Solutions of the classical Yang-Baxter equation for simple superalgebras
type: Item
instance of: scholarly article
title: Solutions of the classical Yang-Baxter equation for simple superalgebras
zbMATH Open document ID: 0537.22014
DOI: 10.1007/BF01031030
published in: Theoretical and Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The equation \([X^{12}(u),X^{13}(u+v)]+[X^{12}(u),X^{23}(v)]+[X^{13}(u+v),X^ {23}(v)]=0,\) where \({\mathfrak G}\) is a Lie (super)algebra and \(X^{ij}:\quad {\mathbb{C}}\to U({\mathfrak G})^ 3\) is called the classical Yang-Baxter equation. Here \(U({\mathfrak G})\) is the universal enveloping algebra of the Lie (super)algebra \({\mathfrak G}\) and \(X^{12}({\mathbb{C}})\subset {\mathfrak G}\otimes {\mathfrak G}\otimes 1, X^{13}({\mathbb{C}})\subset {\mathfrak G}\otimes 1\otimes {\mathfrak G}, X^{23}({\mathbb{C}})\subset 1\otimes {\mathfrak G}\otimes.\) In this paper all nondegenerate solutions of the classical Yang-Baxter equation for simple finite-dimensional Lie superalgebras which are either elliptic or trigonometric or constant functions are found. These solutions are given to within equivalence concerning the following transformations: \(1)\quad X(u)\to X(u)+r,\) where \({\mathfrak h}\) is a Lie subalgebra of \({\mathfrak G}\), \(r\in {\mathfrak h}\otimes {\mathfrak h}, r^{12}=-r^{21}, [r^{12},r^{23}]+[r^{12},r^{13}]+[r^{13},r^{23}]=0; 2)\quad X(u)\to cX(u),\) where \(c\in {\mathbb{C}}\backslash 0\); \(3)\quad X(u_ 1-u- 2)\to(\phi(u_ 1)\otimes \phi(u_ 2))X(u_ 1-u_ 2),\) where \(\phi\) (u)\(\in Aut {\mathfrak G}\).
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 3853390
zbMATH Keywords: Yang-Baxter equation
zbMATH Keywords: Lie superalgebra
zbMATH Keywords: simple roots
author: Dimitry Leites
author: Vera Serganova
reviewed by: A. Fleischer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01031030
OpenAlex ID: W1969355933
description: scientific article; zbMATH DE number 3853390