Quotients of degenerate Sklyanin algebras

Abstract

Abstract. In this paper it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a 3-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.

On the center of 3-dimensional and 4-dimensional Sklyanin algebras

“…In order to find all presentations of four dimensional Sklyanin algebras with the degree 1 part decomposed in four 1-dimensional representations of V 4 , one still needs to allow for base change with the subgroup Aut V4 ( 4 V 1 ). By Schur's lemma, Aut V4 ( 4 V 1 ) ∼ = (C * ) 4 , but the subgroup of scalar matrices C * acts trivially as a Sklyanin algebra is graded. Let…”

“…In addition, it will be shown that Cc 3 is the trivial representation of H 3 , Cc 4 is the trivial representation of H 2 and CΩ 1 + CΩ 2 is isomorphic to the unique simple 2-dimensional representation of H 4 coming from the quotient map H 4 / / / / H 2 that will be explained below. In fact, for c 3 the connection between c 3 and the Sklyanin algebra Q 3 (E, 2τ ) will be explained, finishing the discussion started in [4] and [5].…”