Noncommutative differential geometry related to the Yang-Baxter equation.
type: Item
instance of: scholarly article
title: Noncommutative differential geometry related to the Yang-Baxter equation.
zbMATH Open document ID: 1044.58501
DOI: 10.1007/BF02367232
author: Andrey Radul
author: Volodya Rubtsov
author: Dimitri Gurevich
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-12-05T00:00:00Z
review text: Let \(V\) be a vector space over a (not necessarily commutative) ring \(k\), and consider the ``symmetry'' operator \(S\colon V^{\otimes 2}\to V^{\otimes 2}\), \(S(e_i\otimes e_j)=S^{kl}_{ij} e_k\otimes e_l\), \(e_i\in V\) (with summation over repeated indices), satisfying the Yang-Baxter equation \(S^{12}S^{23}S^{12}=S^{23}S^{12}S^{23}\). If there is a ``column-wise'' inverse \(T\) to \(S\), such that \(S^{kl}_{ij}T^{jq}_{lp}=\delta^q_i\delta^k_p\), then \(S\) is said to be closed. The authors first show how to extend \(S\) to the rigid tensor category \(\mathfrak A(S)\) of vector spaces with symmetries. They then discuss the classification of symmetries (especially so-called Hecke symmetries, defined by \((S-q\cdot\text{id})(S+\text{id})=0\)). Suppose \(V\) is a (noncommutative) ring \(A\) over a field \(k\), with multiplication \(\circ\colon A^{\otimes 2}\to A\) and a symmetry \(S\colon A^{\otimes 2}\to A^{\otimes 2}\) with respect to which \(\circ\) is \(S\)-invariant. \(A\) is said to be \(S\)-commutative if \(\circ\,S=\circ\) (i.e., \(f\circ g=\circ\,S(f\otimes g)\) for \(f,g\in A\)). Define the space of left derivations \(\text{Der}(A)\) as the elements of \(\text{Hom}_k(A,A)\) that satisfy the obvious \(S\)-Leibniz rule; this space is \(S\)-invariant and is a left \(A\)-module. If \(A\) is an associative \(S\)-algebra, the operation \([X,Y]=X\circ Y-\circ\,S(X\otimes Y)\) turns it into an \(S\)-Lie algebra. Requiring that \(S\) be a unitary symmetry, the authors discuss exterior products on an \(S\)-commutative ring \(A\) (over a field \(k\)) and prove many of the properties of this product. At the end they give some examples, including an example of a sheaf of rings in which local solutions of the quantum Yang-Baxter equation are defined but which does not possess a global section.
Mathematics Subject Classification ID: 58B32
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 58A10
zbMATH DE Number: 1373735
MaRDI profile type: Publication
Wikidata QID: Q115392383
cites work: Q3733819
cites work: Q3030982
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cites work: Compact matrix pseudogroups
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full work available at URL: https://doi.org/10.1007/bf02367232
OpenAlex ID: W2041129220
description: scientific article; zbMATH DE number 1373735