arXiv subjects

Stepan Maximov

Publications and source records attributed to Stepan Maximov.

5 recordsLinked to original sources

arXiv2024

Generalized classical Yang-Baxter equation and regular decompositions

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra decompositions of $\mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^m \mathfrak{g}[x]$. The latter decompositions are in bijection with the solutions to the GCYBE. Under appropriate regularity conditions, we obtain a partial classification of such solutions. The paper is concluded with the presentations of the Gaudin-type models associated to these solutions.

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arXiv2024

Regular decompositions of finite root systems and simple Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of $\mathfrak{g}$ and its irreducible root system $\Delta$. A regular decomposition is a decomposition $\mathfrak{g} = \mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_m$, where each $\mathfrak{g}_i$ and $\mathfrak{g}_i \oplus \mathfrak{g}_j$ are regular subalgebras. Such a decomposition induces a partition of the corresponding root system, i.e. $\Delta = \Delta_1 \sqcup \dots \sqcup \Delta_m$, such that all $\Delta_i$ and $\Delta_i \sqcup \Delta_j$ are closed. Partitions of $\Delta$ with $m=2$ were known before. In this paper we prove that the case $m \ge 3$ is possible only for systems of type $A_n$ and describe all such partitions in terms of $m$-partitions of $(n+1)$. These results are then extended to a classification of regular decompositions of $\mathfrak{g}$.

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arXiv2022

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

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 Yang-Baxter equation, correspond to subalgebras of $L$. We discuss their possible utility in the theory of integrable systems.

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arXiv2022

Topological Lie bialgebra structures and their classification over $ \mathfrak{g}[\![x]\!] $

This paper is devoted to a 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 the latter as a special case of the classification of Manin pairs. The classification of topological doubles states that up to some notion of 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. When $F = \mathbb{C}$ the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang-Baxter equation.

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arXiv2020

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

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