Some algebras similar to the 2x2 Jordanian matrix algebras

TL;DR

This paper explores algebras related to the Jordanian matrix algebra, constructing new examples that are birationally equivalent to Weyl algebras, thus advancing understanding of noncommutative ring structures.

Contribution

It introduces a family of algebras constructed as skew polynomial rings that are birationally equivalent to Weyl algebras, expanding the class of known similar algebras.

Findings

Constructed algebras are birationally equivalent to Weyl algebras.

Established a connection between Jordanian deformations and differential operator rings.

Provided new examples of noncommutative algebras with similar properties.

Abstract

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on $2\times 2$ matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.