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Regularity of the four dimensional Sklyanin algebra. (English) Zbl 0758.16001
Let \(\alpha\), \(\beta\) and \(\gamma\) be elements of an algebraically closed field \(k\) of characteristic not two with \(\alpha+\beta+\gamma+\alpha\beta\gamma=0\). The Sklyanin algebra \(S=S(\alpha,\beta,\gamma)\) is the graded \(k\)-algebra with non-commutative generators \(x_ 1\), \(x_ 2\) and \(x_ 3\) of degree one satisfying 6 relations, of which 2 typical ones are \([x_ 0,x_ 1]=\alpha(x_ 2x_ 3+x_ 3x_ 2)\) and \([x_ 1,x_ 2]=x_ 0x_ 3+x_ 3x_ 0\). These algebras were introduced by Sklyanin in connection with his work on the quantum inverse scattering method. The authors’ main theorem concerns properties of \(S\) when two of \(\alpha\), \(\beta\), \(\gamma\) are not 1 and \(-1\) respectively. They prove that \(S\) is a Noetherian domain, its Hilbert series is that of a commutative polynomial ring in 4 variables and \(S\) is a regular graded algebra of dimension 4, i.e., \(S\) has global homological dimension 4, the graded part \(S_ n\) has Gelfand-Kirillov dimension \(\leq n^ r\) for some real number \(r\), and \(S\) is Gorenstein, i.e., \(\text{Ext}^ q_ S(k,A)=\delta_{4,q}k\). The authors set up an algebraic geometric approach, and develop machinery which is analogous to the treatment of 3-dimensional algebras given recently by Artin, Tate and van den Bergh. Some of these geometric ideas involve elliptic curves and theta functions.

MSC:
16E10 Homological dimension in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14H52 Elliptic curves
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