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Yang-Mills algebra

Alain Connes·Michel Dubois-Violette
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Abstract

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s=1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin-Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra A and for the dimension in degree n of the graded Lie algebra of which A is the universal enveloping algebra. In the 4-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie-algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.

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Alain Connes, Michel Dubois-Violette. 2002-09-10. Yang-Mills algebra. https://arxiv.org/abs/math/0206205

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