A view toward the smooth geometry of Sklyanin algebras

TL;DR

This paper investigates the differential smoothness of Sklyanin algebras, showing that three-dimensional cases are smooth while four-dimensional ones lack certain differential structures.

Contribution

It provides a comprehensive analysis of the differential smoothness of Sklyanin algebras in three and four variables, highlighting their geometric properties.

Findings

All non-degenerate three-dimensional Sklyanin algebras are differentially smooth.

Four-dimensional Sklyanin algebras do not admit a suitable connected integrable differential calculus.

The results distinguish geometric properties based on the algebra's dimension.

Abstract

We study the differential smoothness of Sklyanin algebras in three and four variables. We show that all non-degenerate three-dimensional cases are differentially smooth, while none of the four-dimensional Sklyanin algebras admit a connected integrable differential calculus of suitable dimension.