The center of the 3-dimensional and 4-dimensional Sklyanin algebras |
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Authors: | S P Smith and J Tate |
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Institution: | (1) Department of Mathematics, University of Washington, 98195 Seattle, WA, USA;(2) Department of Mathematics, University of Texas, 78712 Austin, TX, USA |
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Abstract: | LetA=A(E, ) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curveE and a point ![tau](/content/x4tp74343131740g/xxlarge964.gif) E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension ofA. It is known thatA is a finite module over its center if and only if is of finite order. Generators and defining relations for the centerZ(A) are given. IfS=Proj(Z(A)) andA is the sheaf ofO
S
-algebras defined byA(S
(f))=Af
–1]0 then the centerL ofA is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result thatSpec
L![cong](/content/x4tp74343131740g/xxlarge8773.gif) 2. However, for the 4-dimensional Sklyanin algebra there is not such a simple result: althoughSpec
L is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus ofA. |
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Keywords: | Sklyanin algebras center |
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