Yang-mills Algebra
نویسنده
چکیده
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 Poincaré 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) selfdual 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. LPT-ORSAY 02-50 Collège de France, 3 rue d’Ulm, 75 005 Paris, and I.H.E.S., 35 route de Chartres, 91440 Bures-sur-Yvette [email protected] Laboratoire de Physique Théorique, UMR 8627, Université Paris XI, Bâtiment 210, F-91 405 Orsay Cedex, France [email protected]
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