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Multilinear forms and graded algebras. (English) Zbl 1141.17010
In the paper under review, the author studies connected graded algebras which are finitely generated in degree $$1$$, finitely presented with relations in degree $$\geq 2$$, of finite global dimension $$D$$ and Gorenstein, plus the condition of being homogeneous and Koszul in the case where $$D \geq 4$$. For $$D = 2$$, it is shown that the algebras are associated to a nondegenerate bilinear form and correspond to the natural quantum spaces for the action of the quantum groups associated to that form in the paper [Phys. Lett., B 245, No. 2, 175–177 (1990; Zbl 1119.16307)], by G. Launer and the author. It is also shown that all homogeneous Koszul-Gorenstein algebras of finite global dimension are associated with multilinear forms which are in particular preregular, a concept introduced in this paper. Homogeneous algebras associated to a multilinear form are introduced and semi-crossed product is investigated for this class of algebras. Generalizing the above mentioned paper with Launer, the author also introduces quantum groups preserving multilinear forms which act on the quantum spaces associated with these forms. The paper analyzes several examples, including the Yang-Mills algebra and the extended $$4$$-dimensional Sklyanin algebra.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W50 Graded rings and modules (associative rings and algebras) 16S37 Quadratic and Koszul algebras
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