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Constructive approach to three dimensional Sklyanin algebras. (English) Zbl 1338.16031

Summary: A three dimensional Sklyanin is the quadratic algebra over a field \(\Bbbk\) with 3 generators \(x,y,z\) given by 3 relations \(xy-ayx-szz=0\), \(yz-azy-sxx=0\) and \(zx-axz-syy=0\), where \(a,s\in\Bbbk\). A generalized Sklyanin algebra is the algebra given by relations \(xy-a_1yx-s_1zz=0\), \(yz-a_2zy-s_2xx=0\) and \(zx-a_3xz-s_3yy=0\), where \(a_i,s_i\in\Bbbk\). In this paper we announce the following results; the complete proofs will appear elsewhere. We determine explicitly the parameters for which these algebras have the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates as well as when these algebras are Koszul and PBW, using constructive combinatorial methods. These provide new direct proofs of results established first by M. Artin, J. Tate, and M. Van den Bergh [Invent. Math. 106, No. 2, 335-388 (1991; Zbl 0763.14001)].

MSC:

16S37 Quadratic and Koszul algebras
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)

Citations:

Zbl 0763.14001
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Full Text: Euclid