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

TL;DR

This paper provides new proofs for the structure of the centers of certain 3D and 4D Sklyanin algebras and revisits Van den Bergh's results on noncommutative quadrics, emphasizing the role of Heisenberg group representations.

Contribution

It offers novel proofs for the centers of quadratic and cubic Sklyanin algebras and revisits Van den Bergh's findings, highlighting the importance of Heisenberg group representations.

Findings

New proofs of the centers of 3D and 4D Sklyanin algebras

Revised proof of Van den Bergh's noncommutative quadrics result

Role of Heisenberg groups in algebraic structure analysis

Abstract

In this article, a new proof is given of the description of the center of quadratic Sklyanin algebras of global dimension three and four and the center of cubic Sklyanin algebras of global dimension three. The representation theory of the Heisenberg groups $H_2$, $H_3$ and $H_4$ will play an important role. In addition a new proof is given of Van den Bergh's result regarding noncommutative quadrics.