 3 we uncover in a canonical way a braiding on \({}_H^H{{{\mathcal {M}}}}_H^H\) for which the strong monoidally equivalence \({}_H^H{{{\mathcal {M}}}}_H^H\cong {}_H^H{\mathcal YD}\) from [2, 27] becomes a braided monoidal equivalence. In Sect. 4, we characterize the Hopf algebras B in \({}_H^H{{{\mathcal {M}}}}_H^H\) as quasi-Hopf algebras with a projection and show that, up to an isomorphism, such a B is nothing but a biproduct quasi-Hopf algebra in the sense of [5]. We should stress that our techniques allow us to show in a more elegant and less computational way that the biproduct is indeed a quasi-Hopf algebra. In addition, we get almost for free the converse of the construction in [5]: if the smash product algebra \(A\# H\) of an algebra A in \({}_H^H{\mathcal YD}\) and H, and the smash product coalgebra 
between the coalgebra A in \({}_H^H{\mathcal YD}\) and H afford a quasi-Hopf algebra structure on \(A\otimes H\), then A is a Hopf algebra in \({}_H^H{\mathcal YD}\). Inspired by the work of Nichols [22], in Sect. 5 we associate to any object \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\) a braided Hopf algebra \(T_H(M)\) within \({}_H^H{{{\mathcal {M}}}}_H^H\), the so-called tensor Hopf algebra of M over H. Furthermore, we describe the quasi-Hopf algebra structure of \(T_H(M)\) and show that it is isomorphic to the biproduct quasi-Hopf algebra of T(V) and H, where V is a certain set of coinvariants of M and T(V) is the tensor Hopf algebra of V built within the braided monoidal category of left H-Yetter–Drinfeld modules. Actually, the construction of T(V) within \({}_H^H{\mathcal YD}\) makes sense for any \(V\in {}_H^H{\mathcal YD}\). This fact is fully exploited in Sect. 6 where a concrete class of quasi-Hopf algebras with a projection is constructed out of a vector space, a cyclic group of order n and a primitive root of unity of degree \(n^2\) in k, \(n\ge 2\).
2 Preliminaries
2.1 Quasi-bialgebras and quasi-Hopf algebras
We work over a field k. All algebras, linear spaces, etc., will be over k; unadorned \(\otimes \) means \(\otimes _k\). Following Drinfeld [10], a quasi-bialgebra is a quadruple \((H, \Delta , \varepsilon , \Phi )\) where H is an associative algebra with unit, \(\Phi \) is an invertible element in \(H\otimes H\otimes H\), and \(\Delta :\ H\rightarrow H\otimes H\) and \(\varepsilon :\ H\rightarrow k\) are algebra homomorphisms satisfying the identities
for all \(h\in H\), where \(\Phi \) is a 3-cocycle, in the sense that
The map \(\Delta \) is called the coproduct or the comultiplication, \(\varepsilon \) is the counit, and \(\Phi \) is the reassociator. As for Hopf algebras, we denote \(\Delta (h)=h_1\otimes h_2\), but since \(\Delta \) is only quasi-coassociative we adopt the further convention (summation understood):
for all \(h\in H\). We will denote the tensor components of \(\Phi \) by capital letters, and the ones of \(\Phi ^{-1}\) by lower case letters, namely
H is called a quasi-Hopf algebra if, moreover, there exists an anti-morphism S of the algebra H and elements \(\alpha , \beta \in H\) such that, for all \(h\in H\), we have:
Our definition of a quasi-Hopf algebra is different from the one given by Drinfeld [10] in the sense that we do not require the antipode to be bijective. In the case where H is finite-dimensional or quasi-triangular, bijectivity of the antipode follows from the other axioms, see [3, 6], so the two definitions are equivalent. Anyway, the bijectivity of the antipode S will be implicitly understood in the case when \(S^{-1}\), the inverse of S, appears is formulas or computations.
It is well known that the antipode of a Hopf algebra is an anti-morphism of coalgebras. For a quasi-Hopf algebra H, there exists an invertible element \(f=f^1\otimes f^2\in H\otimes H\), called the Drinfeld twist or the gauge transformation, such that \(\varepsilon (f^1)f^2=\varepsilon (f^2)f^1=1\) and
for all \(h\in H\), where \(\Delta ^{\mathrm{cop}}(h)=h_2\otimes h_1\). f can be described explicitly: first we define \(\gamma , \delta \in H\otimes H\) by
With this notation f and \(f^{-1}\) are given by the formulas
Moreover, f satisfies the following relations:
We will need the appropriate generalization of the formula \(h_1\otimes h_2S(h_3)=h\otimes 1\) in classical Hopf algebra theory. Following [13, 14], we define
For all \(h\in H\), we then have
and the following relations hold:
2.2 Braided monoidal equivalences
For the definition of a (co)algebra (resp. bialgebra, Hopf algebra) in a monoidal (resp. braided monoidal) category \({{\mathcal {C}}}\) and related topics, we refer to [11, 16, 21]. Usually, for a monoidal category \({{\mathcal {C}}}\), we denote by \(\otimes \) the tensor product, by \(\underline{1}\) the unit object, and by a, l, r the associativity constraint and the left and right unit constraints, respectively.
A strong monoidal functor between two monoidal categories \({{\mathcal {C}}}\), \({{\mathcal {C}}}'\) is a triple \((F, \varphi _2, \varphi _0)\), where \(F:\ {{\mathcal {C}}}\rightarrow {{\mathcal {C}}}'\) is a functor, \(\varphi _0:\ \underline{1}\rightarrow F(\underline{1}')\) is an isomorphism, and \(\varphi _{2, U, V}:\ F(U)\otimes ' F(V)\rightarrow F(U\otimes V)\) is a family of natural isomorphisms in \({{\mathcal {C}}}'\). \(\varphi _0\) and \(\varphi _2\) have to satisfy certain properties, see for example [16, XI.4].
When \(({{\mathcal {C}}}, c)\) and \(({{\mathcal {C}}}', c')\) are (pre)braided monoidal categories, a (pre)braided functor \(F: ({{\mathcal {C}}}, c)\rightarrow ({{\mathcal {C}}}', c')\) is a strong monoidal functor \((F, \varphi _2, \varphi _0):\ {{\mathcal {C}}}\rightarrow {{\mathcal {C}}}'\) compatible with the (pre)braidings c and \(c'\), in the sense that, for any objects \(X, Y\in {{\mathcal {C}}}\), the diagram
commutes.
Finally, for the definition of a natural tensor isomorphism \(\omega \) between two strong monoidal functors \((F, \varphi ^F_2, \varphi ^F_0), (G, \varphi ^G_2, \varphi ^G_0):{{\mathcal {C}}}\rightarrow {{\mathcal {C}}}'\) we refer to [16, Definition XI.4.1]. Note that, according to our terminology, in loc. cit. a tensor functor is nothing but a strong monoidal functor. This is why, for consistency, we will call \(\omega \) as above a natural strong monoidal isomorphism.
We say that F is a strong monoidal equivalence if there exists a strong monoidal functor \(G:{{\mathcal {C}}}'\rightarrow {{\mathcal {C}}}\) such that FG is naturally strongly monoidally isomorphic to \(\mathrm{Id}_{{{\mathcal {C}}}'}\) and GF is naturally strongly monoidally isomorphic to \(\mathrm{Id}_{{\mathcal {C}}}\). If a functor \(F: {{\mathcal {C}}}\rightarrow {{\mathcal {C}}}'\) defines a strong monoidal equivalence between \({{\mathcal {C}}}\) and \({{\mathcal {C}}}'\) we say that the categories \({{\mathcal {C}}}\) and \({{\mathcal {C}}}'\) are strongly monoidally equivalent.
If a functor \(F: {{\mathcal {C}}}\rightarrow {{\mathcal {C}}}'\) defines a strong monoidal equivalence between two (pre)braided categories \({{\mathcal {C}}}\) and \({{\mathcal {C}}}'\) we say that the categories \({{\mathcal {C}}}\) and \({{\mathcal {C}}}'\) are (pre)braided monoidally equivalent, provided that F is a (pre)braided functor, too.
2.3 A strong monoidal equivalence
Let H be a quasi-bialgebra. Then, the category of H-bimodules \({}_H{{{\mathcal {M}}}}_H\) is monoidal, since it can be identified with the category of left modules over the quasi-Hopf algebra \(H^{\mathrm{op}}\otimes H\), where \(H^{\mathrm{op}}\) is the opposite quasi-bialgebra associated to H. Explicitly, \({}_H{{{\mathcal {M}}}}_H\) is monoidal with the following structure. The associativity constraints \(a'_{M, N, P}: (M\otimes N)\otimes P\rightarrow M\otimes (N\otimes P)\) are given by
The unit object is k viewed as an H-bimodule via the counit \(\varepsilon \) of H, and the left and right unit constraints are given by the natural isomorphisms \(k\otimes M\cong M\cong M\otimes k\).
A (co)algebra in \({}_H{{{\mathcal {M}}}}_H\) is called an H-bimodule (co)algebra.
With its regular comultiplication and counit, H is a coalgebra in \({}_H{{{\mathcal {M}}}}_H\). Then, we can define \({}_H^H{{{\mathcal {M}}}}_H^H\) as being the category of H-bicomodules in \({}_H{{{\mathcal {M}}}}_H\). For the explicit definition of an object M in \({}_H^H{{{\mathcal {M}}}}_H^H\), we refer to [2, 8, 27]. Roughly speaking, we have a left H-coaction on M, denoted by \(\lambda _M: M\ni m\mapsto m_{\{-1\}}\otimes m_{\{0\}}\in H\otimes M\), and at the same time a right H-coaction on M, denoted by \(\rho _M: M\ni m\mapsto m_{(0)}\otimes m_{(1)}\in M\otimes H\), which are counital and coassociative up to conjugation by the reassociator \(\Phi \) of H and, moreover, compatible each other, and also with the H-bimodule structure of M, respectively.
\({}_H^H{{{\mathcal {M}}}}_H^H\) is monoidal in such a way that the forgetful functor \({{{\mathcal {U}}}}: {}_H^H{{{\mathcal {M}}}}_H^H\rightarrow ({}_H{{{\mathcal {M}}}}_H, \otimes _H, H)\) is strong monoidal. If \(M, N\in {}_H^H{{{\mathcal {M}}}}_H^H\) then the left and right coactions of H on \(M\otimes _HM\) are defined by those of M and N, and the multiplication of H.
It was proved by Schauenburg in [27] that \({}_H^H{\mathcal M}_H^H\) is monoidally equivalent to the left center of the monoidal category \({}_H{{{\mathcal {M}}}}\). The latter is denoted by \({}_H^H{\mathcal YD}\) and called the category of left Yetter–Drinfeld modules over H. Its objects were described for the first time by Majid in [20]. They are left H-modules M on which H coacts from the left such that \(\varepsilon (m_{[-1]})m_{[0]}=m\) and
for all \(m\in M\). Here, and in what follows, we denote by \(\lambda _M:\ M\rightarrow H\otimes M,~~\lambda _M(m)=m_{[-1]}\otimes m_{[0]}\) the left H-coaction on M. It is compatible with the left H-module structure on M, in the sense that, for all \(h\in H\) and \(m\in M\),
The monoidal structure on \({}_H^H{\mathcal YD}\) is such that the forgetful functor \({}_H^H{\mathcal YD}\rightarrow {}_H{{{\mathcal {M}}}}\) is strong monoidal. The coaction on the tensor product \(M\otimes N\) of two Yetter–Drinfeld modules M and N is given by
for all \(m\in M\) and \(n\in N\).
The strongly monoidally equivalence between \({}_H^H{{{\mathcal {M}}}}_H^H\) and \({}_H^H{\mathcal YD}\) is produced by the following functors, see [2].
Proposition 2.1
Consider the functors \({{{\mathcal {F}}}}: {}^H_H{\mathcal YD}\rightarrow {}_H^H{\mathcal M}_H^H\) and \({{{\mathcal {G}}}}: {}_H^H{{{\mathcal {M}}}}_H^H\rightarrow {}_H^H{\mathcal YD}\) defined as follows:
− For \(M\in {}^H_H{\mathcal YD}\), we have \({{{\mathcal {F}}}}(M)=M\otimes H\in {}_H^H{{{\mathcal {M}}}}_H^H\) with the structure given by
for all \(h, h', h{''}\in H\) and \(m\in M\). If \(f: M\rightarrow N\) is a morphisms in \({}^H_H{\mathcal YD}\) then \({{{\mathcal {F}}}}(f)=f\otimes \mathrm{Id}_H\).
− If \(M\in {}^H_H{{{\mathcal {M}}}}^H_H\) then
the set of alternative coinvariants of M, which belongs to \({}^H_H{\mathcal YD}\) via the structure defined by
for all \(h\in H\) and \(m\in M^{\overline{co(H)}}\), where \(f^{-1}=g^1\otimes g^2\) is the inverse of the Drinfeld’s twist f. On morphisms, we have that \({{{\mathcal {G}}}}(f)=f|_{M^{\overline{co(H)}}}\), a well-defined morphism in \({}_H^H{\mathcal YD}\), for any morphism \(f: M\rightarrow N\) in \({}^H_H{\mathcal M}^H_H\).
Then, \({{{\mathcal {F}}}}\) and \({{{\mathcal {G}}}}\) are inverse strong monoidal equivalence functors.
According to [2], the strong monoidal structure on \({{{\mathcal {F}}}}\) is given, for all \(M, N\in {}_H^H{\mathcal YD}\), \(m\in M\), \(h, h'\in H\) and \(n\in N\), by
and the morphism \(\varphi _0=\mathrm{Id}_H: H\rightarrow F(k)=k\otimes H\cong H\).
The strong monoidal structure on \({{{\mathcal {G}}}}\) is determined by
for all \(M, N\in {}_H^H{{{\mathcal {M}}}}_H^H\), \(m\in M^{\overline{co(H)}}\) and \(n\in N^{\overline{co(H)}}\), and \(\overline{\phi }_0: k\rightarrow {{{\mathcal {G}}}}(H)=k\beta \) defined by \(\overline{\phi }_0(\kappa )=\kappa \beta \), for all \(\kappa \in k\), respectively. Using arguments similar to the ones in the proof of [2, Corollary 3.2], a straightforward computation ensures us that
for all \(M, N\in {}_H^H{{{\mathcal {M}}}}_H^H\) and \(m\otimes _Hn\in (M\otimes _HN)^{\overline{\mathrm{co}(H)}}\), where \(\overline{E}_M: M\rightarrow M^{\overline{co(H)}}\) determined by \(\overline{E}_M(m)=m_{(0)}\cdot \beta S(m_{(1)})\), for all \(m\in M\), is the projection defined in [8].
Furthermore, for all \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\),
is an isomorphism in \({}_H^H{{{\mathcal {M}}}}_H^H\) with inverse \( \overline{\nu }^{-1}_M: M\ni m\mapsto \overline{E}(m_{(0)})\otimes m_{(1)}\in M^{\overline{co(H)}}\otimes H \). The family of all morphisms \(\overline{\nu }_M\) define a natural strong monoidal isomorphism \(\overline{\nu }\) between \({{{\mathcal {F}}}}{{{\mathcal {G}}}}\) and \(\mathrm{Id}_{{}^H_H{{{\mathcal {M}}}}_H^H}\). Likewise, for all \(M\in {}_H^H{\mathcal YD}\),
is an isomorphism in \({}_H^H{\mathcal YD}\) and defines a natural strong monoidal isomorphism \(\zeta \) between \({{{\mathcal {G}}}}{{{\mathcal {F}}}}\) and \(\mathrm{Id}_{{}^H_H{\mathcal YD}}\).
3 A braided monoidal equivalence
In Hopf algebra theory, it is well known that \({}_H^H{{{\mathcal {M}}}}_H^H\) is braided monoidally equivalent to \({}_H^H{\mathcal YD}\). A remarkable braiding on \({}_H^H{{{\mathcal {M}}}}_H^H\) was introduced by Woronowicz [28], and the fact that with respect to this braiding \({}_H^H{{{\mathcal {M}}}}_H^H\) and \({}_H^H{\mathcal YD}\) are braided monoidally equivalent was proved by Schauenburg in [26, Theorem 5.7].
The aim of this section is to generalize the two results above to the quasi-Hopf setting. To this end, we start with a lemma of independent interest.
The results below are stated without proofs in [11, Remark 2.4.10] and [15, Example 2.4]. For the sake of completeness and also for further use, we outline them in what follows.
Lemma 3.1
Let \(F: {{\mathcal {C}}}\rightarrow {{\mathcal {D}}}\) be a functor between two monoidal categories \(({{\mathcal {C}}}, \otimes , a, \underline{1}, l, r)\) and \(({{\mathcal {D}}}, \square , {\mathbf {a}}, \underline{I}, \lambda , \rho )\).
-
(i)
F defines a strong monoidal equivalence if and only if F is strong monoidal and an equivalence of categories.
-
(ii)
If F is as in (i) and \({{\mathcal {C}}}\) is, moreover, braided then there exists a unique braiding on \({{\mathcal {D}}}\) that turns F into a braided monoidal functor. Consequently, a functor defines a braided monoidal equivalence if and only if it is braided and an equivalence of categories.
Proof
If \(F: {{\mathcal {C}}}\rightarrow {{\mathcal {D}}}\) is an equivalence of categories then by the proof of [18, IV.4 Theorem 1] there exist a functor \(G: {{\mathcal {D}}}\rightarrow {{\mathcal {C}}}\) and natural isomorphisms \(\mu : \mathrm{Id}_{{\mathcal {D}}}\rightarrow FG\) and \(\nu : GF\rightarrow \mathrm{Id}_{{\mathcal {C}}}\) such that, for all \(X\in {{\mathcal {C}}}\),
-
(i)
The direct implication is immediate. For the converse, we only indicate the unique strong monoidal structure \((\varphi ^G_2, \varphi ^G_0)\) of G that turns \(\mu \) and \(\nu \) into monoidal transformations. To this end, we denote by \((\varphi ^F_2:=(\varphi ^F_{2, X, Y})_{X, Y\in {{\mathcal {C}}}}, \varphi ^F_0)\) the strong monoidal structure of F, and by \({\widetilde{\varphi }}^F_2\), \({\widetilde{\varphi }}^F_0\) the inverse morphisms of \(\varphi ^F_2\), respectively, \(\varphi ^F_0\). Then, \(\varphi ^G_0=G({\widetilde{\varphi }}^F_0)\nu ^{-1}_{\underline{1}}\) and
$$\begin{aligned} \varphi ^G_{2, U, V}= G((\mu ^{-1}_U\square \mu ^{-1}_V){\widetilde{\varphi }}^F_{2, G(U), G(V)})\nu ^{-1}_{G(U)\otimes G(V)},~\forall ~U,~ V\in {{\mathcal {D}}}. \end{aligned}$$(3.2) -
(ii)
Any braiding c for \({{\mathcal {C}}}\) defines a braiding d on \({{{\mathcal {D}}}}\) as follows. For any objects U, V of \({{{\mathcal {D}}}}\) take \(d_{U, V}\) to be the following composition:
(3.3)
Then, \(({{{\mathcal {D}}}}, d=(d_{U, V})_{U, V\in {{{\mathcal {D}}}}})\) is a braided category and \(F: ({{\mathcal {C}}}, c)\rightarrow ({{{\mathcal {D}}}}, d)\) becomes a braided monoidal functor. \(\square \)
We specialize the above result to the strong monoidal equivalence in Proposition 2.1. Note that, according to [20] the category \({}_H^H{\mathcal YD}\) is braided via the braiding given by \(c=(c_{M, N})_{M, N\in {}_H^H{\mathcal YD}}\), where, for all \(m\in M\) and \(n\in N\),
From now on, throughout the paper H is a quasi-Hopf algebra with bijective antipode. If \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\) then by \(E_M: M\rightarrow M\) we denote the projection on the space of coinvariants of M, defined by \(E_M(m)=X^1\cdot m_{(0)}\cdot \beta S(X^2m_{(1)})\alpha X^3=q^1\cdot \overline{E}_M(m)\cdot S(q^2)\), for all \(m\in M\). Here, \(M\ni m\mapsto \rho _M(m):=m_{(0)}\otimes m_{(1)}\in M\otimes H\) denotes the right coaction of H on M and \(q_R\) is the element in (2.14).
Record from [14] the following properties of \(E_M\):
for all \(m\in M\) and \(h\in H\). Also, recall from [8] that \(M^{\overline{\mathrm{co(H)}}}\) in invariant under the left adjoint action of H, that is 
, for all \(h\in H\) and \(m\in M\), where 
is defined by (2.28). Furthermore, the image of \(\overline{E}_M\) is \(M^{\overline{\mathrm{co(H)}}}\).
Another property of \(\overline{E}_M\) is the following.
Lemma 3.2
Let H be a quasi-Hopf algebra and M a two-sided two-cosided Hopf module over H. Then, for all \(m\in M\), we have that
Proof
For all \(m\in M\in {}_H^H{{{\mathcal {M}}}}_H^H\), we have
as needed. \(\square \)
One can provide now a braiding for \({}_H^H{{{\mathcal {M}}}}_H^H\).
Theorem 3.3
If H is a quasi-Hopf algebra then \({}_H^H{{{\mathcal {M}}}}_H^H\) is a braided monoidal category with the braiding defined by
for all \(M, N\in {}_H^H{{{\mathcal {M}}}}_H^H\).
Furthermore, if we consider \({}_H^H{{{\mathcal {M}}}}_H^H\) braided with the braiding d, then \({}_H^H{{{\mathcal {M}}}}_H^H\) is braided monoidally equivalent to \({}_H^H{\mathcal YD}\), where the braiding on \({}_H^H{\mathcal YD}\) is c as in (3.4).
Proof
Let be the inverse strong monoidal equivalence functors defined in Proposition 2.1. We have that \(\overline{\nu }\) defined by (2.33) is a natural strong monoidal isomorphism between \({{{\mathcal {F}}}}{{{\mathcal {G}}}}\) and \(\mathrm{Id}_{{}_H^H{{{\mathcal {M}}}}_H^H}\), while \(\zeta \) given by (2.34) is a natural strong monoidal isomorphism between \({{{\mathcal {G}}}}{{{\mathcal {F}}}}\) and \(\mathrm{Id}_{{}_H^H{\mathcal YD}}\), respectively.
We prove now that \(({{{\mathcal {F}}}}, \overline{\nu }, \zeta )\) obeys the condition in (3.1), i.e., that \(\overline{\nu }_{{\mathcal F}(M)}={{{\mathcal {F}}}}(\zeta _M): (M\otimes H)^{\overline{\mathrm{co(H)}}}\otimes H\rightarrow M\otimes H\), for all \(M\in {}_H^H{\mathcal YD}\).
To this end, by [8, Remark 2.4] we have \((M\otimes H)^{\overline{\mathrm{co(H)}}}=\{p^1\cdot m\otimes p^2\mid m\in M\}\), where \(p_R=p^1\otimes p^2\) is the element defined in (2.13). Thus, \(\zeta _M(p^1\cdot m\otimes p^2)=m\), for all \(m\in M\), and therefore \({\mathcal F}(\zeta _M)((p^1\cdot m\otimes p^2)\otimes h)=m\otimes h\), for all \(m\in M\) and \(h\in H\).
If \(q_R\) is the element in (2.14) we then compute that
for all \(m\in M\) and \(h\in H\). We conclude that \(\overline{\nu }_{{\mathcal F}(M)}={{{\mathcal {F}}}}(\zeta _M)\), as stated.
It follows by Lemma 3.1 that the braiding c for \({}_H^H{\mathcal YD}\) transports along \({{{\mathcal {F}}}}\) to a braiding d on \({}_H^H{{{\mathcal {M}}}}_H^H\) such that \({{{\mathcal {F}}}}\) becomes a braided monoidal equivalence. It only remains to show that d is as in (3.10). Using (3.3), we see that
for all \(m\in M\) and \(n\in N\), as desired. \(\square \)
4 Hopf algebras within \({}_H^H{{{\mathcal {M}}}}_H^H\)
The aim of this section is to characterize the bialgebras and the Hopf algebras in \({}_H^H{{{\mathcal {M}}}}_H^H\). Inspired by some categorical results of Bespalov and Drabant [1], we show that giving a Hopf algebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) is equivalent to giving a quasi-Hopf algebra projection for H as in [5]. Consequently, we obtain almost for free that quasi-Hopf algebra projections are characterized by the biproduct quasi-Hopf algebras constructed in [5], and therefore by Hopf algebras in \({}_H^H{\mathcal YD}\), too.
A quasi-bialgebra map between two quasi-bialgebras H and A is an algebra map \(i : H\rightarrow A\) which intertwines the quasi-coalgebras structures, respects the counits and satisfies \((i \otimes i \otimes i)(\Phi _H)=\Phi _A\). If H, A are quasi-Hopf algebras then i is a quasi-Hopf algebra map if, in addition, \(i(\alpha _H)=\alpha _A\), \(i(\beta _H)=\beta _A\) and \(S_A\circ i=i\circ S_H\).
For a quasi-Hopf algebra H denote by \(H-\underline{\mathrm{qBialgProj}}\) (resp. \(H-\underline{\mathrm{qHopfProj}}\)) the category whose objects are triples \((A, i, \pi )\) consisting of a quasi-bialgebra (resp. quasi-Hopf algebra) A and two quasi-bialgebra (resp. quasi-Hopf algebra) morphisms 
such that \(\pi i=\mathrm{Id}_H\). A morphism between \((A, i, \pi )\) and \((A', i', \pi ')\) in \(H-\underline{\mathrm{qBialgProj}}\) (resp. \(H-\underline{\mathrm{qHopfProj}}\)) is a quasi-bialgebra (resp. quasi-Hopf algebra) morphism \(\tau : A\rightarrow A'\) such that \(\tau i=i'\) and \(\pi '\tau =\pi \). In what follows, the objects of \(H-\underline{\mathrm{qBialgProj}}\) (resp. \(H-\underline{\mathrm{qHopfProj}}\)) will be called quasi-bialgebra (resp. quasi-Hopf algebra) projections for H.
We also denote by \(\mathrm{Bialg}({}_H^H{{{\mathcal {M}}}}_H^H)\) (resp. \(\mathrm{Hopf}({}_H^H{{{\mathcal {M}}}}_H^H)\)) the category of bialgebras (resp. Hopf algebras) and bialgebra morphisms within \({}_H^H{{{\mathcal {M}}}}_H^H\).
As expected, we next prove that the categories \(\mathrm{Bialg}({}_H^H{{{\mathcal {M}}}}_H^H)\) and \(H-\underline{\mathrm{qBialgProj}}\) (resp. \(\mathrm{Hopf}({}_H^H{{{\mathcal {M}}}}_H^H)\) and \(H-\underline{\mathrm{qHopfProj}}\)) are isomorphic. We first need some lemmas.
Lemma 4.1
Take \(M, N\in {}_H^H{{{\mathcal {M}}}}_H^H\), and the elements \(m, m'\in M\) and \(n, n'\in N\). Then,
Proof
From [14], we have that \(\nu ^{-1}_M: M\ni m\mapsto E_M(m_{(0)})\otimes m_{(1)}\in M^{\mathrm{co(H)}}\otimes H\) is an isomorphism in \({}_H^H{{{\mathcal {M}}}}_H^H\), for all \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\). Here, \(M^{\mathrm{co(H)}}\) is the image of \(E_M\), a left H-module via the structure given by \(h\lnot E_M(m)=E_M(h\cdot m)\), for all \(h\in H\) and \(m\in M\). For a k-space U and \(V\in {}_H{{{\mathcal {M}}}}\) denote by \(\Upsilon _{U, V}: (U\otimes H)\otimes _HV\rightarrow U\otimes V\) the canonical isomorphism. We then have that \(m\otimes _Hn=m'\otimes _Hn'\) if and only if
Thus, if \(m\otimes _Hn=m'\otimes _Hn'\) then
The converse follows easily from (3.7), and we are done. \(\square \)
Now we can construct the functor that gives the desired categorical isomorphism. For the definition of an H-bicomodule algebra \(({\mathcal A}, \lambda , \rho , \Phi _\lambda , \Phi _r, \Phi _{\lambda , r})\), we refer to [13].
Proposition 4.2
Let H be a quasi-Hopf algebra. Then, there is a functor
On objects, \({{{\mathcal {V}}}}\) sends a bialgebra \((B, \underline{m}_B, i: H\rightarrow B, \underline{\Delta }_B, \pi : B\rightarrow H)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\) to the triple \((B, i, \pi )\), where B is considered as a quasi-bialgebra via \(m_B:=\underline{m}_Bq_{B, B}\), \(q_{B, B}: B\otimes B\rightarrow B\otimes _HB\) being the canonical surjection, \(1_B=i(1_H)\),
and \(\Phi _B=(i\otimes i\otimes i)(\Phi )\). \({{{\mathcal {V}}}}\) acts as identity on morphisms.
Proof
We must check that \((B, m_B, 1_B, \Delta _B, \varepsilon _B, \Phi _B)\) is indeed a quasi-bialgebra and, moreover, that \(i, \pi \) become quasi-bialgebra morphisms.
By [2, Lemma 4.9], \((B, \underline{m}_B, i)\) is an algebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) if and only if \((B, m_B, 1_B)\) is a k-algebra and at the same time an H-bicomodule algebra via the original left and right H-coactions and reassociators \(\Phi _\lambda =X^1\otimes X^2\otimes i(X^3)\), \(\Phi _\rho =i(X^1)\otimes X^2\otimes X^3\) and \(\Phi _{\lambda , \rho }=X^1\otimes i(X^2)\otimes X^3\), such that, for all \(h\in H\),
Otherwise stated, i is an H-bicomodule algebra morphism. Furthermore, the H-bimodule structure on B is nothing but the one induced by the restriction of scalars functor defined by i.
Likewise, by [2, Theorem 5.3], we have that \((B, \Delta _B, \varepsilon _B=\varepsilon \pi )\) is a coalgebra in \({}_H\overline{{{\mathcal {M}}}}_H:=({}_H{\mathcal M}_H, \otimes , k, a', l', r')\), i.e., an H-bimodule coalgebra, and \(\pi : B\rightarrow H\) is a coalgebra morphism in \({}_H\overline{{{\mathcal {M}}}}_H\). If we denote \(\Delta _B(b)=b_1\otimes b_2\) we then have
for all \(b\in B\), \(\varepsilon \pi =\varepsilon _B\) and \(\Delta (\pi (b))=\pi (b_1)\otimes \pi (b_2)\), for all \(b\in B\).
The left and right H-coactions on B can be recovered from \(\Delta _B\) and \(\pi \) as
Since i is the unit and \(\pi \) is the counit of the bialgebra B within \({}_H^H{{{\mathcal {M}}}}_H^H\) it follows that \(\pi i=\mathrm{Id}_H\), and therefore, \(\pi \) is surjective. Furthermore, \(\pi : B\rightarrow H\) is an algebra morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\), so \(\pi \) is a k-algebra morphism as well. As we have seen, \(\pi \) intertwines the comultiplications \(\Delta _B\) and \(\Delta \) of B and H, too. If we define \(\Phi _B:=(i\otimes i\otimes i)(\Phi )\), it is clear that \((\pi \otimes \pi \otimes \pi )(\Phi _B)=\Phi \).
Combining (4.3) and (4.5) we get
and therefore \(\pi (i(h)_1)\otimes i(h)_2=\pi (i(h_1))\otimes i(h_2)\), for all \(h\in H\). As \(\pi \) is surjective, we obtain that i intertwines the comultiplications \(\Delta \) and \(\Delta _B\) of H and B, and so \(\Delta _B(1_B)=\Delta _B(i(1_H))=i(1_H)\otimes i(1_H)=1_B\otimes 1_B\). It is also an algebra morphism such that \((i\otimes i\otimes i)(\Phi )=\Phi _B\) and \(\varepsilon _Bi=\varepsilon \).
The most difficult part is to show that \(\Delta _B\) is multiplicative, that is
for all \(b, b'\in B\). Toward this end, observe first that by (3.10) and (4.1) we have that \(\underline{\Delta }_B\) is multiplicative in \({}_H^H{{{\mathcal {M}}}}_H^H\) if and only if
for all \(b, b'\in B\), where, for simplicity, from now on we denote \(E_B\) by E. This allows us to compute that
for all \(b, b'\in B\), as needed. The remaining details are left to the reader. \(\square \)
We can construct an inverse for \({{{\mathcal {V}}}}\) as follows.
Proposition 4.3
Let H be a quasi-Hopf algebra and \((B, i, \pi )\) a quasi-bialgebra projection for it. Then, B is a bialgebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) with the structure given, for all \(h, h'\in H\) and \(b, b'\in B\), by
In this way we have a well-defined functor \({{{\mathcal {T}}}}: H-\underline{\mathrm{qBialgProj}}\rightarrow \mathrm{Bialg}({}_H^H{{{\mathcal {M}}}}_H^H)\). \({{{\mathcal {T}}}}\) acts as identity on morphisms.
Proof
It is easy to see that B is an object in \({}_H^H{{{\mathcal {M}}}}_H^H\) with the structure as in (4.8) and (4.9). Since \((b\cdot h)b'=b(h\cdot b')\), for all \(b, b'\in B\) and \(h\in H\), it follows that \(\underline{m}_B: B\otimes _HB\rightarrow B\) given by \(\underline{m}_B(b\otimes _Hb')=bb'\), for all \(b, b'\in B\), is well defined. By [2, Lemma 4.9] we deduce that \((B, \underline{m}_B, i)\) is an algebra in \({}_H^H{{{\mathcal {M}}}}_H^H\), since
for all \(h\in H\), i.e., i is an H-bicomodule morphism, where the H-bicomodule structure of B is \((B, \lambda , \rho , \Phi _\lambda =X^1\otimes X^2\otimes i(X^3), \Phi _\rho =i(X^1)\otimes X^2\otimes X^3, \Phi _{\lambda , \rho }=X^1\otimes i(X^2)\otimes X^3)\). We should point out that all these facts follow because \(i: H\rightarrow B\) is a quasi-bialgebra morphism.
[2, Theorem 5.3] guarantees that B is a coalgebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) with the structure in (4.11). Thus, it only remains to show that \(\underline{\Delta }_B\) is an algebra morphism, where the algebra structure on \(B\otimes _HB\) is the tensor product algebra one, modulo the braiding in (3.10). We compute
and this shows that, up to the identification given by the unit constraints of the monoidal category \(({}_H{{{\mathcal {M}}}}_H, \otimes _H, H)\), \(\underline{\Delta }_Bi=i\otimes _Hi\).
Due to (4.7) and (4.11), that \(\underline{\Delta }_B\) is multiplicative is equivalent to
for all \(b, b'\in B\). Since, for all \(b, b'\in B\), we have that
it follows that \(\underline{\Delta }_B\) is multiplicative if and only if
The latter equivalence is immediate since \(\Delta _B\) is multiplicative. This ends the proof. \(\square \)
At this point, we can prove one of the main results of this paper.
Theorem 4.4
Let H be a quasi-Hopf algebra. Then, the functors
define a category isomorphism.
They also produce a category isomorphism between \(\mathrm{Hopf}({}_H^H{{{\mathcal {M}}}}_H^H)\) and \(H-\underline{\mathrm{qHopfProj}}\).
Proof
One can check directly that \({{{\mathcal {V}}}}\) and \({{{\mathcal {T}}}}\) are inverse to each other; see also [2, Lemma 4.9 & Corollary 5.4].
Take \((B, i, \pi )\in H-\underline{\mathrm{qHopfProj}}\), and denote by \(S_B\) the antipode of B. We claim that \({{{\mathcal {T}}}}((B, i, \pi ))=B\) is a Hopf algebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) with antipode determined by
Indeed, a technical but straightforward computation ensures that \(\underline{S}\) is a morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\). Then, one can check that
and this fact allows us to compute that
for all \(b\in B\), as required. Similarly, one can easily see that
and from here we get that
for all \(b\in B\). Hence, our claim is proved.
In a similar manner, we can prove that if \(\underline{S}\) is antipode for the bialgebra B in \({}_H^H{{{\mathcal {M}}}}_H^H\) then the quasi-bialgebra \({{{\mathcal {V}}}}(B)\) is actually a quasi-Hopf algebra with antipode determined by
and distinguished elements \(\alpha _B=i(\alpha )\) and \(\beta _B=i(\beta )\). We leave the verification of the remaining details to the reader. \(\square \)
We end this paper by presenting a second characterization for the bialgebras (resp. Hopf algebras) in \({}_H^H{{{\mathcal {M}}}}_H^H\).
By Theorem 3.3, the categories \({}_H^H{\mathcal M}_H^H\) and \({}_H^H{\mathcal YD}\) are braided monoidally equivalent. Therefore, bialgebras (resp. Hopf algebras) in \({}_H^H{{{\mathcal {M}}}}_H^H\) are in a one to one correspondence to bialgebra (resp. Hopf algebra) structures in \({}_H^H{\mathcal YD}\). More precisely, if B is a bialgebra (resp. Hopf algebra) in \({}_H^H{{{\mathcal {M}}}}_H^H\) then \(A:=B^{\overline{\mathrm{co(H)}}}\) is a bialgebra (resp. Hopf algebra) in \({}_H^H{\mathcal YD}\). The inverse of this correspondence associates to any bialgebra (resp. Hopf algebra) A in \({}_H^H{\mathcal YD}\) the bialgebra (resp. Hopf algebra) \({{{\mathcal {F}}}}(A)=A\otimes H\) in \({}_H^H{\mathcal M}_H^H\). Thus, B and \(A\otimes H\) are isomorphic as braided bialgebras (resp. Hopf algebras) in \({}_H^H{{{\mathcal {M}}}}_H^H\). Consequently, \({\mathcal V}(B)\) and \({{{\mathcal {V}}}}(A\otimes H)\) are isomorphic as objects in \(H-\underline{\mathrm{qBialgProj}}\) (resp. H-qHopfProj).
Firstly, \(A\otimes H\) is an object in \({}_H^H{{{\mathcal {M}}}}_H^H\) with the structure as in (2.25)–(2.27). By [2, Theorem 4.11], as an algebra \({{{\mathcal {V}}}}(A\otimes H)=A\# H\), the smash product algebra of A and H from [7]. The multiplication of \(A\# H\) is given by
for all \(a, a'\in A\) and \(h, h'\in H\), and its unit is \(1_A\otimes 1_H\). This contributes to the structure of \({{{\mathcal {V}}}}(A\otimes H)\) with \(j: H\ni h\mapsto 1_A\otimes h\in A\# H\), so far an H-bicomodule algebra morphism, provided that A is an algebra in \({}_H^H{\mathcal YD}\) (see [2, Proposition 4.10] for more details).
Secondly, by [2, Theorem 5.6], as a coalgebra 
, the smash product coalgebra of A and H. More exactly, the comultiplication is defined by
and the counit is \(\varepsilon (a\otimes h)=\varepsilon _A(a)\varepsilon (h)\), for all \(a\in A\) and \(h\in H\). This contributes to the structure of \({{{\mathcal {V}}}}(A\otimes H)\) with 
, so far an H-bimodule coalgebra morphism, provided that A is a coalgebra in \({}_H^H{\mathcal YD}\). As before, \(a\mapsto a_{[-1]}\otimes a_{[0]}\) is the left coaction of H on A, \(\Delta _A(a)=a_1\otimes a_2\) is the comultiplication of A in \({}_H^H{\mathcal YD}\) and \(\varepsilon _A\) is its counit.
Summing up, \({{{\mathcal {V}}}}({{{\mathcal {F}}}}(A))=(A\times H, j, p)\), the biproduct quasi-bialgebra (resp. quasi-Hopf algebra) constructed in [5], provided that A is a bialgebra (resp. Hopf algebra) in \({}_H^H{\mathcal YD}\). Note that, in [5] we gave the coalgebra structure of \(A\times H\) by adapting the one in the Hopf algebra case, and that by hard computations we showed that \(A\times H\) is a quasi-bialgebra (resp. quasi-Hopf algebra), provided that A is a bialgebra (resp. Hopf algebra) in \({}_H^H{\mathcal YD}\). Now we have a more conceptual and less computational proof, and at the same time a converse for the cited result in [5].
Corollary 4.5
Let H be a quasi-Hopf algebra, and B an object of \({}_H^H{\mathcal YD}\) which is at the same time an algebra and a coalgebra in \({}_H^H{\mathcal YD}\). Then, the smash product algebra and the smash product coalgebra afford a quasi-bialgebra (resp. quasi-Hopf algebra) structure on \(A\otimes H\) if and only if A is a bialgebra (resp. Hopf algebra) in \({}_H^H{\mathcal YD}\). If this is the case, then \(A\times H\) is a bialgebra (resp. Hopf algebra) in \({}_H^H{\mathcal M}_H^H\).
Proof
Everything follows from the above comments, and the fact that \({\mathcal F}: {}_H^H{\mathcal YD}\rightarrow {}_H^H{{{\mathcal {M}}}}_H^H\) is a braided monoidal equivalence, and that \({{{\mathcal {T}}}}\), \({{{\mathcal {V}}}}\) are inverse isomorphism functors.
Remark that, the antipode s of the quasi-Hopf algebra \(A\times H\) can be obtained from the antipode \(S_A\) of A in \({}_H^H{\mathcal YD}\) and the antipode S of H as follows. The antipode \(\underline{S}\) of \({{{\mathcal {F}}}}(A)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\) is \({{{\mathcal {F}}}}(S_A)=S_A\otimes \mathrm{Id}_H\), and so we have that
for all \(a\in A\) and \(h\in H\). Clearly, the distinguished elements that together with s define the antipode for \(A\times H\) are \(j(\alpha )=1_A\times \alpha \) and \(j(\beta )=1_A\times \beta \). In this way, we gave an alternative proof for [5, Lemma 3.3]. In the computation above, we wrote \(a\times h\) in place of \(a\otimes h\) in order to distinguish the quasi-bialgebra structure on \(A\otimes H\) given by the biproduct construction. \(\square \)
Collecting the results proved in this section, we get the following.
Theorem 4.6
Let H be a quasi-Hopf algebra. Then, there is a one-to-one correspondence between:
-
bialgebras (resp. Hopf algebras) in \({}_H^H{{{\mathcal {M}}}}_H^H\);
-
quasi-bialgebra (resp. quasi-Hopf algebra) projections for H;
-
bialgebras (resp. Hopf algebras) in \({}_H^H{\mathcal YD}\);
-
biproduct quasi-bialgebra (resp. quasi-Hopf algebra) structures for H.
We end this paper by applying Theorem 4.6 to a class of braided Hopf algebras in \({}_H^H{{{\mathcal {M}}}}_H^H\) obtained from a tensor Hopf algebra type construction.
5 Tensor Hopf algebras within \({}_H^H{{{\mathcal {M}}}}_H^H\)
Let H be a quasi-Hopf algebra and M an object of \({}_H^H{\mathcal M}_H^H\). We show that the tensor algebra \(T_H(M)\) associated to M within \(({}_H^H{{{\mathcal {M}}}}_H^H, \otimes _H, H)\) admits a braided Hopf algebra structure in \(({}_H^H{{{\mathcal {M}}}}_H^H, \otimes _H, H)\) or, equivalently, a quasi-Hopf algebra structure with a projection.
Recall that the tensor algebra \(T_H(M)\) of M within \({}_H^H{\mathcal M}_H^H\) is \(T_H(M)=H\oplus \bigoplus \limits _{n\ge 1}M^{\otimes _Hn}\), where \(M^{\otimes _H1}:=M\) and \(M^{\otimes _Hn}:=M^{\otimes _Hn-1}\otimes _HM\), for all \(n\ge 2\). For \(l<n\), we denote by \(m^{\otimes _Hl+1,n}\) the element \(m^{l+1}\otimes _H\cdots \otimes _Hm^n\in M^{\otimes _Hn-l}\); when \(l=0\) and \(n\ge 1\) we will write \(m^{\otimes _Hn}\) instead of \(m^{\otimes _H1,n}\).
The product \(*\) on \(T_H(M)\) is given by concatenation over H, i.e.,
for all \(h, h'\in H\), \(l\ge 1\), \(n\ge 2\) and \(m^1, \ldots , m^n\in M\). The unit of \(T_H(M)\) is given by the unit 1 of H. As the monoidal category \(({}_H^H{{{\mathcal {M}}}}_H^H, \otimes _H, H)\) is strict, in the writing of an element of \(M^{\otimes _Hn}\) we do not have to pay attention to parenthesis.
Using the monoidal structure on \({}_H^H{{{\mathcal {M}}}}_H^H\) given by \(\otimes _H\), we find that \(T_H(M)\) is an object in \({}_H^H{{{\mathcal {M}}}}_H^H\) via the structure induced by those of H and M, as follows: the H-bimodule structure of \(T_H(M)\) is given by the above product \(*\), while the H-bicomodule structure is defined, for all \(h\in H\), \(l\ge 1\) and \(m^1, \ldots , m^l\in M\), by
5.1 A braided Hopf algebra structure on \(T_H(M)\)
Denote by \(i: H\rightarrow T_H(M)\) and \(j: M\rightarrow T_H(M)\) the canonical embedding maps. It can be easily checked that i is an H-bicomodule algebra map, provided that \(T_H(M)\) is considered as an H-bicomodule algebra via \((*, 1, \lambda , \rho )\) as above and reassociators \(\Phi _\lambda =\Phi _\rho =\Phi _{\lambda , \rho }=X^1\underline{\otimes }X^2\underline{\otimes }X^3\), where, in general, by \(\underline{\otimes }\) we denote the tensor product between \(T_H(M)\) and itself within the category of k-vector spaces. Otherwise stated, \((T_H(M), *, i)\) is an algebra in \(({}_H^H{{{\mathcal {M}}}}_H^H, \otimes _H, H)\) and \(i: H\rightarrow T_H(M)\) is an algebra morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\). Finally, it is immediate that j is a morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\).
Similar to the Hopf case [22], the tensor algebra \(T_H(M)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\) is uniquely determined by the following universal property.
Proposition 5.1
Let H be a quasi-Hopf algebra and \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\). Then for any algebra morphism \(u: A\rightarrow A'\) in \({}_H^H{{{\mathcal {M}}}}_H^H\) and any morphism \(\zeta : M\rightarrow A'\) in \({}_H^H{{{\mathcal {M}}}}_H^H\), there exists a unique morphism \(\overline{\zeta }: T_H(M)\rightarrow A'\) of algebras in \({}_H^H{{{\mathcal {M}}}}_H^H\) such that \(\overline{\zeta }j=\zeta \).
Proof
It is similar to the one given for [22, Proposition 1.4.1]. \(\square \)
The above universal property allows to define a Hopf algebra structure on \(T_H(M)\) as follows. To avoid any possible confusion, by \(\overline{\otimes }\) we denote the tensor product between \(T_H(M)\) and itself within the strict braided monoidal category \(({}_H^H{\mathcal M}_H^H, \otimes _H, H)\).
Proposition 5.2
If H is a quasi-Hopf algebra and \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\) then there exist algebra morphisms \(\underline{\Delta }: T_H(M)\rightarrow T_H(M)~\overline{\otimes }~T_H(M)\) and \(\underline{\varepsilon }: T_H(M)\rightarrow H\) in \({}_H^H{\mathcal M}_H^H\), uniquely determined by
for all \(h\in H\) and \(m\in M\). Furthermore, \((* , 1, \underline{\Delta }, \underline{\varepsilon })\) provides a bialgebra structure on \(T_H(M)\) within \({}_H^H{{{\mathcal {M}}}}_H^H\).
Proof
To define \(\underline{\Delta }\), we apply Proposition 5.1 for \(A=H\), \(A'=T_H(M)~\overline{\otimes }~T_H(M)\), \(\zeta : M\ni m\mapsto m\overline{\otimes }1 + 1\overline{\otimes }m\in A'\) and \(u: A\rightarrow A'\) the unit morphism of \(A'\), where \(A'\) has the tensor product algebra structure of \(T_H(M)\) and itself, within \({}_H^H{{{\mathcal {M}}}}_H^H\). Thus, u is an algebra morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\) and is given by \(u(h)=h\overline{\otimes }1=1\overline{\otimes }h\), for all \(h\in H\). Keeping in mind the monoidal structure of \({}_H^H{{{\mathcal {M}}}}_H^H\), one can easily check that \(\zeta \) is a morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\). Therefore, there is a unique algebra morphism \(\underline{\Delta }: T_H(M)\rightarrow T_H(M)~\overline{\otimes }~T_H(M)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\) such that \(\underline{\Delta }j=\zeta \). Equivalently, \(\underline{\Delta }\) is the algebra morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\) completely determined by
Since \(\underline{\Delta }\) is an algebra morphism, inductively, we can uncover how it acts on an arbitrary element of \(T_H(M)\). For instance, \(\underline{\Delta }(h\otimes _Hm){=}h\underline{\Delta }(m){=}hm\overline{\otimes }1 {+} 1\overline{\otimes }hm{=}\underline{\Delta }(hm)\), and
for all \(h\in H\) and \(m, m^1, m^2\in M\), where, as before, d is the braiding on \({}_H^H{{{\mathcal {M}}}}_H^H\) as in (3.10). And so on.
To define \(\underline{\varepsilon }\), we proceed in a similar manner. This time we apply Proposition 5.1 to \(A=A'=H\), \(u=\mathrm{Id}_H\) and \(\zeta : M\ni m\mapsto 0\in A\), the null morphism. This gives an algebra morphism \(\underline{\varepsilon }: T_H(M)\rightarrow H\), completely determined by \(\underline{\varepsilon }(h)=\underline{\varepsilon }i(h)=u(h)=h\), for all \(h\in H\), and \(\underline{\varepsilon }(m)=\underline{\varepsilon }j(m)=\zeta (m)=0\), for all \(m\in M\). Consequently, for any nonzero natural number n we have
So it remains to prove that \((T_H(M), \underline{\Delta }, \underline{\varepsilon })\) is a coalgebra in \({}_H^H{{{\mathcal {M}}}}_H^H\). To show that \(\underline{\Delta }\) is coassociative we apply again Proposition 5.1, this time to the following datum: \(A=H\), \(A'\) equals the tensor product algebra \(T_H(M)~\overline{\otimes }~T_H(M)~\overline{\otimes }~T_H(M)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\), u equals the unit morphism of \(A'\) and
a morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\). We have, for all \(m\in M\), that
so \((\underline{\Delta }~\overline{\otimes }~\mathrm{Id}_{T_H(M)})\underline{\Delta }j=(\mathrm{Id}_{T_H(M)}~\overline{\otimes }~\underline{\Delta })\underline{\Delta }j=\zeta \). As \((\underline{\Delta }~\overline{\otimes }~\mathrm{Id}_{T_H(M)})\underline{\Delta }\) and \((\mathrm{Id}_{T_H(M)}~\overline{\otimes }~\underline{\Delta })\underline{\Delta }\) are algebras morphisms in \({}_H^H{{{\mathcal {M}}}}_H^H\), it follows from the universal property of \(T_H(M)\) that \((\underline{\Delta }~\overline{\otimes }~\mathrm{Id}_{T_H(M)})\underline{\Delta }=(\mathrm{Id}_{T_H(M)}~\overline{\otimes }~\underline{\Delta })\underline{\Delta }\), as desired.
Up to the identifications given by the left and right unit constraints of \({}_H^H{{{\mathcal {M}}}}_H^H\), we compute that \((\underline{\varepsilon }\otimes _H\mathrm{Id}_{T_H(M)})\underline{\Delta }j=j=(\mathrm{Id}_{T_H(M)}\otimes _H\underline{\varepsilon })\underline{\Delta }j\). \((\underline{\varepsilon }\otimes _H\mathrm{Id}_{T_H(M)})\underline{\Delta }\) and \((\mathrm{Id}_{T_H(M)}\otimes _H\underline{\varepsilon })\underline{\Delta }\) are algebra morphisms in \({}_H^H{{{\mathcal {M}}}}_H^H\), implying \((\underline{\varepsilon }\otimes _H\mathrm{Id}_{T_H(M)})\underline{\Delta }= (\mathrm{Id}_{T_H(M)}\otimes _H\underline{\varepsilon })\underline{\Delta }=\mathrm{Id}_{T_H(M)}\), as required. \(\square \)
Next, we construct the antipode of \(T_H(M)\). It is well known that, in general, the antipode \(\underline{S}\) of a braided Hopf algebra B is an anti-morphism of the algebra B. Otherwise stated, \(\underline{S}\) is an algebra morphism from B to \(B^{\mathrm{op}}\), where \(B^{\mathrm{op}}\) is the opposite algebra associated to B. Coming back to our setting, \(T_H(M)^{\mathrm{op}}\) equals \(T_H(M)\) as object in \({}_H^H{{{\mathcal {M}}}}_H^H\), and is the algebra in \({}_H^H{{{\mathcal {M}}}}_H^H\) having the same unit as \(T_H(M)\) and multiplication \(*_{\mathrm{op}}\) given by \(*_{\mathrm{op}}=*\circ d_{T_H(M), T_H(M)}\). Thus, if \(T_H(M)\) admits an antipode \(\underline{S}\) then it will be completely determined by its restrictions to H and M, since, for all \(z, w\in T_H(M)\),
Theorem 5.3
Let H be a quasi-Hopf algebra and M an object in \({}_H^H{\mathcal M}_H^H\). Then, the tensor product algebra \(T_H(M)\) in \({}_H^H{\mathcal M}_H^H\) admits a Hopf algebra structure within \({}_H^H{{{\mathcal {M}}}}_H^H\).
Proof
We know that \(T_H(M)\) is a braided bialgebra. As in the proof of Proposition 5.2, if we take \(A=H\), \(A'=T_H(M)^{\mathrm{op}}\), u the unit morphism of \(A'\) and \(\zeta : M\ni m\mapsto -m\in A'\), then from the universal property of \(T_H(M)\) we get an algebra morphism \(\underline{S}: T_H(M)\rightarrow T_H(M)^{\mathrm{op}}\) in \({}_H^H{{{\mathcal {M}}}}_H^H\), uniquely determined by
With the help of (5.1), we can see how \(\underline{S}\) acts on an arbitrary element of \(T_H(M)\). To this end, for a fixed natural number \(n\ge 2\), let \(\{s_l=(l, l+1)\mid 1\le l\le n-1\}\) be the set of generators \(s_l\) of the symmetric group \(S_n\) permuting l and \(l+1\). Also, for any \(1\le l\le n-1\), take \(d_l=\mathrm{Id}_{M^{\otimes _Hl-1}}\otimes _Hd_{M, M}\otimes _H\mathrm{Id}_{M^{\otimes _Hn-l-1}}\), an automorphism of \(M^{\otimes _Hn}\), for all \(1\le l\le n-1\). Finally, for \(\sigma \in S_n\) define \(T_\sigma :=d_{l_1}\ldots d_{l_r}\), where \(\sigma =s_{l_1}\ldots s_{l_r}\) is a reduced expression for \(\sigma \) (i.e., r is minimal among all such expressions of \(\sigma \)). Note that, according to [17, Theorem 4.12], \(T_\sigma \) is well defined.
Now, if \(\sigma _0\in S_n\) is given by \(\sigma _0(l)=n-l+1\), for all \(1\le l\le n\), then
for all \(n\ge 2\) and \(m^1, \ldots , m^n\in M\). Observe that \(s_1(s_2s_1)\ldots (s_{n-1}\ldots s_1)\) is a reduced expression for \(\sigma _0\), since \(\sigma _0\) is what is called the longest element of \(S_n\) (for more details see the comments made before [17, Lemma 4.13]). Thus, \(T_{\sigma _0}=d_1(d_2d_1)\cdots (d_{n-1}\cdots d_1)\).
We show that \(\underline{S}\) is antipode for the bialgebra structure of \(T_H(M)\), that is,
for all \(z\in T_H(M)\). Toward this end, remark first that (5.3) is satisfied for any \(z=h\in H\) and \(z=m\in M\). Also, if we define \(*^2=*\) and, in general, \(*^k=* (*^{k-1}\otimes _H\mathrm{Id}_{T_H(M)})\), for all \(k\ge 3\), we have
We used that \(\underline{\Delta }\) is an algebra morphism in the first equality, the fact that \(\underline{S}: T_H(M)\rightarrow T_H(M)^{\mathrm{op}}\) is an algebra morphism in \({}_H^H{{{\mathcal {M}}}}_H^H\) in the second equality, the naturality of the braiding d in the third equality, and the associativity of \(*\) in the last equality.
The above computation says that if the first equality in (5.3) is satisfied by two elements of \(T_H(M)\) then it is also satisfied by their product in \(T_H(M)\). As M generates \(T_H(M)\) as an algebra, this implies that the first equality of (5.3) is satisfied by any \(z\in T_H(M)\). In a similar manner, we can show the second equality in (5.3), so our proof is finished. \(\square \)
5.2 A quasi-Hopf algebra structure on \(T_H(M)\)
It follows now from Theorem 4.6 that \(T_H(M)\) admits also the structure of a quasi-Hopf algebra with a projection or, equivalently, it has the structure of a biproduct quasi-Hopf algebra. More exactly, we have the following.
Proposition 5.4
Let H be a quasi-Hopf algebra and M an object of \({}_H^H{\mathcal M}_H^H\). Then, the tensor algebra \((T_H(M), *, 1)\) within \({}_H^H{{{\mathcal {M}}}}_H^H\) admits the structure of a quasi-Hopf algebra with a projection. Its comultiplication \({\widetilde{\Delta }}\) is given by \({\widetilde{\Delta }}(h)=\Delta (h)\), for all \(h\in H\), and
for all \(m\in M\), extended to the whole \(T_H(M)\) as an algebra morphism from \(T_H(M)\) to \(T_H(M)\underline{\otimes }T_H(M)\), while its counit is determined by \({\widetilde{\varepsilon }}(h)=\varepsilon (h)\), for all \(h\in H\), and \({\widetilde{\varepsilon }}(m)=0\), for all \(m\in M\), extended this time to the whole \(T_H(M)\) as an algebra morphism from \(T_H(M)\) to k. The reassociator of \(T_H(M)\) is \({\widetilde{\Phi }}=X^1\underline{\otimes }X^2\underline{\otimes }X^3\), where \(\Phi =X^1\otimes X^2\otimes X^3\) is the reassociator of H. An antipode \(({\widetilde{S}}, {\widetilde{\alpha }}, {\widetilde{\beta }})\) for \(T_H(M)\) can be obtained from an antipode \((S, \alpha , \beta )\) of H as follows: \({\widetilde{\alpha }}=i(\alpha )=\alpha \), \({\widetilde{\beta }}=i(\beta )=\beta \), \({\widetilde{S}}(h)=S(h)\), for all \(h\in H\), and
for all \(m\in M\), extended to the whole \(T_H(M)\) as an anti-morphism of k-algebras from \(T_H(M)\) to itself, that is, for all \(n\ge 1\) and \(m^1, \ldots , m^n\in M\), we have
(each tensor component over H contains a different copy of \(p_R=p^1\otimes p^2=\cdots =\mathbf{p}^1\otimes \mathbf{p}^2\)).
Finally, via this structure we have quasi-Hopf algebra morphisms 
such that \(\pi i=\mathrm{Id}_H\), where \(\pi =\underline{\varepsilon }\) is the counit of \(T_H(M)\) in \({}_H^H{{{\mathcal {M}}}}_H^H\).
Proof
The unital k-algebra structure on \(T_H(M)\) is given by concatenation, i.e., by \(*\), and 1, the unit of H. The fact that \((T_H(M), *, 1, {\widetilde{\Delta }}, {\widetilde{\varepsilon }}, {\widetilde{\Phi }}, {\widetilde{S}}, \alpha , \beta )\) is a quasi-Hopf algebra is an immediate consequence of Theorems 4.4 and 5.3. \(\square \)
Remarks 5.5
-
(i)
The formula in (5.4) can be also obtained from (5.2) and (4.12), we leave the verification of this fact to the reader.
-
(ii)
The quasi-Hopf algebra structure of \(T_H(M)\) can be deduced as well from the following universal property of \(T_H(M)\): for any k-algebras \(A, A'\), any k-algebra morphisms \(u: H\rightarrow A\), \(v: A\rightarrow A'\) and any H-bimodule morphism \(\zeta : M\rightarrow A'\) there exists a unique k-algebra morphism \(\overline{\zeta }: T_H(M)\rightarrow A'\) which is H-bilinear and such that \(\overline{\zeta }i=vu\) and \(\overline{\zeta }j=\zeta \); here \(A, A'\) are considered H-bimodules via uvu, respectively.
Explicitly, \(\overline{\zeta }(h)=vu(h)\), for all \(h\in H\), and, for all \(n\ge 1\) and \(m^1, \ldots , m^n\in M\),
Now, \({\widetilde{\Delta }}\), \({\widetilde{\varepsilon }}\) and \({\widetilde{S}}\) are uniquely determined by the following data: \((A=H\otimes H, A'=T_H(M)\underline{\otimes }T_H(M), u=\Delta , v=i\otimes i, \zeta =\lambda _M + \rho _M)\), \((A=H, A'=k, u=\mathrm{Id}_H, v=\varepsilon , \zeta =0)\) and \((A=H, A'=T_H(M)^{\mathrm{opp}}, u=\mathrm{Id}_H, v=iS, \zeta : M\ni m\mapsto -S(m_{(0)_{\{-1\}}}p^1)\alpha \cdot m_{(0)_{\{0\}}}\cdot p^2S(m_{(1)})\in T_H(M)^{\mathrm{opp}})\), respectively. Note that \(H\otimes H\) and \(T_H(M)\underline{\otimes }T_H(M)\) are viewed as H-bimodules via \(\Delta \), and \(T_H(M)^{\mathrm{opp}}\) is the opposite k-algebra associated to \(T_H(M)\), regarded as an H-bimodule via the actions \(h*_{op}z*_{op}h'=S(h')*z*S(h)\), for all \(h, h'\in H\) and \(z\in T_H(M)\).
Next, we want to describe, in two equivalent ways, how \({\widetilde{\Delta }}\) acts on an element of \(T_H(M)\). By the universal property of \(\otimes _H\), if A is a k-algebra and an H-bimodule, and \(f_1, f_2: M\rightarrow A\) are H-bimodule morphisms, then we have a well-defined H-bimodule morphism \(f_1\cdot f_2: M\otimes _HM\rightarrow A\) sending \(m^1\otimes _Hm^2\in M\otimes _HM\) to \(f_1(m^1)f_2(m^2)\in A\). We use this simple observation in order to see how \({\widetilde{\Delta }}\) extends to the whole \(T_H(M)\). Actually, we have that \(\lambda _M+\rho _M: M\rightarrow T_H(M)\underline{\otimes }T_H(M)\) is an H-bimodule morphism and
for all \(n\ge 1\) and \(m^1, \ldots , m^n\in M\), where, once more, the product in the right hand side is made in the tensor product algebra \(T_H(M)\underline{\otimes }T_H(M)\) built within the category of k-vector spaces, viewed as an H-bimodule via the monoidal structure of \({}_H{{{\mathcal {M}}}}_H\) given by \(\otimes \). Equivalently,
where, in general, by \(f^n\) we denote the product \(\cdot \) of n copies of f, an H-bimodule morphism from M to a k-algebra that is an H-bimodule, too. Since \(\cdot \) is not commutative, we get that \({\widetilde{\Delta }}\) restricted to \(M^{\otimes _Hn}\) is the sum of \(2^n\) distinct terms, each of them having the form \(f_1\cdot _{\cdots }\cdot f_n\) with \(f_l\in \{\lambda _M, \rho _M\}\), for all \(1\le l\le n\). For instance,
for all \(m^1, m^2\in M\).
A second description for \({\widetilde{\Delta }}\) can be derived from the following result. It is a generalization of [25, Lemma 7] to the quasi-Hopf setting.
Lemma 5.6
For any \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\), we have \(\lambda _M\cdot \rho _M=(\rho _M\cdot \lambda _M)\circ d_{M, M}\).
Proof
For \(m^1, m^2\in M\), we compute
as needed. \(\square \)
For any \(1\le k\le n\), let \(S_{k, n-k}\) be the set of \((k, n-k)\)-shuffles, that is the set of permutations \(\sigma \in S_n\) for which \(\sigma (1)<\cdots \sigma (k)\) and \(\sigma (k+1)<\cdots \sigma (n)\). It can be easily seen that giving an element \(\sigma \in S_{k, n-k}\) is equivalent to giving a subset \(X_k=\{i_1, \ldots , i_k\}\) of \(\{1, \ldots , n\}\): we can assume that the elements of \(X_k\) are arranged in ascending order, and thus, the one-to-one correspondence maps \(X_k\) to the permutation \(\sigma \) given by \(\sigma (1)=i_1, \ldots , \sigma (k)=i_k\) and, for \(j>k\), \(\sigma (j)\) equals the \(j^{\mathrm{th}}\)-element of the set \(\{1, \ldots , n\}\backslash X_k\), ordered in ascending order. Consequently, \(S_{k, n-k}\) has \(\left( \begin{array}{c} n\\ k \end{array}\right) =\frac{n!}{k!(n-k)!} \) elements.
Following [25], \({\widetilde{B}}_{k, n-k}:=\sum \nolimits _{\sigma ^{-1}\in S_{k, n-k}}T_\sigma \), where \(T_\sigma \) is defined by a reduced expression of \(\sigma \) as in the proof of Theorem 5.3.
Corollary 5.7
With the above notation, the comultiplication \({\widetilde{\Delta }}\) of the quasi-Hopf algebra \(T_H(M)\) is given by \({\widetilde{\Delta }}(h)=\Delta (h)\), for all \(h\in H\), and
for all \(n\ge 1\) and \(m^1, \ldots , m^n\in M\).
Proof
Exactly as in the proof of [25, Proposition 6], we can show that, for any two H-bimodule morphisms \(f_1\), \(f_2\) from M to a k-algebra A that is also an H-bimodule, we have
provided that \(f_2\cdot f_1=(f_1\cdot f_2)\circ d_{M, M}\). Our assertion follows now by taking in the above formula \(A=T_H(M)\underline{\otimes }T_H(M)\), \(f_1=\rho _M\) and \(f_2=\lambda _M\). \(\square \)
5.3 \(T_H(M)\) as a biproduct quasi-Hopf algebra
For simplicity, for \(M\in {}_H^H{{{\mathcal {M}}}}^H_H\) we denote \(M^{\overline{\mathrm{co}(H)}}\) by V. Also, by T(V) we denote the k-vector space \(\bigoplus \limits _{n\ge 0}T^{n)}(V)\), where \(T^{0)}(V)=k\), \(T^{\otimes 1)}(V)=V\), \(T^{\otimes 2)}(V)=V\otimes V\) and \(T^{\otimes n)}=V\otimes T^{\otimes n-1)}(V)\), for all \(n\ge 3\). Since \(V\in {}_H^H{\mathcal YD}\) with structure (2.28–2.29) and \({}_H^H{\mathcal YD}\) is monoidal, it follows that T(V) is a left Yetter–Drinfeld module over H. As \({}_H^H{\mathcal YD}\) is not strict monoidal, the order of the parenthesis in the definition of T(V) is essential for the structure of T(V) in \({}_H^H{\mathcal YD}\); the notation \(T^{\otimes n)}(V)\) suggests that we deal with the tensor product of n-copies of V in \({}_H^H{\mathcal YD}\) such that all the closing parentheses are placed on the right-handed side of the last term of \(\otimes \), i.e., \(T^{\otimes n)}(V)=V\otimes (V\otimes (\cdots \otimes (V\otimes V)\cdots ))\), as objects in \({}_H^H{\mathcal YD}\). This also motivates to denote by \(v^{{\mathop {l+1,n}\limits ^{\longleftarrow }}}=v^{l+1}\otimes (v^{l+2}\otimes (\cdots \otimes (v^{n-1}\otimes v^n)\cdots ))\) an element of \(T^{\otimes n-l)}(V)\), for all \(l<n\); in the case when \(l=0\) and \(n\ge 1\), in place of \(v^{{\mathop {1,n}\limits ^{\longleftarrow }}}\) we simply write \(v^{{\mathop {n}\limits ^{\leftarrow }}}\).
We next show that \(T_H(M)^{\overline{\mathrm{co}(H)}}\) and T(V) are isomorphic objects of \({}_H^H{\mathcal YD}\).
Lemma 5.8
Let H be a quasi-Hopf algebra, \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\), \(V=M^{\overline{\mathrm{co}(H)}}\) and \(\overline{E}=\overline{E}_M\). Then, for any \(n\ge 2\), \(\overline{\phi }^{-1}_n: (M^{\otimes _H n})^{\overline{\mathrm{co}(H)}}\rightarrow V^{\otimes n)}\) given by
for all \(m^{\otimes _Hn}\in (M^{\otimes _Hn})^{\overline{\mathrm{co}(H)}}\), is an isomorphism of left Yetter–Drinfeld modules. Consequently, if we set \(\overline{\phi }^{-1}_1=\mathrm{Id}_V\) and \(\overline{\phi }^{-1}_0: H^{\overline{\mathrm{co}(H)}}=k\beta \ni \kappa \beta \rightarrow \kappa \in k\) then
is an isomorphism in \({}_H^H{\mathcal YD}\).
Proof
Observe that \(\overline{\phi }^{-1}_2=\overline{\phi }^{-1}_{2, M, M}\) defined by (2.32), and this justifies our notation. By mathematical induction on \(n\ge 2\), we show that
Actually, \(\overline{\phi }^{-1}_{n+1}=(\mathrm{Id}_V\otimes \overline{\phi }^{-1}_{n-1})\overline{\phi }^{-1}_{2, M, M^{\otimes _Hn}}\), for all \(n\ge 2\), and since
for all \(m^{\otimes _Hn+1}\in (M^{\otimes _Hn+1})^{\overline{\mathrm{co}(H)}}\), it follows that
for all \(m^{\otimes _Hn+1}\in (M^{\otimes _Hn+1})^{\overline{\mathrm{co}(H)}}\), as required. It is clear at this moment that \(\overline{\phi }^{-1}_n\) is an isomorphism in \({}_H^H{\mathcal YD}\), for all \(n\ge 2\). Note that its inverse, denoted by \(\overline{\phi }_n\), is determined by
for all \(v^{{\mathop {n}\limits ^{\leftarrow }}}\in T^{n)}(V)\), where each tensor component contains a distinct copy of \(q_R=q^1\otimes q^1=\cdots =\mathbf{q}^1\otimes \mathbf{q}^2\); also, \(\Phi ^{-1}=x^1\otimes x^2\otimes x^3=\cdots =y^1\otimes y^2\otimes y^3\) appears in the definition of \(\overline{\phi }_{n}\) for \((n-1)\)-times. \(\square \)
\(T_H(M)^{\overline{\mathrm{co}(H)}}\) is a Hopf algebra in the braided category \({}_H^H{\mathcal YD}\), and this induces a Hopf algebra structure on T(V) within \({}_H^H{\mathcal YD}\) as follows.
The comultiplication \(\Delta \) of H is not coassociative, and this forces to introduce the notation \(\Delta _{2)}=\Delta \), \(\Delta _{3)}=(\mathrm{Id}_H\otimes \Delta )\Delta \) and, in general, \(\Delta _{n)}=(\mathrm{Id}_H\otimes \Delta _{n-1)})\Delta \), for all \(n\ge 3\). If \(h\in H\) and \(n\ge 2\), we denote
Proposition 5.9
In the hypothesis of Lemma 5.8, we have that T(V) is a Hopf algebra in \({}_H^H{\mathcal YD}\) via the following structure. The multiplication, denoted by \(\odot \), is given by
for all \(m, n\ge 2\) and \(v^1,\ldots , v^{m+n}\in V\), and the unit equals the unit of the field k.
The comultiplication \(\underline{\underline{\Delta }}\) and the counit \(\underline{\underline{\varepsilon }}\) are defined, for all \(\kappa \in k\) and \(v\in V\), by
and, respectively, by \(\underline{\underline{\varepsilon }}(\kappa )=\kappa \) and \(\underline{\underline{\varepsilon }}(v)=0\), extended to the whole T(V) as algebra morphisms in \({}_H^H{\mathcal YD}\). As before \(\underline{\otimes }\) stands for the tensor product over k between T(V) and itself.
The antipode \(\underline{\underline{S}}\) of T(V) is determined by \(\underline{\underline{S}}(\kappa )=\kappa \) and \(\underline{\underline{S}}(v)=-v\), for all \(\kappa \in k\) and \(v\in V\), extended as an anti-morphism of algebras in \({}_H^H{\mathcal YD}\) between T(V) and itself.
Proof
We show that the structure in the statement is the unique Hopf algebra structure on T(V) within \({}_H^H{\mathcal YD}\) that turns \(\overline{\phi }^{-1}: T_H(M)^{\overline{\mathrm{co}(H)}}\rightarrow T(V)\) into a braided Hopf algebra isomorphism. In this sense, the multiplication \(\odot \) is given by
where \(*\) is the multiplication on \(T_H(M)\) and \({{{\mathcal {G}}}}\) is the functor defined in Proposition 2.1. We have \(\overline{\phi }_1=\mathrm{Id}_V\), and therefore \(v^1\odot v^2=v^1\otimes v^2\), for all \(v^1, v^2\in V\).
For a generic \(v\in V\), let us denote \(w\otimes x^3=q^1x^1_1\cdot v\cdot S(q^2x^1_2)x^2\otimes x^3\). Then, since 
, for all \(v\in V\), we have
This fact allows to compute
for all \(m, n\ge 2\) and \(v^1, \ldots , v^{m+n}\in V\). Now, to get the formula claimed in the statement we must apply (2.19) and (2.1) until we are able to use (2.17). We illustrate this way of computation with few examples: for all \(v^1, \ldots , v^5\in V\) we have
and similarly
In a manner similar to the one above we can show the remaining two relations related to the definition of \(\odot \), we leave the verification of this fact to the reader.
As \(\overline{\phi }_0(\kappa )=\kappa \beta \), for all \(\kappa \in k\), we get \(\kappa \odot \kappa '=\kappa \kappa '\) and \(\kappa \odot v^{{\mathop {n}\limits ^{\leftarrow }}}=v^{{\mathop {n}\limits ^{\leftarrow }}}\odot \kappa = \kappa v^{{\mathop {n}\limits ^{\leftarrow }}}\), for all \(\kappa \), \(\kappa '\in k\) and \(v^1, \ldots , v^n\in V\). In particular, we deduce that \(\odot \) is unital with unit given by the unit of k. This completes the algebra structure of T(V) in \({}_H^H{\mathcal YD}\).
The coalgebra structure \((\underline{\underline{\Delta }}, \underline{\underline{\varepsilon }})\) of T(V) in \({}_H^H{\mathcal YD}\) is obtained from the one of \(T_H(M)^{\overline{\mathrm{co}(H)}}\) as follows: \(\underline{\underline{\varepsilon }}: T(V)~~{\mathop {\longrightarrow }\limits ^{\overline{\phi }}}~~T_H(M)^{\overline{\mathrm{co}(H)}} ~~{\mathop {\longrightarrow }\limits ^{{\mathcal G}(\underline{\varepsilon })}}~~H^{\overline{\mathrm{co}(H)}}~~{\mathop {\longrightarrow }\limits ^{\overline{\phi }^{-1}_0}}~~k\) and
Explicitly, \(\underline{\underline{\varepsilon }}\) and \(\underline{\underline{\Delta }}\) are algebra morphisms in \({}_H^H{\mathcal YD}\), completely determined by \(\underline{\underline{\varepsilon }}(\kappa )=\kappa \), \(\underline{\underline{\varepsilon }}(v)=0\), \(\underline{\underline{\Delta }}(\kappa )= \kappa (\overline{\phi }^{-1}\underline{\otimes }\overline{\phi }^{-1})(\beta \underline{\otimes }\beta )=\kappa \underline{\otimes }1=1\underline{\otimes }\kappa \) and
for all \(\kappa \in k\) and \(v\in V\). In general, for all \(n\ge 2\) and \(v^1, \ldots v^n\in V\) we have \(\underline{\underline{\varepsilon }}(v^{{\mathop {n}\limits ^{\leftarrow }}})=0\) and
where in the right hand side the product is made in the tensor product algebra \(T(V)\underline{\otimes }T(V)\) within \({}_H^H{\mathcal YD}\).
Finally, the formula for the antipode \(\underline{\underline{S}}\) follows from the equality \(\underline{\underline{S}}=\overline{\phi }^{-1}{{{\mathcal {G}}}}(\underline{S})\overline{\phi }\). Note only that it extends to the whole T(V) as an anti-morphism of algebras in \({}_H^H{\mathcal YD}\); thus, the braiding c of \({}_H^H{\mathcal YD}\) plays an important role in this case. \(\square \)
Remark 5.10
Any object V of \({}_H^H{\mathcal YD}\) is the set of right coinvariants of a certain \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\). Thus, the braided tensor Hopf algebra construction T(V) makes sense for any \(V\in {}_H^H{\mathcal YD}\): the structure is the one in Proposition 5.9, of course wth the left adjoint H-action 
replaced by the given left H-action, say \(\cdot \), on V.
By the above results, we get the following.
Theorem 5.11
Let H be a quasi-Hopf algebra, \(M\in {}_H^H{{{\mathcal {M}}}}_H^H\) and \(V=M^{\overline{\mathrm{co}(H)}}\). Then, \(T_H(M)\) is isomorphic to the biproduct quasi-Hopf algebra \(T(V)\times H\).
Proof
We know that \(\overline{\phi }^{-1}: T_H(M)^{\overline{\mathrm{co}(H)}}\rightarrow T(V)\) is an isomorphism of Hopf algebras in \({}_H^H{\mathcal YD}\), and therefore \(\overline{\phi }^{-1}\times \mathrm{Id}_H: T_H(M)^{\overline{\mathrm{co}(H)}}\times H\rightarrow T(V)\times H\) is an isomorphism of quasi-Hopf algebras. But \(\overline{\nu }^{-1}_{T_H(M)}: T_H(M)\rightarrow T_H(M)^{\overline{\mathrm{co}(H)}}\times H\) is a quasi-Hopf algebra isomorphism as well, and from here we conclude that \(\Gamma :=(\overline{\phi }^{-1}\times \mathrm{Id}_H)\overline{\nu }^{-1}_{T_H(M)}: T_H(M)\rightarrow T(V)\times H\) is a quasi-Hopf algebra isomorphism. More precisely, \(\Gamma (h)=1\times h\), \(\Gamma (m)=\overline{E}(m_{(0)})\times m_{(1)}\) and
for all \(h\in H\), \(m\in M\), and \(n\ge 2\) and \(m^1, \ldots , m^n\in M\). The inverse of \(\Gamma \) is \(\Gamma ^{-1}\) given by \(\Gamma ^{-1}(z\times h)=q^1\cdot \overline{\phi }(z)\cdot S(q^2)h\), for all \(z\in T(V)\) and \(h\in H\). \(\square \)
6 An example
Denote by \(C_n\) the cyclic group of order \(n\ge 2\), assume that k contains a primitive root of unity \({\mathfrak q}\) of order \(n^2\) and take \(q:={\mathfrak q}^n\), a primitive root of unity in k of order n (in particular, \(n\not =0\) in k). If g is a generator of \(C_n\) then, for any \(0\le j\le n-1\),
is an idempotent of the group algebra \(k[C_n]\). Furthermore, \(g1_j=q^j1_j\), and so \(g^l1_j=q^{lj}1_j\), for all \(0\le l, j\le n-1\). This implies \(1_l1_j=\delta _{l, j}1_l\), for all \(0\le l, j\le n-1\), and \(\sum \nolimits _{j=0}^{n-1}1_j=\mathbf{1}\), the identity element of \(C_n\).
For a rational number r, denote by [r] the integer part of r. According to [12, Lemma 3.4] or [4, Proposition 5.1], we know that
is a non-trivial normalized 3-cocycle on \(C_n\) (in the Harrison cohomology, we refer to [4] for more details). Thus, we can endow \(k[C_n]\) with a quasi-Hopf algebra structure as follows: the algebra structure is that of the group algebra \(k[C_n]\), the coalgebra structure is given by
for all \(1\le s\le n-1\), the reassociator is \(\Phi \) as above, and the antipode is determined by \(S(g^s)=g^{n-s}\), for all \(0\le s\le n-1\), and distinguished elements \(\alpha =g^{-1}\) and \(\beta =\mathbf{1}\). Otherwise stated, the fact that \(k[C_n]\) is a commutative algebra allows to view the Hopf group algebra \(k[C_n]\) as a quasi-Hopf algebra with reassociator \(\Phi \). We will denote this quasi-Hopf algebra structure on \(k[C_n]\) by \(k_\Phi [C_n]\).
Let now V be a k-vector space. We equip V with a left Yetter–Drinfeld module structure over \(k_\Phi [C_n]\), and then, we construct a Hopf algebra T(V) in the braided category \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\). Thus, \(T(V)\times k_\Phi [C_n]\) is a quasi-Hopf algebra with projection, and our goal is to compute explicitly this quasi-Hopf algebra structure.
Lemma 6.1
With the above notation, V is a left \(k_\Phi [C_n]\)-Yetter–Drinfeld module via the structure given, for all \(v\in V\), by
where \(K:=\sum \limits _{j=0}^{n-1}{\mathfrak q}^j1_j=\sum \limits _{j=0}^{n-1}q^{\frac{j}{n}}1_j\).
Proof
For any \(0\le j, l\le n-1\), we have
We use this equality together with \(1_j\cdot v=\frac{1}{n}\sum \nolimits _{i=0}^{n-1}q^{(n-j)i}g^i\cdot v= \frac{1}{n}\sum \nolimits _{i=0}^{n-1}q^{(n-j+1)i}v=\delta _{j,1}v\), for all \(0\le j\le n-1\) and \(v\in V\), to compute that
for all \(v\in V\). Likewise, since \(1_iK=q^{\frac{i}{n}}1_i\), for all \(0\le i\le n-1\), we see that
for all \(v\in V\), and this shows (2.22). The Yetter–Drinfeld condition in (2.23) is satisfied by our structure since \(k_\Phi [C_n]\) is a commutative algebra. Finally, as \(\varepsilon (1_j)=\delta _{j, 0}\), for all \(0\le j\le n-1\), we deduce that \(\varepsilon (K)=1\), and this finishes our proof. \(\square \)
We start to describe the braided Hopf algebra structure of T(V) by computing its algebra structure in \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\). For this, we need first a lemma.
Lemma 6.2
For any \(m\ge 2\), we have
Proof
We prove the formula by mathematical induction on \(m\ge 2\). To this end, for any natural number p we denote by \(p'\) the remainder of the division of p by n, that is \(p=\left[ \frac{p}{n}\right] n + p'\). Consequently, \(\left[ \frac{p'+l}{n}\right] =\left[ \frac{p+l}{n}\right] -\left[ \frac{p}{n}\right] \), for any natural numbers p, l.
We have \(g^s=g^s\mathbf{1}=\sum \nolimits _{a=0}^{n-1} g^s1_a=\sum \nolimits _{a=0}^{n-1}q^{as}1_a\), for all \(0\le s\le n-1\), and therefore
for all \(0\le i, l\le n-1\), proving the equality in the statement for \(m=2\). Using the mathematical induction and a computation similar to the one above, we get that
as needed. \(\square \)
We can describe now the monoidal algebra structure of T(V).
Proposition 6.3
Let V be the left \(k_\Phi [C_n]\)-Yetter–Drinfeld module defined in Lemma 6.1. Then, T(V) is a left \(k_\Phi [C_n]\)-Yetter–Drinfeld module via the structure given by
for all \(0\le s\le n-1\), \(\kappa \in k\), \(m\ge 1\) and \(v^1, \ldots , v^m\in V\).
Furthermore, T(V) is an algebra in \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\) via the multiplication \(\odot \) determined by
for all \(m, p\ge 2\) and \(v^1, \ldots , v^{m+p}\in V\), and is unital with unit equals the unit of k.
Proof
We specialize the structure in Proposition 5.9 for \(H=k_\Phi [C_n]\) and V as in Lemma 6.1. As any element of \(C_n\) is grouplike, it follows from the monoidal structure on \({}_{k_\Phi [C_n]}{{{\mathcal {M}}}}\) that T(V) is a left \(k_\Phi [C_n]\)-module via the action \(\cdot \) defined above.
We prove the formula concerning the left \(k_\Phi [C_n]\)-coaction on T(V) by mathematical induction on \(m\ge 1\). Let us start by noting that, for all \(m\ge 2\) and \(1\le l\le n-1\), we have
For \(m=1\), we recover the formula for the left \(k_\Phi [C_n]\)-coaction on V. If we assume that it is true for \(m\ge 1\) and for any m elements of V, then it is also true for any \(m+1\) elements \(v^1, \cdots , v^{m+1}\in V\), since, by (2.24) and the fact that \(1_j\cdot v=\delta _{j, 1}v\), for all \(0\le j\le n-1\) and \(v\in V\), we have
as required. In \((*_1)\), we used that \(\Phi ^{-1}=\sum \nolimits _{i, j, l=0}^{n-1}q^{-i\left[ \frac{j+l}{n}\right] }1_i\otimes 1_j\otimes 1_l\), and in \((*_2)\) the facts that \(g^a1_l=q^{la}1_l\), for all \(a\in {\mathbb {N}}\) and \(0\le l\le n-1\), and \(\sum \nolimits _{l=0}^{n-1}1_l=\mathbf{1}\). We have also that \(K^a=\sum \nolimits _{l=0}^{n-1}q^{\frac{al}{n}}1_l\), for all \(a\in {\mathbb {N}}\), and therefore \(K^n=\sum \nolimits _{l=0}^{n-1}q^l1_l=g\). This implies \(g^aK^b=K^{na+b}\), for all \(a, b\in {\mathbb {N}}\), proving the second formula for the left \(k_\Phi [C_n]\)-coaction on T(V) claimed in the statement.
Now, the first two relations defining \(\odot \) follow directly from the definition of \(\odot \), while the third one can be derived from \(1_j\cdot v=\delta _{j, 1}v\), for all \(0\le j\le n-1\) and \(v\in V\), and the formula in Lemma 6.2 as follows:
The proof of the fourth relation involving \(\odot \) is quite technical. Note that (6.2) implies
This leads to the first formula for the \(\odot \) mentioned above, since
for any nonzero natural number a, and therefore
and
The second formula involving the product \(v^{{\mathop {m}\limits ^{\leftarrow }}}\odot v^{{\mathop {m+1, m+p}\limits ^{\longleftarrow }}}\) follows from the first one and the fact that
So our proof is finished. \(\square \)
Next, we complete the algebra structure on T(V) up to a braided Hopf algebra one, making the coalgebra structure of it explicit in terms of the braid group action. Recall that for \(1\le l\le m-1\) by \(S_{l, m-l}\) we denoted the set of \((l, m-l)\)-shuffles. We extend this notation to \(0\le l\le m\), by defining \(S_{0, m}=\{e\}=S_{m, 0}\), where e is the identity permutation of \(S_m\). In what follows, by \(S_m\) we understand the symmetric group of \(\{2, \ldots , m+1\}\). Finally, the length of a permutation \(\sigma \in S_{l, m-l}\) is the length of any reduced expression for \(\sigma \) in terms of the generators \(s_l=(l, l+1)\), \(1\le l\le m-1\). We will denote it by \(r(\sigma )\); by convention, \(r(e)=0\).
Proposition 6.4
The algebra T(V) in \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\) built in Proposition 6.3 admits a Hopf algebra structure in the braided category \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\). The coalgebra structure is defined by the comultiplication \(\underline{\underline{\Delta }}\), \(\underline{\underline{\Delta }}(\kappa )=\kappa \underline{\otimes }1=1\underline{\otimes }\kappa \), for all \(\kappa \in k\), and
for all \(m\ge 1\) and \(v^1, \ldots , v^m\in V\) (if a component of the \(\underline{\otimes }\) monomial does not make sense then it is equal to the unit of k), and counit \(\underline{\underline{\varepsilon }}\) determined by \(\underline{\underline{\varepsilon }}(\kappa )=\kappa \), for all \(\kappa \in k\), and \(\underline{\underline{\varepsilon }}(v^1\otimes (\cdots \otimes (v^{m-1}\otimes v^m)\cdots ))=0\), for all \(m\ge 1\) and \(v^1, \ldots , v^m\in V\).
The antipode \(\underline{\underline{S}}\) of T(V) is completely determined by \(\underline{\underline{S}}(\kappa )=\kappa \), for all \(\kappa \in k\), and
for all \(m\ge 1\) and \(v^1, \ldots , v^m\in V\).
Proof
We specialize Proposition 5.9 for \(H=k_\Phi [C_n]\) and V as in Lemma 6.1. The defining relations for \(\underline{\underline{\varepsilon }}\) are immediate, as well as that for \(\underline{\underline{\Delta }}\) restricted to k. We prove now by mathematical induction on \(m\ge 1\) that \(\underline{\underline{\Delta }}\) restricted to \(T^{\otimes m)}(V)\) has the form stated in (6.4). For \(m=1\), this reduces to \(\underline{\underline{\Delta }}(v)=v\underline{\otimes }1+ 1\underline{\otimes }v\), for all \(v\in V\), which is just the definition of \(\underline{\underline{\Delta }}\) restricted to V. To see that m implies \(m+1\), we proceed as follows. Firstly, from (3.4) and \(K\cdot (v^2\otimes (\cdots \otimes (v^m\otimes v^{m+1})\cdots ))= q^{\frac{m'}{n}}v^2\otimes (\cdots \otimes (v^m\otimes v^{m+1})\cdots )\), for all \(m\ge 1\) and \(v^2,\ldots , v^{m+1}\in V\), we get that
for all \(m\ge 1\) and \(v^1, \ldots , v^m\in V\). Secondly, by the definition of \(\Phi \) and the above formula for c we deduce that
for all \(m, p\ge 1\) and \(v^1, \ldots , v^{m+p}\in V\). Once more, the product is made in the tensor product algebra \(T(V)\underline{\otimes }T(V)\), built within the braided category \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\).
Now we use that \(\underline{\underline{\Delta }}\) is an algebra morphism in \({}_{k_\Phi [C_n]}^{k_\Phi [C_n]}{\mathcal YD}\) and the mathematical induction to compute that
So we have two double sums, each of them having \(2^m\) summands. Now, for the first double sum we can write its general term under the form
where \(\sigma _1^{-1}:=\left( \begin{matrix} 1&{} 2&{}\cdots &{} m+1\\ 1&{}\sigma ^{-1}(2)&{}\cdots &{}\sigma ^{-1}(m+1) \end{matrix}\right) \). It is clear that \(\sigma _1^{-1}\in S_{l+1, m-l}\) with \(r(\sigma _1^{-1})=r(\sigma ^{-1})\). Also,
where the congruence modulo n is due to (6.3).
Analogously, we can write a summand of the second double sum under the general form
with \(\sigma _2^{-1}=\left( \begin{matrix} 1&{}\cdots &{}l&{}l+1&{}l+2&{}\cdots &{}m+1\\ \sigma ^{-1}(2)&{}\cdots &{}\sigma ^{-1}(l+1)&{}1&{}\sigma ^{-1}(l+2)&{}\cdots &{}\sigma ^{-1}(m+1) \end{matrix}\right) {\in } S_{l, m+1-l} \).
By [17, Lemma 4.7], for \(w\in S_n\) and \(1\le l\le n-1\) we have \(r(ws_l)=r(w)+1\) if and only if \(w(l)<w(l+1)\). By using inductively this result and the fact that
we get that \(r(\sigma _2^{-1})=r(\sigma ^{-1})+l\), and consequently, a reduced expression for \(\sigma _2^{-1}\) can be obtained by multiplying to the right a reduced expression for \(\sigma ^{-1}\) with \(s_1\cdots s_{l}\) in \(S_{m+1}\). Hence, we have that
Otherwise stated, we have proved that all the summands of the two double sums considered above are also summands of the double sum
which means that the latter double sum contains the two mentioned double sums. Actually, it is the sum of the two because in both cases we have \(2^{m+1}\) summands. This completes the induction.
Finally, by definition \(\underline{\underline{S}}(\kappa )=\kappa \), for all \(\kappa \in k\). We have \(\underline{\underline{S}}(v)=-v\), for all \(v\in V\), and
for all \(m\ge 1\) and \(v^1, \ldots , v^{m+1}\in V\). Thus, the formula in (6.5) is a consequence of the mathematical induction and of the explicit definition of \(\odot \) in the statement. \(\square \)
By using the biproduct quasi-Hopf algebra construction, to the triple \((V, C_n, {\mathfrak q})\) we associate a quasi-Hopf algebra with projection \(H(n, q, V):=T(V)\times k_\Phi [C_n]\), where \(q={\mathfrak q}^n\) and \(\Phi \) is as in (6.1). We next describe this structure.
Recall that, for \(\kappa \in k\backslash \{0\}\) and \(a\in {\mathbb {N}}\backslash \{0\}\), \((a)_\kappa := \sum \limits _{j=0}^{a-1}\kappa ^j=\left\{ \begin{array}{cl} a&{},\text{ if } \kappa =1\\ \frac{\kappa ^a-1}{\kappa -1}&{},\text{ if } \kappa \not =1 \end{array}\right. \).
If \(v^{{\mathop {m}\limits ^{\leftarrow }}}=v^1\otimes (v^2\otimes (\cdots \otimes (v^{m-1}\otimes v^m)\cdots ))\) then \(v^{{\mathop {m_\tau }\limits ^{\leftarrow }}}:=v^m\otimes (v^{m-1}\otimes (\cdots \otimes (v^2\otimes v^1)\cdots ))\). Also, the Heaviside symbol \([i>j]\) stands for the integer 1 if \(i>j\) and for 0 otherwise.
Theorem 6.5
Let k be a field containing a primitive root of unity \({\mathfrak q}\) of degree \(n^2\), \(n\ge 2\), V a k-vector space and \(C_n\) the cyclic group of order n generated by g. If \(q={\mathfrak q}^n\) then the quasi-Hopf algebra structure of \(H(n, q, V)=T(V)\times k_\Phi [C_n]\) is the following.
The multiplication is given by
for all \(m, p\in {\mathbb {N}}\) and \(0\le s, t\le n-1\), where, by convention, \(v^{{\mathop {0}\limits ^{\leftarrow }}}=1\), the unit of k. It is unital with unit \(1\times \mathbf{1}\).
The comultiplication \(\Delta \) is completely determined by
and is counital with counit given by \(\underline{\underline{\varepsilon }}(v^{{\mathop {m}\limits ^{\leftarrow }}}\times g^j)=\delta _{m, 0}v^{{\mathop {m}\limits ^{\leftarrow }}}\), for all \(m\in {\mathbb {N}}\), \(j\in \{0, \ldots , n-1\}\) and \(v^1, \ldots , v^m\in V\).
The antipode s is defined, for all \(v^1,\ldots , v^m\in V\) and \(0\le l\le n-1\), by
The distinguished elements \(\alpha \) and \(\beta \) that together with s define the antipode of H(n, q, V) are \(1\times g^{-1}\) and \(1\times \mathbf{1}\), respectively.
Proof
We have
and therefore
From here, we get
and since
we conclude that
as stated. For the computation of \(\Delta (v^{{\mathop {m}\limits ^{\leftarrow }}}\times g^s)\), we proceed in a similar manner. First, we use (4.13) to calculate
This leads to the claimed formula for \(\Delta (v^{{\mathop {m}\limits ^{\leftarrow }}}\times g^j)\) since \(g^j=\sum \nolimits _{i=0}^{n-1}q^{ij}1_i\).
So it remains to prove the formula for s. For this, notice that \(S(1_l)=1_{n-l}\), for all \(0\le l\le n-1\), where by convention \(1_n=1_0\). Therefore, the element \(p_R\) in \(k_\Phi [C_n]^{\otimes 2}\) is
where in the last but one equality we used the fact that \(g\sum \nolimits _{i=0}^{n-1}q^{-i}1_i=\mathbf{1}\) in \(k_\Phi [C_n]\). Thus,
Remark also that K is invertible with inverse \(K^{-1}=\sum \nolimits _{j=0}^{n-1}q^{-j}1_j\), and this allows to prove that \(K^a=\sum \nolimits _{j=0}^{n-1}q^{aj}1_j\), for any integer number a. Hence, for \(a\in {\mathbb {Z}}\) and \(v^1,\ldots , v^m\in V\),
Finally, for \(m\in {\mathbb {N}}\) and \(j\in \{0,\ldots , n-1\}\) the equation \((m'+t)'=j\) has a unique solution in \(\{0, \ldots , n-1\}\). Namely, if \(j\in \{0,\ldots , m'-1\}\) then \(t=n + j - m'\), and if \(j\in \{m',\ldots , n+m'-1\}\) then \(t=j-m'\). By using all these facts and the formula for the antipode s of a biproduct quasi-Hopf algebra found in Corollary 4.5, we get that
as stated (for the third equality we used that \((1\times 1_0)(v^{{\mathop {m}\limits ^{\leftarrow }}}\times 1_0)=\delta _{m', 0} v^{{\mathop {m}\limits ^{\leftarrow }}}\times 1_0\)). \(\square \)
We end by specializing Theorem 6.5 for \(V=kv\), a one dimensional vector space. In this situation \(\{v_m\}_{m\in {\mathbb {N}}}\) is a basis for T(V), where \(v_m:=v\otimes (v\otimes (\cdots \otimes (v\otimes v)\cdots ))\in T^{m)}(V)\); by convention \(v_0=1\), the unit of k. It follows that \(\{v_mg^l\mid m\in {\mathbb {N}},~0\le l\le n-1\}\) is a basis for H(n, q, kv), where we identify \(v_m\equiv v_m\times \mathbf{1}\) and \(g^l\equiv 1\times g^l\), and therefore \(v^m\times g^l=(v_m\times \mathbf{1})(1\times g^l)\equiv v_mg^l\). With these identifications in mind, we have that
for all \(m\in {\mathbb {N}}\) and \(0\le l\le n-1\). Therefore, H(n, q, kv) is the unital associative algebra generated by \(\{v_m\}_{m\in {\mathbb {N}}}\) and g with relations
for all \(m\in {\mathbb {N}}\) and \(0\le l, a, b\le n-1\). The unit is \(1=v_0\).
In order to give a nicer form for the comultiplication of H(n, q, kv), we need a preliminary result. We believe that it was proved already somewhere else, but because we were not able to find a reference we decided to include its proof here. Recall that \((0)!_{\mathfrak q}:=1\) and \((p)!_{\mathfrak q}=(1)_{\mathfrak q}(2)_{\mathfrak q}\cdots (p)_{\mathfrak q}\) is the \({\mathfrak q}\)-factorial of p, \(p\in {\mathbb {N}}\), and that \(\left( \begin{array}{c} p\\ s \end{array} \right) _{\mathfrak q}=\frac{(p)!_{\mathfrak q}}{(s)!_{\mathfrak q}(p-s)!_{\mathfrak q}} \), with \(0\le s\le p\), are the so-called Gauss polynomials.
Lemma 6.6
We have \(\sum \limits _{w\in S_{l, m-l}}{\mathfrak q}^{r(w)}=\left( \begin{array}{c} m\\ m-l \end{array} \right) _{\mathfrak q} \), for all \(m\in {\mathbb {N}}\) and \(0\le l\le m\).
Proof
For simplicity, denote \(\lambda _m({\mathfrak q}, l):=\sum \limits _{w\in S_{l, m-l}}{\mathfrak q}^{r(w)}\). As we observed, any \((l, m-l)\) shuffle is completely determined by a subset \(\{i_1,\ldots , i_l\}\) of \(\{1, \ldots , m\}\), arranged in ascending order. Actually, any \((l, m-l)\) shuffle is of the form
for some \(1\le i_1<\cdots<i_k<\cdots <i_l\le m\). Consequently, the inversions of it are
and so these are in number of \(i_1+\cdots i_l- \frac{l(l+1)}{n}\). According to [17, Lemma 4.7], the length of a permutation is equal to the number of its inversions, and so
for all \(m\in {\mathbb {N}}\) and \(1\le l\le m\). This recurrence together with \(\lambda _m({\mathfrak q}, 0)=1\) and the Pascal identity, see [16, Proposition IV.2.1],
valid for any \(0\le k\le n\) in \({\mathbb {N}}\), allows to obtain in an inductive way the formula for \(\lambda _m({\mathfrak q}, l)\) stated above. We leave this detail to the reader. \(\square \)
One can present now the quasi-Hopf algebra structure of H(n, q, kv).
Corollary 6.7
For any \(n\in {\mathbb {N}}\), \(n\ge 2\) and \({\mathfrak q}\), a primitive root of unity of degree \(n^2\) in k denote by \(H_\aleph (n, q)\) the k-algebra generated by \(\{v_m\}_{m\in {\mathbb {N}}}\) and g with relations (6.6), (6.7) and unit \(1=v_0\), where \(q={\mathfrak q}^{n}\). Then, \(H_\aleph (n, q)\) is a quasi-Hopf algebra with projection, via the quasi-coalgebra structure given, for all \(m\in {\mathbb {N}}\), by
and antipode s determined by \(s(g)=g^{-1}\) and
for all \(m\in {\mathbb {N}}\), and distinguished elements \(\alpha =g^{-1}\) and \(\beta =1\).
Proof
We have \(H_\aleph (n, q)=H(n, q, kv)\), so everything follows from the comments made after Theorem 6.5 and the formula in Lemma 6.6.\(\square \)
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Bulacu, D. Quasi-quantum groups obtained from tensor braided Hopf algebras. J Algebr Comb 52, 405–453 (2020). https://doi.org/10.1007/s10801-019-00907-5
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DOI: https://doi.org/10.1007/s10801-019-00907-5