Topological Manin pairs and $(n,s)$-type series

Abstract

Lie subalgebras of $L = {\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]$, complementary to the diagonal embedding $\Delta$ of ${\mathfrak {g}}[\![x]\!]$ and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series ${\mathfrak {g}}[\![x]\!]$. In this work we consider arbitrary subspaces of L complementary to $\Delta$ and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$. Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

1 Introduction

Let $F$ be an algebraically closed field of characteristic $0$ equipped with the discrete topology and ${\mathfrak {g}}$ be a simple Lie algebra over $F$. We define the Lie algebra ${\mathfrak {g}}[\![x]\!]$ to be the space ${\mathfrak {g}}\otimes F[\![x]\!]$ with the bracket

$$\begin{aligned}{}[a \otimes f, b \otimes g] = [a,b] \otimes fg \end{aligned}$$

and we equip it with the $(x)$-adic topology. The continuous dual of ${\mathfrak {g}}[\![x]\!]$ is denoted by ${\mathfrak {g}}[\![x]\!]'$ and it is endowed with the discrete topology.

A topological Manin pair is a pair $(L, {\mathfrak {g}}[\![x]\!])$ where

• 1. $L$ is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form $B$;

• 2. ${\mathfrak {g}}[\![x]\!] \subset L$ is a Lagrangian subalgebra with respect to $B$;

• 3. for any continuous functional $T :{\mathfrak {g}}[\![x]\!] \rightarrow F$ there is $f \in L$ such that $T = B(f, -)$.

Topological Manin pairs were classified in [1] using the tools from [8]. More precisely, if $(L, {\mathfrak {g}}[\![x]\!])$ is a topological Manin pair, then $L$ is isomorphic, as a Lie algebra with form, to either $L(\infty )$ or $L(n, \alpha )$. In this paper we consider only the ”non-degenerate” case, namely $L \cong L(n,\alpha )$.

As a Lie algebra

$$\begin{aligned} L(n,\alpha ) = {\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]. \end{aligned}$$

The bilinear form B on $L(n,\alpha )$ is completely determined by the sequence $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$. For example, when $n = 0$ the form is given by

$$\begin{aligned} B(a \otimes f, b \otimes g) = \kappa (a,b)\text {res}_0 \left\{ \alpha (x) fg \right\} , \end{aligned}$$

where $\kappa$ is the Killing form on ${\mathfrak {g}}$ and $\alpha (x) :=1 + \alpha _{-2}x + \alpha _{-3}x^2 + \dots \in F(\!(x)\!)$. In case $n > 0$ the form is given by a similar formula; see Sect. 2.

It was established in [1], that the following objects are in one-to-one correspondence

• Lagrangian subalgebras $W \subseteq L(n,0)$, $0 \le n \le 2$, complementary to the diagonal

  $$\begin{aligned} \Delta :=\{ (f,[f]) \mid f \in {\mathfrak {g}}[\![x]\!] \}, \end{aligned}$$

  i.e. $\Delta \dotplus W = L(n,0)$;

• non-degenerate topological Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ and

• formal solutions to the classical Yang-Baxter equation (CYBE) in the form

  $$\begin{aligned} \frac{y^n \Omega }{x-y} + g(x,y) = \Omega \sum _{k \ge 0} x^{-k - 1} y^{k+n} + g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!], \end{aligned}$$

  (1)

  where $\Omega \in {\mathfrak {g}}\otimes {\mathfrak {g}}$ is the quadratic Casimir element and $g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$.

Furthermore, the proof of the above-mentioned correspondence reveals that series Eq. (1) can be viewed as a generating series for the corresponding subalgebra W. The present paper can be thus considered as a continuation of [1], where we extend the preceding correspondence using series of type (n, s).

To define a series of type (n, s) fix a basis $\{b_i \}_{i=1}^{d}$ of ${\mathfrak {g}}$, orthonormal with respect to its Killing form $\kappa$, and interpret $y^n\Omega /(x-y)$ as a series

$$\begin{aligned} \begin{aligned} \frac{y^n\Omega }{x-y} = \sum _{k = 0}^{\infty }\sum _{i = 1}^d w_{k,i} \otimes b_i y^k \in \big (({\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]) \otimes {\mathfrak {g}}\big )[\![y]\!]. \end{aligned} \end{aligned}$$

(2)

This expression might be understood as a Taylor series expansion. Elements $w_{k,i} \in {\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]$ are presented explicitly in Eq. (20). A series of type $(n,s)$ is a series of the form

$$\begin{aligned} \frac{s(x)y^n \Omega }{x-y} + g(x,y) \in \big (({\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]) \otimes {\mathfrak {g}}\big )[\![y]\!], \end{aligned}$$

(3)

where $s \in F[\![x]\!]^\times$ and $g \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$; See Definition 3.2. For each series $r$ of type $(n,s)$ we define another series ${\overline{r}}$ of the same type as follows

$$\begin{aligned} {\overline{r}} :=\frac{s(y)x^n\Omega }{x-y} - \tau (g(y,x)), \end{aligned}$$

(4)

where $\tau$ is the $F[\![x,y]\!]$-linear extension of the map $a \otimes b \mapsto b \otimes a$.

Each series of type (n, s) produces a subspace of ${\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]$ complementary to the diagonal embedding $\Delta$ of ${\mathfrak {g}}[\![x]\!]$. The following results generalize the above-mentioned correspondence from [1].

Theorem A

Let $n \in {\mathbb {Z}}_{\ge 0}$ and $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ be an arbitrary sequence with the corresponding series $\alpha (x) :=x^{-n} + \alpha _{n-2}x^{-n+1} + \dots + \alpha _0 x^{-1} + \dots \in F(\!(x)\!)$. For any $(n,s)$-type series

$$\begin{aligned} r = \sum _{k = 0}^{\infty }\sum _{i = 1}^d f_{k,i} \otimes b_i y^k \in \big (({\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]) \otimes {\mathfrak {g}}\big )[\![y]\!] \end{aligned}$$

(5)

define the space

$$\begin{aligned} W(r) :=\text {span}_F\{ f_{k,i} \mid k \ge 0, \, 1 \le i \le d \} \subseteq {\mathfrak {g}}(\!(x)\!) \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]. \end{aligned}$$

(6)

The following results are true:

• 1. $W$ defines a bijection between series of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ and subspaces $V \subset L(n,\alpha )$ complementary to the diagonal $\Delta$, i.e. $L(n,\alpha ) = \Delta \dotplus V;$

• 2. For any series $r$ of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ we have $W(r)^\perp = W({\overline{r}})$ inside $L(n, \alpha )$;

• 3. Any series $r$ of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ satisfies $\text {GCYB}(r) = \psi$ (see Definition 3.5 for the meaning of $\text {GCYB}(r)$), where $\psi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![x_1,x_2,x_3]\!]$ is defined by

  $$\begin{aligned}B(v_1 \otimes v_2 \otimes v_3,\psi ) = B(v_1,[v_2,v_3]) \end{aligned}$$

  for all $v_1 \in W({\overline{r}}),v_2,v_3 \in W(r)$.

In particular, considering the case when $r$ is skew-symmetric, meaning $r = {\overline{r}}$, or when $\psi = 0$ we get the following correspondences.

Corollary B

Let $n \in {\mathbb {Z}}_{\ge 0}$, $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ and $W$ be the map from Theorem A. Then

• 1. $W$ defines a bijection between skew-symmtric $\left( n,\frac{1}{x^n \alpha (x)}\right)$-type series and Lagrangian subspaces $V \subseteq L(n,\alpha )$, complementary to the diagonal $\Delta$;

• 2. $W$ defines a bijection between $\left( n,\frac{1}{x^n \alpha (x)}\right)$-type series solving the GCYBE and subalgebras $V \subseteq L(n,\alpha )$ complementary to the diagonal $\Delta$.

Observe that an (n, s)-type series produces a subspace of $L(n,\alpha )$ for any sequence $\alpha$. However, to obtain the compatibility with the form, given by $\alpha$, we need the equality $s(x) = 1/(x^n\alpha (x))$. In this case, the components $f_{k,i}$ and $b_iy^k$ of the series become dual bases for W(r) and $\Delta$ respectively.

The requirement on a series $r$ of type $(n,s)$ to solve the CYBE is equivalent to being skew-symmetric and to solve the GCYBE. Together with Corollary B this implies that Lagrangian subalgebras $W \subset L(n,\alpha )$, satisfying $W \dotplus \Delta = L(n,\alpha )$, are in bijection with $(n,1/(x^n \alpha (x)))$-type series solving the classical Yang-Baxter equation. These correspondences are schematically depicted in Fig. 1.

Fig. 1

Series-subspaces correspondence

Remark 1.1

Let r be a series of type (n, s). Applying the projection $(a,b) \otimes c \mapsto a \otimes c$ onto the left component to r we obtain the series

$$\begin{aligned} r_{\text {proj}} = \frac{s(x)y^n \Omega }{x-y} + g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!]. \end{aligned}$$

(7)

Conversely, starting with a series $r_{\text {proj}}$ of the form Eq. (7), we can obtain an (n, s)-type series r by taking two Taylor series expansions of $r_{\text {proj}}$ at $x = 0$ and $y = 0$ respectively and then constructing r by combining the coefficients of $b_i y^k$, $k \ge 0$, in these expansions. These two constructions are inverse to each other and hence both r and its projection $r_{\text {proj}}$ contain exactly the same information. Consequently, all the statements made for (n, s)-type series can be stated in terms of their projections onto the left component and vice versa. In cotrast with [1], in this paper we give preference to series of type (n, s) rather than to their projections, because the statement that series of type (n, s) generate subspaces of $L(n,\alpha )$ becomes transparent.

Reinterpreting the results of [1] in terms of $(n,s)$-type series we see that skew-symmetric series of type $(n, 1/(x^n \alpha (x))$, that also solve the GCYBE, exist only for $n = 0, 1$ and $n = 2$ with $\alpha _0 = 0$.

Lagrangian subalgebras of $L(n,\alpha )$, complementary to $\Delta$, correspond to topological Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$. If we instead consider Lagrangian subspaces (not necessarily subalgebras) of $L(n,\alpha )$, we get so called (non-degenerate) topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$. A topological quasi-Lie bialgebra structure on ${\mathfrak {g}}[\![x]\!]$ consists of

• a skew-symmetric continuous linear map $\delta :{\mathfrak {g}}[\![x]\!] \rightarrow ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ and

• a skew-symmetric element $\varphi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y,z]\!]$,

which are subject to the following three conditions

• $\delta ([a,b]) = [a \otimes 1 + 1 \otimes a, \delta (b)] - [b \otimes 1 + 1 \otimes b, \delta (a)]$, i.e. $\delta$ is a $1$-cocycle;

• $\frac{1}{2}\text {Alt}((\delta \otimes 1)\delta (a)) = [a \otimes 1 \otimes 1 + 1 \otimes a \otimes 1 + 1 \otimes 1 \otimes a, \varphi ]$;

• $\text {Alt}((\delta \otimes 1 \otimes 1)\varphi ) = 0$,

where $\text {Alt}(x_1 \otimes \dots \otimes x_n) :=\sum _{\sigma \in S_n} \text {sgn}(\sigma )x_{\sigma (1)} \otimes \dots \otimes x_{\sigma (n)}$.

Following [5] we prove the following direct relation between $\delta$, $\varphi$ and skew-symmetric $(n,s)$-type series $r$.

Proposition C

There is a bijection between topological quasi-Lie bialgebras and skew-symmetric $(n,s)$-type series. Let $r$ be the $(n,s)$-type series corresponding to $({\mathfrak {g}}[\![x]\!], \delta , \varphi )$, then, under the identification ${\mathfrak {g}}[\![x]\!] \cong \Delta$, we have the following identities:

• $[a \otimes 1 + 1 \otimes a, r] = -\delta (a)$ for any $a \in {\mathfrak {g}}[\![x]\!]$ and

• $\text {CYB}(r) = -\varphi$.

The same is true if $r$ is interpreted as an element in $({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!]$.

In view of this result we call skew-symmetric $(n,s)$-type series quasi-$r$-matrices.

Repeating the ideas from [7] and [5] we show that topological quasi-Lie bialgebras can be twisted similar to topological Lie bialgebras. More precisely, if $\delta$ is a quasi-Lie bialgebra structure on ${\mathfrak {g}}[\![x]\!]$, given by the Lagrangian subspace $W$, and $s :=\sum _{i} a_i \otimes b^i \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ is an arbitrary skew-symmetric tensor, then

$$\begin{aligned} W_s :=\left\{ \sum _{i} B(b^i, w)a_i - w \mid w \in W \right\} \end{aligned}$$

(8)

is another (twisted) Lagrangian subspace complementary to the diagonal. This observation implies, that in order to classify all topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ up to twisting it is enough to find one single Lagrangian subspace within each $L(n,\alpha )$. Moreover, it was shown in [1] that substitutions of the form $x \mapsto x + a_2 x^2 + a_3 x^3 + \dots$, $a_i \in F$, allow us to assume that $\alpha$ has the form

$$\begin{aligned} \alpha = (\dots , 0, \alpha _0, 0, \dots , 0). \end{aligned}$$

Lagrangian subspaces for such $L(n,\alpha )$ are constructed in Sect. 4.1.

Using Theorem A and Proposition C we explain how twisting of a Lagrangian subspace $W \subset L(n,\alpha )$ is seen at the level of $\delta$ and the corresponding quasi-$r$-matrix $r$.

Corollary D

Let $({\mathfrak {g}}[\![x]\!], \delta , \varphi )$ be a topological quasi-Lie bialgebra structure corresponding to the quasi-$r$-matrix $r$. If we twist $W(r)$ with a skew-symmetric tensor $s$ we obtain another topological quasi-Lie bialgebra $({\mathfrak {g}}[\![x]\!], \delta _s, \varphi _s)$, such that

• 1. $W(r)_s = W(r-s)$;

• 2. $\delta _s = \delta + ds$;

• 3. $\varphi _s = \varphi + \text {CYB}(s) - \frac{1}{2}\text {Alt}((\delta \otimes 1)s)$.

Therefore, to describe all quasi-$r$-matrices up to twisting it is enough to find one single quasi-$r$-matrix for each $L(n,\alpha )$. We achieve that goal in Sect. 4.2 by writing out explicitly series of type $(n,s)$ for subspaces from Sect. 4.1.

The results above, in particular, show that if $r$ is a quasi-$r$-matrix and $\delta (a) :=[a \otimes 1 + 1 \otimes a, r]$, then the condition

$$\begin{aligned} \text {Alt}((\delta \otimes 1 \otimes 1) \text {CYB}(r)) = 0 \end{aligned}$$

(9)

is trivially satisfied.

We conclude the paper by using Theorem A for construction of Lie algebra splittings $\Delta \dotplus W = L(n,\alpha )$ and the corresponding $(n,s)$-type series, which we call generalized $r$-matrices. These constructions are important in the theory of integrable systems because of their use in the Adler-Konstant-Symes (AKS) scheme and the so-called $r$-matrix approach; see [4, 6]. The subalgebra splittings of $L(0,0)$ as well as their physical applications were considered in e.g. [9, 10].

Our first result shows that in order to obtain new generalized $r$-matrices from subalgebra splittings $L(n,\alpha ) = \Delta \dotplus W$ with $n > 2$, the subalgebra $W$ must be unbounded. Otherwise the situation can be reduced to the splitting of $L(2,\alpha )$.

Proposition E

Let $L(n,\alpha ) = \Delta \dotplus W$ for some subalgebra $W \subset L(n,\alpha )$ and $n > 2$. Assume $W$ is bounded, i.e. there is an integer $N > 0$ such that

$$\begin{aligned} x^{-N}{\mathfrak {g}}[x^{-1}] \subseteq W_+ \subseteq x^N{\mathfrak {g}}[x^{-1}], \end{aligned}$$

where $W_+$ is the projection of $W \subset L(n,\alpha ) = {\mathfrak {g}}(\!(x)\!) \oplus {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]$ on the first component ${\mathfrak {g}}(\!(x)\!)$. Then we have the inclusion

$$\begin{aligned}\{0\} \times [x^2]{\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x] \subseteq W\end{aligned}$$

and the image ${\widetilde{W}}$ under the canonical projection $L(n,\alpha ) \rightarrow L(2,\alpha )$ is a subalgebra satisfying $L(2,\alpha ) = \Delta \dotplus {\widetilde{W}}$.

Despite this result we think that bounded subalgebras $W \subset L(n,\alpha )$ complementary to $\Delta$ are still interesting, because in the case $\alpha \ne 0$ they lead to unbounded orthogonal complements $W^\perp$ which are also important in view of the AKS scheme. We give examples of subalgebras of $L(n,\alpha )$ with unbounded orthogonal complements.

2 Topological Manin pairs

Let $F$ be an algebraically closed field of characteristic $0$, ${\mathfrak {g}}$ be a finite-dimensional simple $F$-Lie algebra and ${\mathfrak {g}}[\![x]\!] :={\mathfrak {g}}\otimes F[\![x]\!]$ be the Lie algebra with the bracket defined by

$$\begin{aligned}{}[a \otimes f, b \otimes g] :=[a,b] \otimes fg, \end{aligned}$$

for all $a,b \in {\mathfrak {g}}$ and $f,g \in F[\![x]\!]$. From now on, we always endow $F$ with the discrete topology and view ${\mathfrak {g}}[\![x]\!]$ as a topological Lie algebra with the $(x)$-adic topology.

A topological Manin pair is a pair $(L, {\mathfrak {g}}[\![x]\!])$, where $L$ is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form $B$, such that

• 1. ${\mathfrak {g}}[\![x]\!] \subseteq L$ is a Lagrangian Lie subalgebra with respect to $B$;

• 2. for any continuous functional $T :{\mathfrak {g}}[\![x]\!] \rightarrow F$ there exists an element $f \in L$ such that $T = B(f, -)$.

The statements of [8, Proposition 2.9] and [1, Proposition 3.12] give a description of all topological Manin pairs. For precise formulation we need to repeat the definitions of some specific Lie algebras with forms from [1, Section 3.2] and [8, Section 2].

Definition 2.1

We define the Lie algebra $L(\infty ) :={\mathfrak {g}}\otimes A(\infty )$, where $A(\infty )$ is the unital commutative algebra with underlying space $\sum _{i \ge 0} Fa_i \dotplus F[\![x]\!]$ and multiplication given by

$$\begin{aligned} a_i a_j :=0, \ a_ix^j :=a_{i-j} \text { for } i \ge j \text { and } a_i x^j :=0 \text { otherwise}. \end{aligned}$$

Let ${{\,\textrm{t}\,}}:A \rightarrow F$ be the functional, given by ${{\,\textrm{t}\,}}(a_0) :=1$, ${{\,\textrm{t}\,}}(a_i) :=0$, $i \ge 1$ and ${{\,\textrm{t}\,}}(F[\![x]\!]) :=0$. We equip $L(\infty )$ with the symmetric non-degenerate invariant bilinear form

$$\begin{aligned}{} & {} B\left( a \otimes \left( \sum _{i \ge 0} c_i a_i, f(x)\right) , b \otimes \left( \sum _{i \ge 0} t_i a_i, g(x)\right) \right) \nonumber \\{} & {} \quad :=\kappa (a,b){{\,\textrm{t}\,}}\left( g(x)\sum _{i \ge 0} c_i a_i + f(x)\sum _{i \ge 0} t_i a_i\right) . \end{aligned}$$

(10)

Definition 2.2

Let $n \ge 1$ and $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ be an arbitrary sequence. Consider the algebra

$$\begin{aligned} A(n, \alpha ) :=F(\!(x)\!) \oplus F[x]/(x^n). \end{aligned}$$

Abusing the notation we denote the element $x^{-n} + \alpha _{n-2}x^{-n+1} + \dots + \alpha _0 x^{-1} + \dots \in F(\!(x)\!)$ with the same letter $\alpha$. Define the functional ${{\,\textrm{t}\,}}:A(n, \alpha ) \rightarrow F$ by

$$\begin{aligned} {{\,\textrm{t}\,}}(f, [p]) :=\text {res}_0 \left\{ \alpha (f- p) \right\} . \end{aligned}$$

Taking the tensor product of $A(n, \alpha )$ with ${\mathfrak {g}}$ we get the Lie algebra

$$\begin{aligned} L(n,\alpha ) :={\mathfrak {g}}\otimes A(n,\alpha ) \end{aligned}$$

(11)

which we equip with the form

$$\begin{aligned} B(a \otimes (f, [p]), b \otimes (g,[q])) :=\kappa (a,b){{\,\textrm{t}\,}}(fg,[pq]). \end{aligned}$$

(12)

It is known that the bilinear form $B$ is symmetric non-degenerate and invariant.

Definition 2.3

Take an arbitrary sequence $\alpha = (\alpha _i \in F \mid -\infty < i \le -2)$ and let $A(0, \alpha ) :=F(\!(x)\!)$. We define the functional ${{\,\textrm{t}\,}}:A(0, \alpha ) \rightarrow F$ by

$$\begin{aligned} {{\,\textrm{t}\,}}(f) :=\text {res}_0 \left\{ \alpha f \right\} , \end{aligned}$$

where $\alpha = 1 + \alpha _{-2}x + \dots \in F(\!(x)\!)$. We equip the Lie algebra $L(0, \alpha ) :={\mathfrak {g}}\otimes A(0, \alpha )$ with the bilinear form

$$\begin{aligned} B(a \otimes f, b \otimes g) :=\kappa (a,b){{\,\textrm{t}\,}}(fg), \end{aligned}$$

(13)

which is again symmetric non-degenerate and invariant. From now on we identify $F(\!(x)\!)$ with $F(\!(x)\!) \times \{ 0 \}$ and write $(f,0)$ for elements in $A(0, \alpha )$.

Definition 2.4

A series of the form $\varphi = x + a_2 x^2 + a_3 x^3 + \dots \in F[\![x]\!]$ is called a coordinate transformation. Coordinate transformations form a group $\text {Aut}_0 F[\![x]\!]$ under substitution which we view as a subgroup of automorphisms of $F[\![x]\!]$.

An element $\varphi \in \text {Aut}_0 F[\![x]\!]$ induces an automorphism of $A(n, \alpha )$ by $f/g \mapsto \varphi (f)/\varphi (g)$ and $[p] \mapsto [\varphi (p)]$ that changes the functional ${{\,\textrm{t}\,}}$ to ${{\,\textrm{t}\,}}\circ \, \varphi$. We write $A(n,\alpha )^{(\varphi )}$ for the algebra $A(n,\alpha )$ with the functional ${{\,\textrm{t}\,}}\circ \, \varphi$. It is not hard to see that for any $\varphi \in \text {Aut}_0 F[\![x]\!]$ there is a sequence $\beta$ such that $A(n,\alpha )^{(\varphi )} = A(n, \beta )$.

Let $(L, {\mathfrak {g}}[\![x]\!])$ be a topological Manin pair. According to [8, Proposition 2.9] as a Lie algebra with form $L \cong L(\infty )$ or $L \cong L(n, \alpha )$, for some $n \ge 0$ and some sequence $\alpha$. Here we identify ${\mathfrak {g}}[\![x]\!]$ with the diagonal

$$\begin{aligned} \Delta :=\{ (f, [f]) \mid f \in {\mathfrak {g}}[\![x]\!] \} \subset L(n, \alpha ). \end{aligned}$$

Moreover, we can assume that all the elements $\alpha _i$ in the sequence $\alpha$, except maybe $\alpha _0$, are $0$ by virtue of the following result.

Proposition 2.5

[1, Proposition 3.12] Let $n \ge 0$ and $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ be a sequence. There exists a $\varphi \in \text {Aut}_0F[\![x]\!]$ such that $A(n,\alpha ) \cong A(n,\beta )^{(\varphi )}$, where $\beta$ is the sequence satisfying $\beta _i = 0$ for all $i \ne 0$ and $\beta _0 = \alpha _0$.

Remark 2.6

Observe that the result of Proposition 2.5 can be interpreted in terms of a formal differential equation. Consider an arbitrary $\alpha (x) = x^{-n} + \alpha _{n-2}x^{-n+1} + \dots + \alpha _0x^{-1} + \dots \in F(\!(x)\!)$ and $\beta (x) = x^{-n} + \alpha _0x^{-1}$. Then the functionals ${{\,\textrm{t}\,}}_\alpha$ and ${{\,\textrm{t}\,}}_\beta$ defined on $A(n,\alpha )$ and $A(n,\beta )$ respectively are given by

$$\begin{aligned} {{\,\textrm{t}\,}}_{\alpha }(f,[p]) = \text {res}_0\{\alpha (f- p)\} \ \ \text { and } \ \ {{\,\textrm{t}\,}}_{\beta }(f,[p]) = \text {res}_0\{\beta (f- p)\}) \end{aligned}$$

The equality $A(n,\alpha )^{(\varphi )} = A(n,\beta )$ can be expressed as

$$\begin{aligned} \text {res}_0\{\beta (x)f(x)\} = \text {res}_0\{\alpha (x)f(\varphi (x))\} = \text {res}_0\{\alpha (\psi (x))f(x)\psi '(x)\}, \end{aligned}$$

(14)

where $\psi \in \text {Aut}_0(F[\![x]\!])$ is the compositional inverse of $\varphi$, i.e. $\varphi (\psi (x)) = x$. Since the residue pairing is non-degenerate on $F(\!(x)\!)$, we obtain

$$\begin{aligned} \alpha (\psi (x))\psi '(x) = \beta (x). \end{aligned}$$

(15)

In particular, the transformation $\varphi$ is the compositional inverse of the solution to Eq. (15).

3 Series of type $(n,s)$ and subspaces of $L(n,\alpha )$

Let $\{b_i\}_{i = 1}^d$ be an othonormal basis of ${\mathfrak {g}}$ with respect to the Killing form $\kappa$. We write $\Omega$ for the quadratic Casimir element $\sum _{i = 1}^d b_i \otimes b_i \in {\mathfrak {g}}\otimes {\mathfrak {g}}$. It satisfies the identity $[a \otimes 1 + 1 \otimes a, \Omega ] = 0$ for all $a \in {\mathfrak {g}}$.

In this section we describe a bijection between subspaces $W \subset L(n, \alpha )$ complementary to $\Delta$ and certain series. The following definition introduces convenient spaces containing these series.

Definition 3.1

We put $A_1(n,\alpha ) :=A(n, \alpha ) = F(\!(x_1)\!) \oplus F[x_1]/(x_1^n)$ and then define inductively the algebras

$$\begin{aligned} A_m(n, \alpha ) :=A_{m-1}(n, \alpha )(\!(x_m)\!) \oplus A_{m-1}(n, \alpha )[x_m]/x_m^nA_{m-1}(n, \alpha ), \ m > 1. \end{aligned}$$

(16)

The functional ${{\,\textrm{t}\,}}$ defined on $A(n, \alpha )$ extends inductively to a functional on $A_m(n, \alpha )$. More precisely,

$$\begin{aligned} {{\,\textrm{t}\,}}\left( \sum _{k \ge -N} f_k x_m^k, \sum _{\ell = 0}^{n-1} [g_\ell x_m^\ell ] \right) :=\sum _{k \ge -N} {{\,\textrm{t}\,}}(f_k){{\,\textrm{t}\,}}(x_m^k, 0) + \sum _{\ell = 0}^{n-1}{{\,\textrm{t}\,}}(g_\ell ){{\,\textrm{t}\,}}(0, [x_m]^\ell ),\nonumber \\ \end{aligned}$$

(17)

where $f_k,g_\ell \in A_{m-1}(n,\alpha )$. Since ${{\,\textrm{t}\,}}(x^n F[\![x]\!]) = 0$, the sum on the right-hand side of Eq. (17) is finite and well-defined. This allows us to extend the form $B$ on $L(n, \alpha )$ to a symmetric non-degenerate bilinear form on the ${\mathfrak {g}}$-module

$$\begin{aligned} L_m(n, \alpha ) :={\mathfrak {g}}^{\otimes m} \otimes A_m(n, \alpha ) \end{aligned}$$

(18)

by letting

$$\begin{aligned} B((a_1 \otimes \dots \otimes a_m)\otimes f,(b_1 \otimes \dots \otimes b_m)\otimes g) :={{\,\textrm{t}\,}}(fg) \prod _{k=1}^m \kappa (a_k, b_k), \end{aligned}$$

(19)

for all $a_1,\dots ,a_m,b_1,\dots ,b_m \in {\mathfrak {g}}$ and $f,g \in A_{m}(n,\alpha )$.

Fix some integer $n \ge 0$. We interpret the quotient $y^n \Omega / (x-y)$ in the following way

$$\begin{aligned} \begin{aligned}&\frac{y^n\Omega }{x-y} = \sum _{k = 0}^{n-1}\sum _{i = 1}^d b_i(0,-[x]^{(n-1)-k}) \otimes b_i(y^k, [y]^k) \\&+ \sum _{k = n}^\infty \sum _{i = 1}^d b_i(x^{(n-1)-k},0) \otimes b_i(y^k, 0)\\&= \sum _{k = 0}^{\infty }\sum _{i = 1}^d w_{k,i} \otimes b_i(y^k,[y]^k) \in \left( L(n, \alpha ) \otimes {\mathfrak {g}}\right) [\![(y, [y])]\!] \subset L_2(n, \alpha ), \end{aligned} \end{aligned}$$

(20)

where $\alpha$ is an arbitrary sequence and we write $b_i(x^\ell , [x]^m)$ meaning $b_i \otimes (x^\ell , [x]^m)$.

Definition 3.2

Since $\left( L(n, \alpha ) \otimes {\mathfrak {g}}\right) [\![(y,[y])]\!]$ is an $F[\![x]\!] \cong F[\![(x, [x])]\!]$-module and

$$\begin{aligned} ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!] \cong (\Delta \otimes {\mathfrak {g}})[\![(y, [y])]\!] \subset \left( L(n, \alpha ) \otimes {\mathfrak {g}}\right) [\![(y,[y])]\!] \end{aligned}$$

the series

$$\begin{aligned} r(x,y) = \frac{s(x) y^n \Omega }{x-y} + g(x,y), \end{aligned}$$

(21)

where $g \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ and $s \in F[\![x]\!]^\times$, is also inside $\left( L(n, \alpha ) \otimes {\mathfrak {g}}\right) [\![(y,[y])]\!]$. Series of the form Eq. (21) are called series oftype $(n,s)$.

Remark 3.3

Every series

$$\begin{aligned} r(x,y) = \frac{h(x,y)\Omega }{x-y} + g(x,y) \in L_2(n, \alpha ), \end{aligned}$$

where $h \in F[\![x,y]\!]$, $h(x,x) \ne 0$ and $g \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$, has a unique representation as a series of type $(n,s)$. Indeed, write $h(x,x) = x^n s(x)$ for some $s \in F[\![x]\!]^\times$. Then $h(x,y) - y^n s(x) = (x-y)f(x,y)$ for some $f \in F[\![x,y]\!]$. This implies that we can rewrite $r$ in the $(n, s)$ form

$$\begin{aligned} \begin{aligned} r(x,y)&= \frac{s(x) y^n\Omega }{x-y} + f(x,y)\Omega + g(x,y). \end{aligned} \end{aligned}$$

(22)

In the construction of $f$ we are using the fact that for any $F$-vector space $V$ and any element $h \in V[\![x,y]\!]$

$$\begin{aligned} h(z,z) = 0 \implies h(x,y) = (x-y)f(x,y) \end{aligned}$$

(23)

for some $f\in V[\![x,y]\!]$.

Definition 3.4

For each series $r$ of type $(n, s)$ we define another series ${\overline{r}}$ of the same type $(n,s)$ by

$$\begin{aligned} {\overline{r}}(x,y) :=\frac{s(y)x^n \Omega }{x-y} - \tau (g(y,x)) \in \left( L(n,\alpha )\otimes {\mathfrak {g}}\right) [\![(y,[y])]\!], \end{aligned}$$

(24)

where $\tau$ is the $F[\![x,y]\!]$-linear extension of the map $a \otimes b \mapsto b \otimes a$. To see that this is an $(n,s)$-type series its enough to apply the argument from Remark 3.3. Series of type $(n,s)$, satisfying $r = {\overline{r}}$, are called skew-symmetric.

Definition 3.5

The generalized classical Yang-Baxter equation (GCYBE) is the equation for an $(n,s)$-type series of the form

$$\begin{aligned} \text {GCYB}(r):= & {} [r^{12}(x_1,x_2),r^{13}(x_1,x_3)] + [r^{12}(x_1,x_2),r^{23}(x_2,x_3)] \nonumber \\{} & {} + [r^{13}(x_1,x_3),{\overline{r}}^{23}(x_2,x_3)] = 0. \end{aligned}$$

(25)

Here $(-)^{13} :L_2(n, \alpha ) \rightarrow (U({\mathfrak {g}}) \otimes U({\mathfrak {g}}) \otimes U({\mathfrak {g}}))\otimes A_3(n, \alpha )$ is the inclusion map given by

$$\begin{aligned}{} & {} a \otimes b \otimes \left( \sum _{k \ge -N} F(x_1,[x_1])x_2^k, \sum _{m = 0}^{n-1} G(x_1,[x_1])[x_2]^m \right) \\ {}{} & {} \quad \mapsto a \otimes 1 \otimes b \otimes \left( \sum _{k \ge -N} F(x_1,[x_1])x_3^k, \sum _{m = 0}^{n-1} G(x_1,[x_1])[x_3]^m \right) . \end{aligned}$$

Other inclusions are defined in a similar manner. The commutators are then taken in the associative $A_3(n,\alpha )$- algebra $(U({\mathfrak {g}}) \otimes U({\mathfrak {g}}) \otimes U({\mathfrak {g}}))\otimes A_3(n, \alpha )$.

Before formulating the main theorem of the section we note that if $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ is an arbitrary sequence and $\alpha (x) = x^{-n} + \alpha _{n-2}x^{-n+1} + \dots + \alpha _0x^{-1} + \dots \in F(\!(x)\!)$ is the corresponding series, then $x^n \alpha (x) \in F[\![x]\!]^\times$.

Theorem 3.6

Let $n \in {\mathbb {Z}}_{\ge 0}$ and $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ be an arbitrary sequence with the corresponding series $\alpha (x) \in F(\!(x)\!)$. Consider the map

$$\begin{aligned} W :L_2(n,\alpha ) \longrightarrow \{V \subset L(n,\alpha )\mid V \text { is a subspace}\} \end{aligned}$$

given by

$$\begin{aligned}{} & {} \sum _{i, j} b_i \otimes b_j \otimes \left( \sum _{k \ge -N_i} (f^{ij}_k, [p^{ij}_k])x^k, \sum _{m=0}^{n-1} (g^{ij}_m, [q^{ij}_m]) [x]^m \right) \nonumber \\{} & {} \quad \mapsto \text {span}_F \left\{ b_i(f^{ij}_k, [p^{ij}_k]) \mid k \ge -N, 1 \le i, j \le d \right\} . \end{aligned}$$

The following results are true:

• 1. $W$ defines a bijection between series of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ and subspaces $V \subseteq L(n,\alpha )$ complementary to the diagonal $\Delta$, i.e. $L(n,\alpha ) = \Delta \dotplus V;$

• 2. For any series $r$ of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ we have $W(r)^\perp = W({\overline{r}})$ inside $L(n, \alpha )$;

• 3. Any series $r$ of type $\left( n,\frac{1}{x^n \alpha (x)}\right)$ satisfies $\text {GCYB}(r) = \psi$, where

  $$\begin{aligned} \psi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![(x_1, [x_1]),(x_2, [x_2]),(x_3, [x_3])]\!] \end{aligned}$$

  is defined by

  $$\begin{aligned}B(v_1 \otimes v_2 \otimes v_3,\psi ) = B(v_1,[v_2,v_3])\end{aligned}$$

  for all $v_1 \in W({\overline{r}}),v_2,v_3 \in W(r)$.

Proof

Fix an $\left( n,\frac{1}{x^n \alpha (x)}\right)$-type series inside $(L(n,\alpha ) \otimes {\mathfrak {g}})[\![(y,[y])]\!]$

$$\begin{aligned} \begin{aligned} r(x,y)&= \frac{1}{x^n \alpha (x)} \frac{y^n \Omega }{x-y} + g(x,y) \\&= \sum _{k = 0}^{\infty } \sum _{i = 1}^d s_{k,i} \otimes b_i(y^k,[y]^k) + \sum _{k = 0}^{\infty } \sum _{i = 1}^d g_{k,i} \otimes b_i(y^k,[y]^k). \end{aligned} \end{aligned}$$

It is easy to see that

$$\begin{aligned} U :=\text {span}_{F} \{ w_{k,i} \mid k \ge 0, 1 \le k \le d \} \subset L(n, \alpha ), \end{aligned}$$

where $w_{k,i}$ are defined in Eq. (20), satisfies the condition $\Delta \dotplus U = L(n, \alpha )$. Since $s :=\frac{1}{x^n \alpha (x)}$ is invertible, we have $sU \dotplus s\Delta = sU \dotplus \Delta = L(n, \alpha )$. In other words, the space

$$\begin{aligned} sU = \text {span}_{F}\{ s_{k,i} = sw_{k,i} \mid k \ge 0, 1 \le k \le d \} \subset L(n, \alpha ) \end{aligned}$$

(26)

is also complementary to the diagonal. Finally, since $g_{k,i} \in \Delta$ the space

$$\begin{aligned} W(r) = \text {span}_F\{ sw_{k,i} + g_{k,i} \mid k \ge 0, 1 \le k \le d \} \subset L(n, \alpha ) \end{aligned}$$

is complementary to the diagonal. Conversely, if $V \subset L(n, \alpha )$ satisfies $V \dotplus \Delta = L(n, \alpha )$, then for each $k\ge 0$ and $1 \le i \le d$ we can find a unique $g_{k,i} \in \Delta$ such that $s w_{k,i} + g_{k,i} \in V$. Define the $(n,s)$ series $r_V$ by

$$\begin{aligned} r_V(x,y) = \sum _{k \ge 0} \sum _{i = 1}^d (s w_{k,i} + g_{k,i}) \otimes b_i (y^k,[y]^k). \end{aligned}$$

It is now clear, that $W(r_V) = V$. These constructions establish the bijection in part 1.

To prove the second statement, observe that

$$\begin{aligned} B(s w_{k,i},b_j (y^\ell ,[y]^\ell )) = \delta _{i,j}\delta _{k,\ell }. \end{aligned}$$

(27)

Furthermore, the straightforward calculation shows that

$$\begin{aligned} \begin{aligned} B(s w_{k,i},s w_{\ell ,j})&= {\left\{ \begin{array}{ll} - \text {res}_0 \left\{ s x^{(n-1) - k-\ell - 1} \right\} &{} \text { if } \, i = j \, \text { and } \, 0 \le k,\ell \le n-1, \\ \text {res}_0 \left\{ s x^{(n-1) - k-\ell - 1} \right\} &{} \text { if } \, i = j \, \text { and } \, k,\ell \ge n, \\ 0 &{} \text { otherwise}, \end{array}\right. } \\&= {\left\{ \begin{array}{ll} - s_{k + \ell - n + 1} &{} \text { if } \, i = j, \, 0 \le k,\ell \le n-1 \, \text { and } \, k + \ell \ge n-1, \\ s_{k + \ell - n + 1} &{} \text { if } \, i = j \, \text { and }k \,,\ell \ge n, \\ 0 &{} \text { otherwise}, \end{array}\right. } \end{aligned} \end{aligned}$$

where $s(x) = \sum _{k = 0}^\infty s_k x^k$. We write

$$\begin{aligned} \begin{aligned} {\overline{r}}(x,y)&= \frac{s(y)x^n \Omega }{x-y} - \tau (g(y,x)) = \frac{s(x)y^n \Omega }{x-y} - \frac{(s(x)y^n - s(y)x^n)\Omega }{x-y} -\tau (g(y,x)) \\&= \sum _{k \ge 0} \sum _{i = 1}^d (s w_{k,i} + {\overline{g}}_{k,i}) \otimes b_i (y^k,[y]^k). \end{aligned} \end{aligned}$$

Consider the quotient

$$\begin{aligned} \begin{aligned}&\frac{(s(x)y^n-s(y)x^n)\Omega }{x-y} = \frac{y^n(s(x) - s(y))\Omega }{x-y} - \frac{s(y)(x^n - y^n) \Omega }{x-y}\\&= \sum _{k \ge 0} \sum _{i = 1}^d s_k \left( \sum _{\ell = 1}^k b_i (x^{k-\ell }, [x]^{k - \ell }) \otimes b_i (y^{(n-1)+\ell }, [y]^{(n-1)+\ell })\right. \\&- \left. \sum _{\ell = 1}^n b_i (x^{n-\ell }, [x]^{n-\ell }) \otimes b_i(y^{k+\ell - 1}, [y]^{k+\ell - 1}) \right) . \end{aligned} \end{aligned}$$

The coefficient of $b_i(x^{k}, [x]^k) \otimes b_i (y^{\ell }, [y]^{\ell })$ in the expression above is

$$\begin{aligned} \begin{aligned} -s_{k+\ell -(n-1)}&\ \text { if } \, 0 \le k, \ell \le n-1 \, \text { and } \, k + \ell \ge n-1, \\ s_{k+\ell -(n-1)}&\ \text { if } \, k, \ell \ge n, \end{aligned} \end{aligned}$$

which coincides with $B(s w_{k,i}, s w_{\ell , i})$. If we now expand the coefficients $g_{k,i}$ in the following way

$$\begin{aligned} g_{k,i} = \sum _{\ell \ge 0} \sum _{j = 1}^d g_{k,i}^{\ell ,j} b_j (x^\ell ,[x]^\ell ), \end{aligned}$$

the coefficients ${\overline{g}}_{k,i}$ can be rewritten as

$$\begin{aligned} {\overline{g}}_{k,i} = - \sum _{\ell \ge 0} \sum _{j = 1}^d (g^{k,i}_{\ell , j} + B(s w_{k,i}, s w_{\ell , j})) b_i (x^k, [x]^k) \otimes b_j (y^\ell , [y]^\ell ). \end{aligned}$$

Combining all the results above we obtain the desired equality

$$\begin{aligned} \begin{aligned} B(s w_{k,i} + g_{k,i},s w_{\ell ,j} + {\overline{g}}_{\ell ,j})&= B(sw_{k,i}, sw_{\ell , j})+ B(sw_{k,i},{\overline{g}}_{\ell ,j})\\&\quad +B(g_{k,i}, sw_{\ell , j}) +B(g_{k,i},{\overline{g}}_{\ell ,j}) \\&=B(sw_{k,i}, sw_{\ell , j}) {+} ({-}g_{k,i}^{\ell ,j} {-} B(sw_{k,i}, sw_{\ell , j})) {+} g_{k,i}^{\ell ,j} {+} 0 \\&=0 \end{aligned} \end{aligned}$$

which completes the proof of the second statement.

Using the same technique as in [2, Section 1], one can prove that

$$\begin{aligned} \psi :=\text {GCYB}(r) \in (\Delta \otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![(x_2,[x_2]), (x_3,[x_3])]\!] \end{aligned}$$

for any series $r$ of type $(n,s)$. Define $r_{k,i} :=s w_{k,i} + g_{k,i}$ and ${\overline{r}}_{k,i} :=s w_{k,i} + {\overline{g}}_{k,i}$ and rewrite $\text {GCYB}(r)$ as

$$\begin{aligned} \psi= & {} \sum _{k, \ell \ge 0} \sum _{i,j = 1}^d [r_{k,i}, r_{\ell , j}] \otimes b_i (x_2^k,[x_2]^k) \otimes b_j (x_3^\ell , [x_3]^\ell ) \nonumber \\{} & {} + \sum _{k \ge 0} \sum _{i = 1}^d r_{k,i} \otimes \left( [b_i (x_2^k, [x_2]^k) \otimes (1,1), r(x_2, x_3)]\right. \nonumber \\{} & {} \left. + [(1,1) \otimes b_i(x_3^k, [x_3]^k), {\overline{r}}(x_2,x_3)]\right) . \end{aligned}$$

(28)

Applying $B({\overline{r}}_{k_1,i_1} \otimes r_{k_2,i_2} \otimes r_{k_3,i_3},-)$ to the equation above, we get

$$\begin{aligned} B({\overline{r}}_{k_1,i_1} \otimes r_{k_2,i_2} \otimes r_{k_3,i_3}, \psi ) = B({\overline{r}}_{k_1,i_1},[r_{k_2,i_2},r_{k_3,i_3}]). \end{aligned}$$

(29)

This gives the last statement because $W(r)$ and $W({\overline{r}})$ are generated by $r_{k,i}$ and ${\overline{r}}_{k,i}$ respectively. $\square$

Corollary 3.7

Let $n \in {\mathbb {Z}}_{\ge 0}$, $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ and $W$ be as in Theorem 3.6. Then

• 1. $W$ defines a bijection between skew-symmtric $\left( n,\frac{1}{x^n \alpha (x)}\right)$-type series and Lagrangian subspaces $V \subseteq L(n,\alpha )$ complementary to the diagonal $\Delta$;

• 2. $W$ defines a bijection between $\left( n,\frac{1}{x^n \alpha (x)}\right)$-type series solving GCYBE and subalgebras $V \subseteq L(n,\alpha )$ complementary to the diagonal $\Delta$.

As we can see from the proof of Theorem 3.6 the element $\psi$ in $\text {GCYB}(r) = \psi$ represents the obstruction for $W(r)$ from being a Lie subalgebra. This observation raises an interesting question that we do not consider in this paper: what elements $\psi$ can appear on the right-hand side of the above-mentioned equation.

Observe that if $r$ is a series of type $(n,s)$ and it satisfies

$$\begin{aligned} \begin{aligned} \text {CYB}(r) :=&\, [r^{12}(x_1,x_2),r^{13}(x_1,x_3)] + [r^{12}(x_1,x_2),r^{23}(x_2,x_3)] \\ {}&+ [r^{13}(x_1,x_3),{r}^{23}(x_2,x_3)] = \psi \end{aligned}\end{aligned}$$

(30)

for some $\psi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y,z]\!]$, then $r$ is automatically skew-symmetric and hence solves the first equation as well. To prove that one can e.g. repeat the argument from [1, Lemma 5.2]. In other words, for a fixed $\psi$ solutions to $\text {CYB}(r) = \psi$ form a subclass of solutions to $\text {GCYB}(r) = \psi$. In particular, solutions to $\text {CYB}(r) = 0$. are exactly the skew-symmetric solutions to $\text {GCYB}(r) = 0$. We call the equation $\text {CYB}(r) = \psi$ Manin-Yang-Baxter equation.

Remark 3.8

As our notation suggest, we could have interpreted $y^n\Omega / (x-y)$ as

$$\begin{aligned} \frac{y^n \Omega }{x-y} = \sum _{k \ge 0} \sum _{i=1}^d b_i x^{-k-1} \otimes b_i y^{n + k} \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!] \end{aligned}$$

and performed all the arithmetic calculations in this form. To restore an $(n,s)$-type series from

$$\begin{aligned} \frac{s(x)y^n \Omega }{x-y} + g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!] \end{aligned}$$

(31)

we can simply view $s(x) \in F[\![x]\!]^\times$ and $g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ as elements in $F[\![(x,[x])]\!]^\times$ and $({\mathfrak {g}}\otimes {\mathfrak {g}})[\![(x,[x]),(y,[y])]\!]$ respectively and reinterpret the singular part $y^n \Omega / (x-y)$ as it was done in Eq. (20).

Conversely, to get a series of the form Eq. (31) from a series of type $(n,s)$ we can just project the latter onto the first component.

In other words, we have a bijection between $(n,s)$-type series in $L_2(n, \alpha )$ and their projections Eq. (31) onto the first component given by different interpretations of the singular part $y^n \Omega / (x-y)$.

Although, all arithmetic operations can be performed in the form Eq. (31), the construction of $W(r)$ and statements like $\Delta \cap W(r) = 0$ require us to pass to the interpretation Eq. (20). This is our main motivation to work directly with $(n,s)$-type series in $L_2(n,\alpha )$ instead of their projections.

In view of Remark 3.8, we have a new proof of [1, Corollary 5.5].

Corollary 3.9

Classical (formal) $r$-matrices, i.e. skew-symmetric elements

$$\begin{aligned} \frac{s(x)y^n \Omega }{x-y} + g(x,y) = \frac{1}{x^n \alpha (x)}\frac{y^n \Omega }{x-y} + g(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!], \end{aligned}$$

(32)

solving GCYBE, are in bijection with skew-symmetric series of type $(n,s)$ solving GCYBE and hence in bijection with Lagrangian Lie subalgebras of $L(n, \alpha )$ complementary to the diagonal $\Delta$.

The result of [1, Theorem 5.6] can be now formulated in the following way.

Corollary 3.10

Skew-symmetric series of type $\left( n, \frac{1}{x^n \alpha (x)}\right)$ that also solve GCYBE exist only for $n = 0, 1$ and $n = 2$ with $\alpha _0 = 0$.

4 Quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$

We remind that $F$ is a discrete algebraically closed field of characteristic $0$ and ${\mathfrak {g}}[\![x]\!]$ is an $F$-Lie algebra equipped with the $(x)$-adic topology.

As we now know, series of type $\left( n, 1/(x^n \alpha (x))\right)$ solving CYBE Eq. (30) are in bijection with Lagrangian subalgebras $W \subset L(n,\alpha )$ complementary to the diagonal. On the other hand, such Lagrangian subalgebras are in bijection with non-degenerate topological Lie bialgebra structures. See [1] for their definition and classification.

It turns out, that if we drop the condition on $W$ being a subalgebra, we get so called (non-degenerate) topological quasi-Lie bialgebras. This section is devoted to their classification up to topological twists and coordinate transformations.

Definition 4.1

A topological quasi-Lie bialgebra structure on ${\mathfrak {g}}[\![x]\!]$ consists of

• a skew-symmetric continuous linear map $\delta :{\mathfrak {g}}[\![x]\!] \rightarrow ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ and

• a skew-symmetric element $\varphi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y,z]\!]$,

which are subject to the following conditions

• 1. $\delta ([a,b]) = [a \otimes 1 + 1 \otimes a, \delta (b)] - [b \otimes 1 + 1 \otimes b, \delta (a)]$, i.e. $\delta$ is a $1$-cocycle;

• 2. $\frac{1}{2}\text {Alt}((\delta \otimes 1)\delta (a)) = [a \otimes 1 \otimes 1 + 1 \otimes a \otimes 1 + 1 \otimes 1 \otimes a, \varphi ]$;

• 3. $\text {Alt}((\delta \otimes 1 \otimes 1)\varphi ) = 0$,

where $\text {Alt}(x_1 \otimes \dots \otimes x_n) :=\sum _{\sigma \in S_n} \text {sgn}(\sigma )x_{\sigma (1)} \otimes \dots \otimes x_{\sigma (n)}$.

Lemma 4.2

There is a one-to-one correspondence between triples $(L, {\mathfrak {g}}[\![x]\!], W)$, where $(L, {\mathfrak {g}}[\![x]\!])$ is a topological Manin pair and $W \subset L$ is a Lagrangian subspace satisfying $W \dotplus {\mathfrak {g}}[\![x]\!] = L$, and quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$.

Proof

We start with a topological Manin pair $(L,{\mathfrak {g}}[\![x]\!])$. If $W \subset L$ is a Lagrangian subspace complementary to ${\mathfrak {g}}[\![x]\!]$, then it is easy to see that $W \cong {\mathfrak {g}}[\![x]\!]'$. Therefore, we have an isomorphism of vector spaces

$$\begin{aligned} L \cong {\mathfrak {g}}[\![x]\!] \dotplus {\mathfrak {g}}[\![x]\!]'. \end{aligned}$$

The form on $L$ under this isomorphism becomes standard evaluation form $\langle -,-\rangle$ on ${\mathfrak {g}}[\![x]\!] \dotplus {\mathfrak {g}}[\![x]\!]'$. We fix such an isomorphism.

Let us define two linear functions

$$\begin{aligned} p_1 :{\mathfrak {g}}[\![x]\!]' \otimes {\mathfrak {g}}[\![y]\!]' \rightarrow {\mathfrak {g}}[\![x]\!] \ \text { and } \ p_2 :{\mathfrak {g}}[\![x]\!]' \otimes {\mathfrak {g}}[\![y]\!]' \rightarrow {\mathfrak {g}}[\![x]\!]' \end{aligned}$$

by $[f,g] = p_1(f \otimes g) + p_2(f \otimes g)$. We put

$$\begin{aligned} \delta :=p_2^{\vee } :({\mathfrak {g}}[\![x]\!]')^\vee \cong {\mathfrak {g}}[\![x]\!] \rightarrow ({\mathfrak {g}}[\![x]\!]' \otimes {\mathfrak {g}}[\![y]\!]')^\vee \cong ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!], \end{aligned}$$

and let $\psi \in ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y,z]\!]$ be the unique element satisfying the condition

$$\begin{aligned} \langle h, [f,g]\rangle = \langle h, p_1(f \otimes g) \rangle = \langle f \otimes g \otimes h, \psi \rangle \ \text { for all } f,g,h \in {\mathfrak {g}}[\![x]\!]'.\qquad \quad \end{aligned}$$

(33)

The skew-symmetry of $p_2$ implies the skew-symmetry of $\delta$, whereas the skew-symmetry of $p_1$ and the invariance of the evaluation form yield the skew-symmetry of $\psi$.

Next, we observe that for all $a,b \in {\mathfrak {g}}[\![x]\!]$ and $f,g \in {\mathfrak {g}}[\![x]\!]'$ we have

$$\begin{aligned} \begin{aligned}&\langle [a,f], g \rangle = \langle a, [f,g] \rangle = \langle a, p_2(f \otimes g) \rangle = \langle \delta (a), f \otimes g \rangle = \langle (f \otimes 1)\delta (a), g \rangle , \\&\langle [a,f], b \rangle = - \langle f, [a,b] \rangle = - \langle f \circ \mathord {{{\,\textrm{ad}\,}}}_a, b \rangle . \end{aligned} \end{aligned}$$

In other words, the invariance of the form forces the following equality to hold

$$\begin{aligned}{}[a,f] = - f \circ \mathord {{{\,\textrm{ad}\,}}}_a + (f \otimes 1)\delta (a). \end{aligned}$$

(34)

Using Eq. (34) and non-degeneracy of the form we show that $\delta$ is a $1$-cocycle:

$$\begin{aligned} \begin{aligned} \langle \delta ([a,b]), f \otimes g \rangle&= \langle [a,b], p_2(f \otimes g) \rangle = \langle [a,b], [f, g] \rangle = \langle [[a,b],f], g \rangle \\&= \langle -[[b,f],a] - [[f,a],b], g \rangle \\&= \langle [f \circ \mathord {{{\,\textrm{ad}\,}}}_b - (f \otimes 1)\delta (b),a] - [f \circ \mathord {{{\,\textrm{ad}\,}}}_a - (f \otimes 1)\delta (a),b], g \rangle \\&= - \langle a, [f \circ \mathord {{{\,\textrm{ad}\,}}}_b, g] \rangle + \langle b, [f \circ \mathord {{{\,\textrm{ad}\,}}}_a, g] \rangle + \langle (f \otimes \mathord {{{\,\textrm{ad}\,}}}_a) \delta (b), g \rangle \\&\quad - \langle (f \otimes \mathord {{{\,\textrm{ad}\,}}}_b) \delta (a), g \rangle \\&= \langle [a \otimes 1 + 1 \otimes a, \delta (b)] - [b \otimes 1 + 1 \otimes b, \delta (a)], f \otimes g \rangle . \end{aligned}\nonumber \\ \end{aligned}$$

(35)

The $1$-cocycle condition implies that $\delta$ is continuous as it was noted in [1, Remark 3.16].

For conditions 2 and 3 from the definition of a topological quasi-Lie bialgebra consider the Jacobi identity for $f,g,h \in {\mathfrak {g}}[\![x]\!]'$:

$$\begin{aligned} 0&= [p_1(f \otimes g), h] + [p_1(g \otimes h), f] + [p_1(h \otimes f), g] \nonumber \\&\quad + p_1(p_2(f \otimes g)\otimes h) + p_1(p_2(g \otimes h)\otimes f) + p_1(p_2(h \otimes f)\otimes g) \nonumber \\&\quad + p_2(p_2(f \otimes g)\otimes h) + p_2(p_2(g \otimes h)\otimes f) + p_2(p_2(h \otimes f)\otimes g). \end{aligned}$$

(36)

We denote by $\circlearrowleft$ the summation over circular permutations of symbols $f,g$ and $h$, e.g. $\circlearrowleft \langle p_1(f \otimes g), h \rangle = \langle p_1(f \otimes g), h \rangle + \langle p_1(g \otimes h), f \rangle + \langle p_1(h \otimes f), g \rangle$. Applying $\langle -, a \rangle$ to Eq. (36) for an arbitrary $a \in {\mathfrak {g}}[\![x]\!]$ gives

$$\begin{aligned} \begin{aligned} \langle p_2(p_2 \otimes 1)(\circlearrowleft f \otimes g\otimes h), a \rangle&= - \langle \circlearrowleft [p_1(f \otimes g), h], a \rangle \\ \langle p_2 \otimes 1(\circlearrowleft f \otimes g \otimes h), \delta (a) \rangle&= \circlearrowleft \langle -h\circ \mathord {{{\,\textrm{ad}\,}}}_a, p_1(f \otimes g) \rangle \\ \langle \circlearrowleft f \otimes g \otimes h, (\delta \otimes 1)\delta (a) \rangle&= \circlearrowleft \langle f \otimes g \otimes (-h\circ \mathord {{{\,\textrm{ad}\,}}}_a), \psi \rangle \\ \langle f \otimes g \otimes h, \text {Alt} ((\delta \otimes 1)\delta (a))/2 \rangle&= - \langle f \otimes g \otimes h, [1 \otimes 1 \otimes a + 1 \otimes a \otimes 1 \\&\quad + a \otimes 1 \otimes 1,\psi ] \rangle , \end{aligned} \end{aligned}$$

where the very last identity holds because of the skew-symmetry of $\psi$. Multiplying this equality by $2$ we get the relation

$$\begin{aligned} \langle f \otimes g \otimes h, \text {Alt} ((\delta \otimes 1)\delta (a)) + 2 [1 \otimes 1 \otimes a + 1 \otimes a \otimes 1 + a \otimes 1 \otimes 1,\psi ] \rangle = 0. \end{aligned}$$

Letting $\varphi :=- \psi$ we obtain the second identity from the definition of a topological quasi-Lie bialgebra structure. Applying instead $\langle s, - \rangle$, $s \in {\mathfrak {g}}[\![x]\!]'$ to the Jacobi identity Eq. (36) we get the desired

$$\begin{aligned} \text {Alt}((\delta \otimes 1 \otimes 1) \psi ) = 0. \end{aligned}$$

Therefore, $({\mathfrak {g}}[\![x]\!], \delta , \varphi )$ is a topological quasi-Lie bialgebra.

For the converse direction, we put $L :={\mathfrak {g}}[\![x]\!] \dotplus {\mathfrak {g}}[\![x]\!]'$ with the standard evaluation form; we let $p_1$ be the unique element in $\text {Hom}_{F-\text {Vect}}({\mathfrak {g}}[\![x]\!]' \otimes {\mathfrak {g}}[\![x]\!]', {\mathfrak {g}}[\![x]\!])$ satisfying Eq. (33) with $\psi :=- \varphi$; we define $p_2 :=\delta '$, i.e. the dual map of $\delta$. The Lie bracket between two elements in ${\mathfrak {g}}[\![x]\!]'$ is given by the sum $p_1 + p_2$. Defining $[a, f]$ as in Eq. (34) the evaluation form becomes invariant and we get a topological Manin pair $(L, {\mathfrak {g}}[\![x]\!])$ with the Lagrangian subspace ${\mathfrak {g}}[\![x]\!]'$. These constructions are clearly inverse to each other. $\square$

Combining the classification of Manin pairs mentioned in Sect. 2 with Corollary 3.7 and Lemma 4.2 we get the following description of all topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$.

Lemma 4.3

There is a bijection between topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ and Lagrangian subspaces $W \subset L(n, \alpha )$ or $L(\infty )$ complementary to the diagonal $\Delta$, where $\alpha = (\alpha _i \in F \mid -\infty < i \le n-2)$ is an arbitrary sequence and $n \ge 0$. Moreover, such Lagrangian subspaces $W \subset L(n, \alpha )$ are in bijection with skew-symmetric sequences of type $(n, 1/(x^n \alpha (x)))$.

In view of this result we call skew-symmetric series of type $(n,s)$ as well as their projections onto the first component quasi-$r$-matrices. Quasi-Lie bialgebra structures can also be described using their associated quasi-$r$-matrices in the following way.

Proposition 4.4

Assume $({\mathfrak {g}}[\![x]\!], \delta , \varphi )$ is a topological quasi-Lie bialgebra and let $r \in L_2(n, \alpha )$ be the corresponding quasi-$r$-matrix given by the bijection from Lemma 4.3. Under the identification ${\mathfrak {g}}[\![(x,[x])]\!] \cong {\mathfrak {g}}[\![x]\!]$ we have the following identities:

• $[a \otimes 1 + 1 \otimes a, r] = -\delta (a)$ for any $a \in {\mathfrak {g}}[\![x]\!]$ and

• $\text {CYB}(r) =-\varphi .$

The same is true for the projection $r \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!]$.

Proof

We start, as in the proof of Lemma 4.2, by fixing an identification $L(n, \alpha ) = \Delta \dotplus W(r) \cong {\mathfrak {g}}[\![x]\!] \dotplus {\mathfrak {g}}[\![x]\!]'$. Let $\{ v_{k,i} \}$ be a basis for ${\mathfrak {g}}[\![x]\!]'$ dual to $\{\varepsilon _{k,i} :=b_i y^k \}$. Then $r = \sum _{k \ge 0} \sum _{i = 1}^d v_{k,i} \otimes \varepsilon _{k,i}$ and we have

$$\begin{aligned}{}[a \otimes 1 + 1 \otimes a, r]= & {} \sum _{k \ge 0} \sum _{i=1}^d [a, v_{k,i}] \otimes \varepsilon _{k,i} + v_{k,i} \otimes [a, \varepsilon _{k,i}] \\= & {} \sum _{k \ge 0} \sum _{i=1}^d (-v_{k,i} \circ \mathord {{{\,\textrm{ad}\,}}}_a + (v_{k,i} \otimes 1)\delta (a)) \otimes \varepsilon _{k,i} + v_{k,i} \otimes [a, \varepsilon _{k,i}]. \end{aligned}$$

Applying $\langle v_{\ell ,j} \otimes v_{m, t}, - \rangle$ to the equality above we get

$$\begin{aligned} \begin{aligned} \langle v_{\ell ,j} \otimes v_{m, t}, [a \otimes 1 + 1 \otimes a, r] \rangle&= \sum _{k \ge 0} \sum _{i=1}^d \langle v_{\ell ,j} \otimes v_{m, t}, (v_{k,i} \otimes 1)\delta (a) \otimes \varepsilon _{k,i} \rangle \\&= \langle v_{\ell ,j}, (v_{m,t} \otimes 1)\delta (a) \rangle \\&= \langle v_{\ell , j} \otimes v_{m,t}, -\delta (a) \rangle . \end{aligned} \end{aligned}$$

Applying instead $\langle \varepsilon _{\ell ,j} \otimes v_{m, t}, - \rangle$ to the same equality we obtain

$$\begin{aligned} \begin{aligned} \langle \varepsilon _{\ell ,j} \otimes v_{m, t}, [a \otimes 1 + 1 \otimes a, r] \rangle&= \sum _{k \ge 0} \sum _{i=1}^d \langle \varepsilon _{\ell ,j} \otimes v_{m, t}, (-v_{k,i} \circ \mathord {{{\,\textrm{ad}\,}}}_a) \otimes \varepsilon _{k,i} \\&\quad + v_{k,i} \otimes [a, \varepsilon _{k,i}] \rangle \\&= - \langle \varepsilon _{\ell ,j}, v_{m,t} \circ \mathord {{{\,\textrm{ad}\,}}}_a \rangle + \langle v_{m,t}, [a, \varepsilon _{\ell ,j}] \rangle \\&= 0. \end{aligned} \end{aligned}$$

This implies the desired equality $[a \otimes 1 + 1 \otimes a, r] = -\delta (a)$. The identity $\text {CYB}(r) = -\varphi$ follows from the skew-symmetry of $r$, Theorem 3.6 and the fact that $\varphi = -\psi$ according to the proof of Lemma 4.2. $\square$

Remark 4.5

Assume $r \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!]$ is a series such that

$$\begin{aligned}{}[f(x) \otimes 1 + 1 \otimes f(y), r(x,y)] \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!] \end{aligned}$$

(37)

for all $f \in {\mathfrak {g}}[\![x]\!]$. Write $r = s(x^{-1},y) + g(x,y)$, where $s \in x^{-1}({\mathfrak {g}}\otimes {\mathfrak {g}})[x^{-1}][\![y]\!]$ and $g \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$. Then, because of Eq. (37), we must have

$$\begin{aligned}{}[a \otimes 1 + 1 \otimes a, s(x^{-1}, y)] = 0 \end{aligned}$$

for all $a \in {\mathfrak {g}}$. Since the ${\mathfrak {g}}$-invariant elements of ${\mathfrak {g}}\otimes {\mathfrak {g}}$ are precisely the multiples of the quadratic Casimir element $\Omega$, we have the identity $s(x^{-1}, y) = p(x^{-1},y) \Omega$ for some $p \in x^{-1}F[x^{-1}][\![y]\!]$. Furthermore, the condition

$$\begin{aligned}{}[ax \otimes 1 + 1 \otimes ay, p(x^{-1}, y) \Omega ] = [a(x-y) \otimes 1, p(x^{-1}, y) \Omega ] \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!] \end{aligned}$$

implies $(x-y)p(x^{-1}, y) \in F[\![x,y]\!]$, meaning that there exists an $s \in F[\![y]\!]$ such that $p(x^{-1}, y) = s(y) / (x-y)$. In other words, $r$ has the form Eq. (21). This result can be considered as another motivation to study series of type $(n,s)$.

Observe that if we know one Lagrangian subspace $W_0$ inside $L \cong {\mathfrak {g}}[\![x]\!] \dotplus {\mathfrak {g}}[\![x]\!]'$ then any other Lagrangian subspace can be constructed from $W_0$ through twisting. More precisely, if $s = \sum _{i} a_i \otimes b^i \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$ is a skew-symmetric tensor, then we can associate with it a (twisted) Lagrangian subspace

$$\begin{aligned} W_s :=\left\{ \sum _{i} B(b^i, w)a_i - w \mid w \in W \right\} \subseteq L \end{aligned}$$

(38)

complementary to ${\mathfrak {g}}[\![x]\!]$. The converse is also true; for proof see [3]. In other words, the following statement holds.

Lemma 4.6

There is a bijection between Lagrangian subspaces $W \subseteq L(n, \alpha )$ or $L(\infty )$ and skew-symmetric tensors in $({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$.

Combining Lemma 4.4, Eq. (38) and the algorithm for constructing a quasi-$r$-matrix from a Lagrangian subspace $W \subset L(n, \alpha )$, $W \dotplus \Delta = L(n, \alpha )$, we obtain the following twisting rules for Lagrangian subspaces, quasi-Lie bialgebra structures and quasi-$r$-matrices.

Lemma 4.7

Let $({\mathfrak {g}}[\![x]\!], \delta , \varphi )$ be a topological quasi-Lie bialgebra structure corresponding to the quasi-$r$-matrix $r$. If we twist $W(r)$ with a skew-symmetric tensor $s$ as described in Eq. (38) we obtain another topological quasi-Lie bialgebra $({\mathfrak {g}}[\![x]\!], \delta _s, \varphi _s)$, such that

• 1. $W(r)_s = W(r-s)$;

• 2. $\delta _s = \delta + ds$;

• 3. $\varphi _s = \varphi + \text {CYB}(s) - \frac{1}{2}\text {Alt}((\delta \otimes 1)s)$,

where $ds(a) :=[a \otimes 1 + 1 \otimes a, s]$.

Remark 4.8

Since any quasi-$r$-matrix $r$ defines a topological quasi-Lie bialgebra structure $\delta (a) = [a \otimes 1 + 1 \otimes a, r]$ on ${\mathfrak {g}}[\![x]\!]$, the third condition in Definition 4.1 is trivially satisfied. In other words,

$$\begin{aligned} \text {Alt}((\delta \otimes 1 \otimes 1)\text {CYB}(r)) = 0\end{aligned}$$

for any quasi-$r$-matrix $r$.

Lemma 4.6 and Lemma 4.7 state that, in order to obtain a description of topological quasi-Lie bialgebra structures on ${\mathfrak {g}}[\![x]\!]$ up to twisting it is enough to find a single Lagrangian subspace $W_0$, complementary to ${\mathfrak {g}}[\![x]\!]$, inside $L(\infty )$ and each $L(n, \alpha )$. The same is true for the associated quasi-$r$-matrices

The case $L(\infty )$ is trivial, because by definition ${\mathfrak {g}}[\![x]\!]' = \bigoplus _{j \ge 0} {\mathfrak {g}}\otimes a_j \subseteq L(\infty )$ is a Lagrangian subalgebra (see Lemma 2.1). Similar to the Lie bialgebra case, topological quasi-Lie bialgebras corresponding to the Manin pair $(L(\infty ), {\mathfrak {g}}[\![x]\!])$ are called degenerate.

Let us now focus on non-degenerate topological quasi-Lie bialgebra structures, i.e. the ones corresponding to the Manin pairs $(L(n, \alpha ), \Delta )$. By Proposition 2.5 for each Manin pair $(L(n, \alpha ), \Delta )$ there exists an appropriate coordinate transformation that makes it into $(L(n, \beta ), \Delta )$, where $\beta _0 = \alpha _0$ and all other $\beta _i = 0$. This means, that to classify all non-degenerate topological quasi-Lie bialgebras on ${\mathfrak {g}}[\![x]\!]$, up to coordinate transformations and twisting, it is enough to construct a Lagrangian subspace $W_0$ within each $L(n, \alpha _0) :=L(n, (\dots , 0, \alpha _0, 0, \dots , 0))$ complementary to $\Delta$. Equivalently, it is enough to find a quasi-$r$-matrix of type $(n, \alpha _0)$ for any $n \ge 0$ and $\alpha _0 \in F$.

4.1 Lagrangian subspaces of $L(n, \alpha _0)$

As before we let $\{ b_i \}_{i=1}^d$ be an orthonormal basis for ${\mathfrak {g}}$ with respect to the Killing form $\kappa$. The form $B$ on $L(n, \alpha _0)$ has the following explicit form

$$\begin{aligned}{} & {} B(a \otimes (f, [p]), b \otimes (g, [q])) \nonumber \\{} & {} \qquad = {\left\{ \begin{array}{ll} \kappa (a,b)\left\{ \text {coeff}_{n-1}(fg {-} pq) {-} \alpha _0 \text {coeff}_{0}(fg - pq)\right\} &{} \, \text { if } \, n {\ge } 2, \\ \kappa (a,b)\text {coeff}_{n-1}(fg {-} pq) &{} \, \text { if } \, n {=} 0, 1. \end{array}\right. }\nonumber \\ \end{aligned}$$

(39)

We now present an explicit construction for a Lagrangian subspace of $L(n, \alpha _0)$ complementary to $\Delta$ for arbitrary $n \ge 0$ and $\alpha _0 \in F$. Using the twisting procedure from Lemma 4.7, this subspace can be twisted in order to obtain all other Lagrangian subspaces of $L(n,\alpha _0)$ complementary to $\Delta$.

n = 0: When $n = 0$, the subalgebra $W_0 :=x^{-1}{\mathfrak {g}}[\![x^{-1}]\!] \subseteq {\mathfrak {g}}(\!(x)\!)$ is known to be Lagrangian.

n = 1: For $n = 1$ it is easy to see that the subspace

$$\begin{aligned} W_0 :=\text {span}_F\{ b_i(1,-1), b_i(x^{-k}, 0) \mid k \ge 1, \, 1 \le i \le d \} \subset L(1, \alpha _0) \end{aligned}$$

(40)

is Lagrangian and complementary to the diagonal $\Delta$.

n = 2k: For even $n \ge 2$ and arbitrary $\alpha _0 \in F$ the subspace $W_0 \subset L(n, \alpha _0)$ spanned by the elements

$$\begin{aligned} \begin{aligned} b_i&\left\{ (x^{(n{-}1)-m}, 0) {-} \alpha _0 (x^{2(n-1) {-} m},0) {+} \alpha _0^2 (x^{3(n-1) {-} m},0) {-} \alpha _0^3 (x^{4(n-1) - m},0) {+} \dots \right\} , \\&\ 0 \le m \le \frac{n}{2}-1, \\ b_i&\left( 0, -[x]^{(n-1)-\ell } \right) , \ \frac{n}{2} \le \ell < n-1, \\ b_i&\left( 0, -1 + \frac{\alpha _0}{2}[x]^{n-1} \right) , \\ b_i&(x^{-k}, 0), k \ge 1, \end{aligned} \end{aligned}$$

is Lagrangian and complementary to the diagonal.

n = 2k + 1: Modifying slightly the basis for even case we obtain the following basis for $W_0 \subset L(n, \alpha _0)$ with odd $n \ge 3$:

$$\begin{aligned} \begin{aligned} b_i&\left\{ (x^{(n{-}1)-m}, 0) {-} \alpha _0 (x^{2(n-1) {-} m},0) {+} \alpha _0^2 (x^{3(n-1) {-} m},0) {-} \alpha _0^3 (x^{4(n-1) {-} m},0) {+} \dots \right\} , \\&\ 0 \le m \le \frac{n-1}{2}-1, \\ b_i&\left\{ (x^{\frac{n-1}{2}}, -[x]^{\frac{n-1}{2}}) - \alpha _0(x^{\frac{3(n-1)}{2}}, 0) + \alpha _0^2(x^{\frac{5(n-1)}{2}}, 0) - \alpha _0^3(x^{\frac{7(n-1)}{2}}, 0) + \dots \right\} , \\ b_i&( 0, -[x]^{(n-1)-\ell } ), \ \frac{n-1}{2}+1 \le \ell < n-1, \\ b_i&\left( 0, -1 + \frac{\alpha _0}{2}[x]^{n-1} \right) , \\ b_i&(x^{-k}, 0), k \ge 1. \end{aligned} \end{aligned}$$

The subspaces above were constructed by ”guessing”. However, there is an abstract procedure that produces Lagrangian subspaces for arbitrary $n$ and $\alpha$. We present it here for completeness.

The easiest skew-symmetric $(n,s)$-type series is given by

$$\begin{aligned} r(x,y)&:=\frac{1}{2}\left( \frac{s(x)y^n\Omega }{x-y} + \frac{s(y)x^n\Omega }{x-y}\right) = \frac{s(x)y^n\Omega }{x-y} + \frac{\Omega }{2}\left( \frac{s(y)x^n-s(x)y^n}{x-y}\right) \\&= \frac{s(x)y^n\Omega }{x-y} - \frac{1}{2}\sum _{k,\ell = 0}^\infty \sum _{i,j = 1}^d B(sw_{k,i},sw_{\ell ,j})b_i(x,[x])^k \otimes b_j(y,[y])^\ell , \end{aligned}$$

where we recall that

$$\begin{aligned} \begin{aligned} B(s w_{k,i},s w_{\ell ,j}) = {\left\{ \begin{array}{ll} - s_{k + \ell - n + 1} &{} \text { if } \, i = j, \, 0 \le k,\ell \le n-1 \, \text { and } \, k + \ell \ge n-1, \\ s_{k + \ell - n + 1} &{} \text { if } \, i = j \, \text { and }k \,,\ell \ge n, \\ 0 &{} \text { otherwise}. \end{array}\right. } \end{aligned} \end{aligned}$$

By Corollary 3.7 the subspace

$$\begin{aligned} \begin{aligned} W(r)&= \text {span}_F\left\{ sw_{k,i} - \frac{1}{2}\sum _{\ell = 0}^\infty B(sw_{\ell ,i},sw_{k,i})b_i(x,[x])^\ell \, \bigg | \, k \ge 0, 1 \le d \le n\right\} \\ {}&= \text {span}_F\left\{ sw_{k,i} + \frac{1}{2}\left( \sum _{\ell = 0}^{n-1}s_{k+\ell - n + 1}b_i(x,[x])^\ell - \sum _{\ell = n}^\infty s_{k+\ell - n +1}b_i(x,[x])^\ell \right) \, \bigg | \, \right. \\&\quad \left. k \ge 0, 1 \le d \le n\right\} \end{aligned} \end{aligned}$$

is Lagrangian and complementary to the diagonal. Here we used the convention that $s_k = 0$ for $k < 0$. Calculating the basis explicitly for some particular $s$ requires some effort and it may not look as friendly as the ones given above.

4.2 Quasi-$r$-matrices

The goal of this section is to describe the quasi-$r$-matrices corresponding to the Lagrangian subspaces described in the previous section. The twisting procedure from Lemma 4.7 then yields all other quasi-$r$-matrices.

The proof of Theorem 3.6 gives us an algorithm for constructing a series of type $\left( n, s(x) :=1/(x^n \alpha (x)) \right)$ from a subspace $W \subset L(n, \alpha )$ complementary to the diagonal. More precisely, the desired series is given by

$$\begin{aligned} \sum _{k \ge 0} \sum _{i = 1}^d v_{k,i} \otimes b_i (y^k, [y]^k), \end{aligned}$$

(41)

where

$$\begin{aligned} W = \text {span}_F \{ v_{k,i} \mid k \ge 0, \, 1 \le i \le d \} \, \text { and } \, B(v_{k,i}, b_j (y^{\ell },[y]^\ell )) = \delta _{i,j}\delta _{k,\ell }, \end{aligned}$$

i.e. $\{ v_{k,i} \}$ is a basis of $V$ dual to $\{b_i (y^k,[y]^k) \}$. Indeed, non-degeneracy of the form $B$ then implies that $v_{k,i}$ has the desired form $v_{k,i}= sw_{k,i} + g_{k,i}$ for some $g_{k,i} \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[\![x,y]\!]$.

Applying this idea to $W_0$’s constructed in the preceding section we get the following series.

n = 0: The classical $r$-matrix (equivalently $(0,1)$-type series) corresponding to $W_0 :=x^{-1}{\mathfrak {g}}[\![x^{-1}]\!] \subseteq {\mathfrak {g}}(\!(x)\!)$ is the Yang’s matrix $\Omega /(x-y)$.

n = 1: The quasi-$r$-matrix corresponding to $\text {span}_F\{ b_i(1,-1), b_i(x^{-k}, 0) \mid k \ge 1, \, 1 \le i \le d \} \subset L(1, \alpha _0)$ is

$$\begin{aligned}{} & {} \frac{y \Omega }{x-y} + \frac{1}{2} \sum _{i=1}^d b_i (1, -1) \otimes b_i (1,1) \in L_2(1, 1) \, \text { with the projection } \\{} & {} \frac{y \Omega }{x-y} + \frac{1}{2} \Omega \in ({\mathfrak {g}}\otimes {\mathfrak {g}})(\!(x)\!)[\![y]\!]. \end{aligned}$$

n = 2k: For even $n \ge 2$ and arbitrary $\alpha _0 \in F$ we have the following quasi-$r$-matrix

$$\begin{aligned} \begin{aligned} \frac{1}{1 + \alpha _0 x^{n-1}} \frac{y^n \Omega }{x-y}&+ \frac{\Omega }{1+\alpha _0 x^{n-1}} \sum _{0 \le m< \frac{n}{2}} x^{(n-1)-m}y^m \\&+ \frac{\alpha _0 \Omega }{(1+\alpha _0x^{n-1})(1+\alpha _0y^{n-1})} \left( y^{2(n-1)}\right. \\&\quad +\left. \sum _{\frac{n}{2} \le \ell < n-1} x^{(n-1)-\ell }y^{(n-1)+\ell } - \frac{1}{2}x^{n-1}y^{n-1} \right) . \end{aligned} \end{aligned}$$

n = 2k + 1: In the odd case $n \ge 3$ the series corresponding to $W_0 \subset L(n, \alpha _0)$ is

$$\begin{aligned} \begin{aligned} \frac{1}{1 + \alpha _0 x^{n-1}} \frac{y^n \Omega }{x-y}&+ \frac{\Omega }{1+\alpha _0 x^{n-1}} \left( x^{\frac{n-1}{2}}y^{\frac{n-1}{2}} + \sum _{0 \le m< \frac{n-1}{2}} x^{(n-1)-m}y^m \right) \\&+ \frac{\alpha _0 \Omega }{(1+\alpha _0x^{n-1})(1+\alpha _0y^{n-1})} \left( y^{2(n-1)} \right. \\&\left. + \sum _{\frac{n-1}{2}< \ell < n-1} x^{(n-1)-\ell }y^{(n-1)+\ell } - \frac{1}{2}x^{n-1}y^{n-1} \right) . \end{aligned} \end{aligned}$$

5 Lie algebra splittings of $L(n,\alpha )$ and generalized $r$-matrices

By Corollary 3.7 we have a bijection between subalgebras of $L(n, \alpha )$ and series of type $(n, 1/(x^n \alpha (x)))$ solving GCYBE. Therefore, we can construct new solutions to GCYBE by finding subalgebras of $L(n, \alpha )$ complementary to the diagonal. However, as the following result shows, the most interesting new solutions should arise from unbounded subalgebras of $L(n,\alpha )$, $n > 2$.

Proposition 5.1

Let $L(n,\alpha ) = \Delta \dotplus W$ for some subalgebra $W \subset L(n,\alpha )$ and $n > 2$. Assume $W$ is bounded, i.e. there is an integer $N > 0$ such that

$$\begin{aligned} x^{-N}{\mathfrak {g}}[x^{-1}] \subseteq W_+ \subseteq x^N{\mathfrak {g}}[x^{-1}], \end{aligned}$$

where $W_+$ is the projection of $W \subset L(n,\alpha ) = {\mathfrak {g}}(\!(x)\!) \oplus {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]$ on the first component ${\mathfrak {g}}(\!(x)\!)$. Then there is an element $\sigma \in \text {Aut}_{F[x]-\text {LieAlg}}({\mathfrak {g}}[x])$ such that

$$\begin{aligned}\{0\} \times [x^2]{\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x] \subseteq (\sigma \times \sigma )W \subseteq x{\mathfrak {g}}[x^{-1}] \times {\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x]\end{aligned}$$

and the image ${\widetilde{W}}$ under the canonical projection $L(n,\alpha ) \rightarrow L(2,\alpha )$ is a subalgebra satisfying $L(2,\alpha ) = \Delta \dotplus {\widetilde{W}}$.

In the language of $(n,s)$-type series: Let

$$\begin{aligned}r = \frac{s(x) y^n \Omega }{x-y} + g(x,y)\end{aligned}$$

be the generalized $r$-matrix corresponding to a bounded $W \subset L(n,\alpha ), n \ge 2$. Then there is $p(x,y) \in ({\mathfrak {g}}\otimes {\mathfrak {g}})[x,y]$ of degree at most one in x and an element $\sigma \in \text {Aut}_{F[x]-\text {LieAlg}}({\mathfrak {g}}[x])$ such that

$$\begin{aligned} (\sigma (x) \otimes \sigma (y))r(x,y) = y^{n-2} \Big (\underbrace{ \frac{s(x)y^2 \Omega }{x-y} + p(x,y)}_{r'(x,y)} \Big ), \end{aligned}$$

where $r'$ is a generalized $r$-matrix in $L_2(2,\alpha )$.

Proof

The condition $x^{-N}{\mathfrak {g}}[x^{-1}] \subseteq W_+ \subseteq x^N{\mathfrak {g}}[x^{-1}]$ means exactly that $W_+$ is an order. Moreover, since $W$ is complementary to the diagonal, we have $W_+ + {\mathfrak {g}}[x] = {\mathfrak {g}}[x,x^{-1}]$. It was shown in [11] that such orders, up to the action of some $\sigma \in \text {Aut}_{F[x]-\text {LieAlg}}({\mathfrak {g}}[x])$, are contained in a maximal order ${\mathfrak {M}}$ associated to the so called fundamental simplex $\Delta _{\text {st}}$. These maximal orders are explicitly described in [11] and satisfy ${\mathfrak {M}} \subseteq x{\mathfrak {g}}[x^{-1}]$. Therefore, we have $\sigma W_+ \subseteq {\mathfrak {M}} \subseteq x{\mathfrak {g}}[x^{-1}]$. Moreover, we have the identity

$$\begin{aligned} (\sigma \times \sigma )W \dotplus \Delta = L(n,\alpha ), \end{aligned}$$

implying the inclusion $\{0\} \times [x^2]{\mathfrak {g}}[x]/x^n{\mathfrak {g}}[x] \subseteq (\sigma \times \sigma )W$. The remaining parts follow straightforward from the construction Theorem 3.6. $\square$

Unfortunately, we have not found a new example of an unbounded subalgebra of $L(n,\alpha )$. However, we present an infinite family of bounded subalgebras. We believe these examples are still interesting because their orthogonal complements, which are important in the view of Adler-Kostant-Symes scheme, are unbounded if $\alpha \ne 0$.

Consider the subspaces of $L(n,\alpha _0)$, $n > 0$:

$$\begin{aligned} \begin{aligned} W_0&= \text {span}_F \{ b_i(x^{-k},0), b_i(1,0), b_i(0, -[x]^\ell ) \mid k \ge 1, \, 1 \le \ell \le n-1 \}, \\ W_1&= \text {span}_F \{ b_i(x^{-k},0), b_i(0,-1), b_i(0, -[x]^\ell ) \mid k \ge 1, \, 1 \le \ell \le n-1 \}. \end{aligned} \end{aligned}$$

These are clearly subalgebras. The corresponding generalizerd $r$-matrices are

$$\begin{aligned} \begin{aligned} r_0&= \frac{1}{1+\alpha _0 x^{n-1}} \frac{y^n\Omega }{x-y} + \frac{y^{n-1} \Omega }{(1 + \alpha _0 x^{n-1})(1 + \alpha _0 y^{n-1})} \\&\ \ \ +\frac{\alpha _0 \Omega }{(1 + \alpha _0 x^{n-1})(1 + \alpha _0 y^{n-1})}\left( y^{2(n-1)} + \sum _{ 0 \le \ell< n-1} x^{(n-1)-\ell } y^{(n-1)+\ell } \right) \\&= \frac{y^{n-1}}{1+\alpha _0 y^{n-1}} \left( \frac{y\Omega }{x-y} + \Omega \right) ,\\ r_1&= \frac{1}{1+\alpha _0 x^{n-1}} \frac{y^n\Omega }{x-y} +\frac{\alpha _0 \Omega }{(1 + \alpha _0 x^{n-1})(1 + \alpha _0 y^{n-1})}\\&\quad \times \left( y^{2(n-1)} + \sum _{ 0< \ell < n-1} x^{(n-1)-\ell } y^{(n-1)+\ell } \right) \\ {}&= \frac{1}{1+\alpha _0 y^{n-1}}\frac{y^n\Omega }{x-y}. \end{aligned} \end{aligned}$$

By considering decompositions ${\mathfrak {g}}= {\mathfrak {s}}_1 \dotplus {\mathfrak {s}}_2$ of ${\mathfrak {g}}$ into direct sums of subalgebras we can get an infinite family of generalized $r$-matrices ”in between” $r_0$ and $r_1$. More precisely, let $\{s_{1,i}\}_{i=1}^{d_1}$ and $\{s_{2,j}\}_{j=1}^{d_2}$ be bases for ${\mathfrak {s}}_1$ and ${\mathfrak {s}}_2$ respectively. Such a decomposition leads to another subalgera of $L(n, \alpha _0)$:

$$\begin{aligned} \begin{aligned} W_{01}&:=\,\text {span}_F \Big \{ b_i(x^{-k},0), s_{1,m}(1,0), s_{2,j}(0,1), b_i(0, -[x]^\ell ) \mid k {\ge } 1, \, 1 {\le } \ell {\le } n-1, \, 1 {\le } i \le d, \\&\qquad 1 \le m \le d_1, \, 1 \le j \le d_2 \Big \}. \end{aligned} \end{aligned}$$

Rewrite the elements $b_i$ in terms of $s_{1,m}$ and $s_{2,j}$:

$$\begin{aligned} b_i = \sum _{m=1}^{d_1} \lambda _{1,m}^i s_{1,m} + \sum _{j=1}^{d_2} \lambda _{2,j}^i s_{2,j}, \end{aligned}$$

where $\lambda _{1,m}^i, \lambda _{2,j}^i \in F$. Finding a basis in $W_{12}$ dual to $\{ b_i(y^m, [y]^m) \} \subset \Delta$ and then projecting the generating series for $W_{01}$ onto the first component we obtain the following generalized $r$-matrix

$$\begin{aligned} r_{01}{} & {} = \frac{1}{1+\alpha _0 x^{n-1}} \frac{y^n\Omega }{x-y} +\frac{\alpha _0 \Omega }{(1 + \alpha _0 x^{n-1})(1 + \alpha _0 y^{n-1})}\nonumber \\{} & {} \quad \times \left( y^{2(n-1)} + \sum _{ 0< \ell < n-1} x^{(n-1)-\ell } y^{(n-1)+\ell } \right) \nonumber \\{} & {} \quad + \frac{y^{n-1}}{1 + \alpha _0 y^{n-1}} \sum _{i=1}^d \sum _{m=1}^{d_1} \lambda ^i_{1,m} s_{1,m} \otimes b_i \nonumber \\ {}{} & {} = \frac{y^{n-1}}{1+\alpha _0y^{n-1}}\left( \frac{y\Omega }{x-y} + \sum _{i=1}^d \sum _{m=1}^{d_1} \lambda ^i_{1,m} s_{1,m} \otimes b_i\right) . \end{aligned}$$

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Clearly $r_{01}$ coincides with $r_0$ when ${\mathfrak {s}}_1 = {\mathfrak {g}}$ and $r_1$ if ${\mathfrak {s}}_2 = {\mathfrak {g}}$. The corresponding orhogonal complements are

$$\begin{aligned} \begin{aligned} W_0^{\perp }&= \,W(\overline{r_0}) {=}\, \text {span}_{F} \left\{ b_i\left( 0,[x]^{n-1}\right) , b_i\left( \frac{x^{-k(n-1)-m}}{1+\alpha _0 x^{n-1}}, 0\right) \mid k \ge -1, 0< m< n-1 \right\} , \\ ~\\ W_1^{\perp }&= W(\overline{r_1}) = \text {span}_{F} \left\{ b_i\left( \frac{x^{-k(n-1)-m}}{1+\alpha _0 x^{n-1}}, 0\right) \mid k \ge -1, 0 \le m< n-1 \right\} , \\~\\ W_{01}^{\perp }&= W(\overline{r_{01}}) = {\mathfrak {s}}_1^\perp \left( \frac{x^{n-1}}{1+\alpha _0 x^{n-1}},0\right) \dotplus {\mathfrak {s}}_2^\perp (0, [x]^{n-1}) \\&\qquad \qquad \qquad \qquad \quad \dotplus \text {span}_{F} \left\{ b_i\left( \frac{x^{-k(n-1)-m}}{1+\alpha _0 x^{n-1}}, 0\right) \mid k \ge -1, 0< m < n-1 \right\} , \end{aligned}\nonumber \\ \end{aligned}$$

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which are unbounded because of the factor $1/ (1+\alpha _0x^{n-1})$.

Note that a series of type $(n,s)$ defines a subspace inside $L(n,\alpha )$ for any $\alpha$, because the subalgebra property is not affected by the form. With the previous examples in mind we can prove the following statement.

Lemma 5.2

Let $B_0$ and $B_\alpha$ be the bilinear forms on $L(n, 0)$ and $L(n, \alpha )$ respectively. For a series $r$ of type $(n,s)$ we have

$$\begin{aligned} W\left( r\right) ^{\perp _{B_\alpha }} = \frac{1}{x^n \alpha (x)} W(r)^{\perp _{B_0}} \subset L(n, \alpha ). \end{aligned}$$

(44)

Proof

Set $u(x) :=1/(x^n \alpha (x))$. Write

$$\begin{aligned} r {=} \sum _{k {\ge } 0} \sum _{i = 1}^d (s w_{k,i} {+} g_{k,i}) \otimes b_i(y^k,[y]^k) \ \text { and } \ {\overline{r}} {=} \sum _{k \ge 0} \sum _{i = 1}^d (sw_{k,i} {+} \overline{g_{k,i}}) \otimes b_i(y^k,[y]^k). \end{aligned}$$

Then by Theorem 3.6 and definition Eq. (12) $B_{\alpha }(sw_{k,i} + g_{k,i}, u( sw_{\ell ,j} + \overline{g_{\ell ,j}})) = B_0 (sw_{k,i} + g_{k,i}, sw_{\ell ,j} + \overline{g_{\ell ,j}}) = 0$. $\square$