Representation theory of three-dimensional Sklyanin algebras

TL;DR

This paper classifies the irreducible representations of three-dimensional Sklyanin algebras, linking algebraic structures to vacua in deformed supersymmetric theories and exploring their polynomial identities and Calabi-Yau geometry.

Contribution

It determines the dimensions of irreducible representations and polynomial identity degrees, advancing understanding of Sklyanin algebras in mathematical physics.

Findings

Dimensions of irreducible representations are classified.

Polynomial identity degree for certain Sklyanin algebras is established.

Calabi-Yau geometric properties are discussed.

Abstract

We determine the dimensions of the irreducible representations of the Sklyanin algebras with global dimension 3. This contributes to the study of marginal deformations of the N=4 super Yang-Mills theory in four dimensions in supersymmetric string theory. Namely, the classification of such representations is equivalent to determining the vacua of the aforementioned deformed theories. We also provide the polynomial identity degree for the Sklyanin algebras that are module finite over their center. The Calabi-Yau geometry of these algebras is also discussed.