Yang-Mills algebra
type: Item
instance of: scholarly article
title: Yang-Mills algebra
zbMATH Open document ID: 1028.53025
DOI: 10.1023/A:1020733628744
author: Alain Connes
author: Michel Dubois-Violette
published in: Letters in Mathematical Physics
publication date: 2003-04-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0206205
review text: The cubic algebra \({\mathcal A}\) generated by the covariant derivatives of a Yang-Mills connection over the \((s+1)\)-dimensional pseudo-Euclidean space is Koszul of global dimension 3 and Gorenstein (Theorem 1) except in the two-dimensional case \((s=1)\) where it is the universal enveloping algebra of the Heisenberg Lie algebra and a cubic Artin-Schelter regular algebra [see \textit{R. Berger}, J. Algebra 239, 705-734 (2001; Zbl 1035.16023)]. An explicit formula for the Poincaré series of \({\mathcal A}\) and for the dimension in degree \(n\) of the corresponding graded Lie algebra is given. In the four-dimensional Euclidean case \((s=3)\) a quotient of \({\mathcal A}\) is the quadratic algebra generated by a self-dual connection. This algebra is Koszul of global dimension 2 but is not Gorenstein (Theorem 2). It is the universal enveloping algebra of the semi-direct product of the free Lie algebra with three generators of degree 1 by a derivation of degree 1 [see also \textit{M. Dubois-Violette}, K-Theory 14, 371-404 (1998; Zbl 0918.18008), \textit{M. Dubois-Violette} and \textit{I. T. Todorov}, Lett. Math. Phys. 48, 323-338 (1999; Zbl 0949.81067), and \textit{M. Wambst}, K-Theory 23, 377-397 (2001; Zbl 0996.18011)].
reviewed by: Maido Rahula
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 16S37
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 81R60
zbMATH DE Number: 1901321
zbMATH Keywords: duality
zbMATH Keywords: Homogeneous algebras
zbMATH Keywords: Koszul algebras
zbMATH Keywords: \(N\)-complexes
zbMATH Keywords: Yang-Mills algebra
MaRDI profile type: Publication
arXiv ID: math/0206205
OpenAlex ID: W1713945898
description: scientific article