Symmetric noncommutative birational transformations

TL;DR

This paper explores symmetric noncommutative birational transformations, demonstrating their invertibility and extending the framework to include noncommutative quadrics via $ abla$-algebras, enriching the understanding of noncommutative algebraic geometry.

Contribution

It establishes the symmetry of inverse birational transformations and extends the theory to $ abla$-algebras, integrating noncommutative quadrics into the framework.

Findings

Inverse birational transformations are of the same type as the original.

Extension to $ abla$-algebras incorporates noncommutative quadrics.

Symmetry of transformations enhances understanding of noncommutative geometry.

Abstract

In a previous paper (arXiv:1410.5207) certain birational transformations were constructed between the noncommutative schemes associated to quadratic and cubic three dimensional Sklyanin algebras. In the current paper we consider the inverse birational transformations and show that they are of the same type. Moreover we extend everything to the $\mathbb{Z}$-algebras context, which allows us to incorporate the noncommutative quadrics introduced by Van den Bergh.