An analogue of the classical Yang-Baxter equation for general algebraic structures
type: Item
instance of: scholarly article
title: An analogue of the classical Yang-Baxter equation for general algebraic structures
zbMATH Open document ID: 0828.17019
DOI: 10.1007/BF01293592
author: Xiaoping Xu
published in: Monatshefte für Mathematik
publication date: 1995-12-19T00:00:00Z
full work available at URL: https://eudml.org/doc/178690
review text: The author introduces an analogue of the classical Yang-Baxter equation for a general complex nonassociative algebra \({\mathcal A}\) with operation `\(\cdot\)'. Under certain conditions this equation is equivalent to the equation \[ R(z_1+ z_2) (x\cdot R( -z_1) (y))+ R(z_2) (R(z_1) (x)\cdot y)= R(z_1+ z_2) (x)\cdot R(z_2) (y) \] for \(x\), \(y\) in \({\mathcal A}\) and \(z_1, z_2\) in \(\mathbb{C}\), where the solution \(R\) is an analytic function with values in \(\text{End } {\mathcal A}\). The author gives several ways to construct solutions in the case when \({\mathcal A}\) is graded by an abelian group. As an example the author gives some unitary nondegenerate trigonometric solutions of the classical Yang-Baxter equation for affine Lie algebras. It is noted that the above equation can be modified for the case when \({\mathcal A}\) is a vertex operator algebra and that solutions can be constructed in a similar way.
reviewed by: Mirko Primc
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 757762
zbMATH Keywords: classical Yang-Baxter equation
zbMATH Keywords: complex nonassociative algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 757762