Multilinear forms and graded algebras

Details Multilinear forms and graded algebras Multilinear forms and graded algebras

2006-04-12

arxiv.org/abs/math/0604279v3

Mathematics

Quantum Algebra

38 pages; references added, final version to appear in Journal of Algebra

Scientific paper

In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang-Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated.

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