  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.
