Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]

Abstract

The main result of the paper is classification of topological Lie bialgebra structures on the Lie algebra ${\mathfrak {g}}[\![x]\!]$, where ${\mathfrak {g}}$ is a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0. We introduce the notion of a topological Manin pair $(L, {\mathfrak {g}}[\![x]\!])$ and present their classification by relating them to trace extensions of $F[\![x]\!]$. Then we recall the classification of topological doubles of Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ and view it as a special case of the classification of Manin pairs. The classification of topological doubles states that up to an appropriate equivalence there are only three non-trivial doubles. It is proven that topological Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ are in bijection with certain Lagrangian Lie subalgebras of the corresponding doubles. We then attach algebro-geometric data to such Lagrangian subalgebras and, in this way, obtain a classification of all topological Lie bialgebra structures with non-trivial doubles. For $F = {\mathbb {C}}$ the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang–Baxter equation.

Appendices

Appendix A. List of notations

Symbol	Meaning
$F$	Algebraically closed field of characteristic 0
${\mathfrak {g}}$	Finite-dimensional simple Lie algebra over $F$
$\kappa$	The Killing form on ${\mathfrak {g}}$
$\Omega$	The quadratic Casimir element in ${\mathfrak {g}}\otimes {\mathfrak {g}}$
$dr$	The differential of $r \in {\mathfrak {g}}\otimes {\mathfrak {g}}$, given by $dr(a) = [a \otimes 1 + 1 \otimes a, r]$
$V^{\vee }$	Dual of a vector space $V$
$V'$	Continuous dual of a topological vector space $V$
$\mathord {{{\,\textrm{ad}\,}}}$	The adjoint representation of ${\mathfrak {g}}$ on itself, $\mathord {{{\,\textrm{ad}\,}}}_x(y) = [x,y]$
$\mathord {{{\,\textrm{ad}\,}}}^*$	The co-adjoint representation of ${\mathfrak {g}}$ on ${\mathfrak {g}}^{\vee }$, $\mathord {{{\,\textrm{ad}\,}}}^*_x(f)(y) = -f([x,y])$
$V[x]$	Polynomials in one variable with coefficients in a vector space $V$
$V[\![x]\!]$	Formal Taylor power series in one variable with coefficients in a vector space $V$
$V(\!(x)\!)$	Formal Laurent power series in one variable with coefficients in a vector space $V$
$\dotplus$	Direct sum of vector spaces
$\text {Alt}$	Cyclic sum of triple tensors: $\text {Alt}(a\otimes b \otimes c) = a \otimes b \otimes c + b \otimes c\otimes a + c \otimes a \otimes c$
$\text {CYB}$	Left-hand side of the classical Yang–Baxter equation in (4.2)
${\mathfrak {D}}(L,\delta )$	(Topological) double of a (topological) Lie bialgebra $(L,\delta )$
$r_i$	$i$-th standard $r$-matrix according to Eq. (5.2), $i \in \{0,1,2,3\}$
$\delta _i$	$i$-th standard Lie bialgebra structure defined by $r_i$, $i \in \{0,1,2,3\}$
${\mathfrak {D}}_i$	Topological double of $({\mathfrak {g}}[\![x]\!],\delta _i)$ : ${\mathfrak {D}}_i :={\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^{i-1}{\mathfrak {g}}[x]$, $i \in \{1,2,3\}$
$B_i$	Bilinear form on ${\mathfrak {D}}_i$ given in Eq. (1.8), $i \in \{1,2,3\}$
${\mathcal {K}}_i$	Bilinear form of ${\mathfrak {g}}(\!(x)\!)$ defined in Eq. (3.30), $i \in \{1,2,3\}$
${\overline{\otimes }}$	Tensor product, completed with respect to the projective tensor product topology
$A(n,\alpha ),A(\infty )$	Trace extensions of ${\mathfrak {g}}(\!(x)\!)$ from Example 3.6, Example 3.5 and Example 3.7
$\text {Aut}_{R-\text {Alg}}(L)$	$R$-linear Lie algebra automorphisms of a Lie algebra $L$ over a ring $R$
$\text {Aut}_0(F[\![x]\!])$	The group of coordinate transformations $x \mapsto x + a_2 x^2 + \cdots$

Appendix B. Different types of equivalence

Now we give a brief description of different equivalence notions used in this paper. We denote by $F$ an algebraically closed field of characteristic $0$.

$(L, \delta _1) \cong (L, \delta _2)$	Isomorphism of two topological Lie bialgebra structures	There exists $\varphi \in \text {Aut}_{F\text {-LieAlg}}(L)$ such that $\varphi$ and its dual $\varphi '$ are homeomorphisms and $(\varphi \,{\overline{\otimes }}\,\varphi )\delta _1 = \delta _2 \varphi$
$(L, \delta _1) \sim (L, \delta _2)$	Equivalence of two topological Lie bialgebra structures	There exists a constant $\xi \in F^\times$ such that $(L, \xi \delta _1) \cong (L, \delta _2)$
$(L, L_+, L_-) \cong (M, M_+, M_-)$	Isomorphism of two topological Manin triples	There exists a Lie algebra isomorphism $\varphi :L \rightarrow M$ such that $\varphi$ is a homeomorphism, $\varphi (L_\pm ) = M_\pm$ and it intertwines the corresponding forms
$(L, L_+, L_-) \sim (M, M_+, M_-)$	Equivalence of two topological Manin triples	There exists a constant $\xi \in F^\times$ such that the Manin triple $(L, L_+, L_-)$ with form $\xi B_L$ is isomorphic to $(M, M_+, M_-)$ with form $B_M$
$(A_1,t_1) \sim (A_2,t_2)$	Equivalence of two trace extensions of $F[\![x]\!]$	There exists an algebra isomorphism $T :A_1 \rightarrow A_2$, identical on $F[\![x]\!]$, and a constant $\xi \in F^\times$ such that $t_2 \circ T = \xi t_1$

Lie bialgebra isomorphisms of ${\mathfrak {g}}[\![x]\!]$, given by $x \mapsto a_1x + a_2x^2 + \cdots$ with $a_i \in F$ and $a_1 \ne 0$, are called coordinate transformations. We show in Theorem 3.3 that any Lie algebra automorphism of ${\mathfrak {g}}[\![x]\!]$ decomposes uniquely into a coordinate transformation and an $F[\![x]\!]$-linear Lie algebra automorphism of ${\mathfrak {g}}[\![x]\!]$.

In general, scaling and coordinate transformations do not preserve the topological double of a topological Lie bialgebra: they change the corresponding form. These equivalences are used to place each topological Lie bialgebra structure into a particular topological double. After that we work primarily with formal isomorphisms, i.e. $F[\![x]\!]$-linear Lie algebra automorphisms of ${\mathfrak {g}}[\![x]\!]$ because they leave topological doubles invariant.

Topological twists – topological analogue of classical twists – are introduced in Sect. 4. This notion allows to reduce the classification of topological Lie bialgebra structures to the classification of twists within certain topological doubles. We call two topological twists $s_1, s_2 \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ of $\delta$ formally isomorphic if the corresponding Lie bialgebra structures $\delta + ds_1$ and $\delta + ds_2$ are formally isomorphic.

In Sect. 3.4 we explain that there are only three non-trivial topological doubles of topological Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$. They are denoted by ${\mathfrak {D}}({\mathfrak {g}}[\![x]\!], \delta _i)$, $i \in \{1,2,3 \}$. Each topological twist of $\delta _i$ is completely determined by a certain Lagrangian Lie subalgebra of ${\mathfrak {D}}({\mathfrak {g}}[\![x]\!], \delta _i)$. Conversely, every Lagrangian Lie subalgebra of ${\mathfrak {D}}({\mathfrak {g}}[\![x]\!], \delta _i)$, complementary to ${\mathfrak {g}}[\![x]\!]$, defines a topological twist of $\delta _i$. Two Lagrangian Lie subalgebras $W_1, W_2 \subseteq {\mathfrak {D}}({\mathfrak {g}}[\![x]\!], \delta _i)$ are said to be formally isomorphic if there is an $F[\![x]\!]$-linear automorphism $\phi$ of ${\mathfrak {g}}[\![x]\!]$ such that

$$\begin{aligned} W_2 = (\phi \times [\phi ])(W_1). \end{aligned}$$

Another important notion tightly related to topological twists is the notion of a formal $r$-matrix; see Sect. 5. Again, each topological twist of $\delta _i$ gives rise to a formal $r$-matrix $-r_i + t$ and conversely, any formal $r$-matrix of the form $-r_i + t$ defines a topological twist of $\delta _i$. Two formal $r$-matrices $r_1$ and $r_2$ are called formally (resp. polynomially) gauge equivalent if there is an element $\varphi$ in ${{\,\textrm{Aut}\,}}_{F[\![x]\!]\text {-LieAlg}}({\mathfrak {g}}[\![x]\!])$ (resp. in ${{\,\textrm{Aut}\,}}_{F[x]\text {-LieAlg}}({\mathfrak {g}}[x])$) such that

$$\begin{aligned} (\varphi (x) \otimes \varphi (y))r_1(x,y) = r_2(x,y). \end{aligned}$$

Similarly, classical $r$-matrices $r_1$ and $r_2$ are called holomorphically gauge equivalent if there is a holomorphic function $\varphi :U \subseteq {\mathbb {C}}\rightarrow {{\,\textrm{Aut}\,}}({\mathfrak {g}})$ such that $(\varphi (x) \otimes \varphi (y))r_1(x,y) = r_2(x,y)$.

The statement of Theorem 5.9 tells us that the relations between the three objects mentioned above preserve equivalences.