NDLI: Yang–Mills Algebra

Content Provider	Springer Nature Link
Author	Connes, Alain Dubois Violette, Michel
Copyright Year	2002
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 Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-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 Poincaré series of this algebra $$\mathcal{A}$$ and for the dimension in degree n of the graded Lie algebra of which $$\mathcal{A}$$ is the universal enveloping algebra. In the four-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.
Starting Page	149
Ending Page	158
Page Count	10
File Format	PDF
ISSN	03779017
Journal	Letters in Mathematical Physics
Volume Number	61
Issue Number	2
e-ISSN	15730530
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2002-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Group Theory and Generalizations Geometry Mathematical and Computational Physics Statistical Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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