Automorphism groupoids in noncommutative projective geometry

TL;DR

This paper introduces a groupoid framework for understanding automorphisms of noncommutative projective spaces, revealing rigidity in dimension 1 and describing automorphisms via point scheme geometry in higher dimensions, with applications to quantum spaces.

Contribution

It defines a noncommutative projective space automorphism groupoid and relates it to point scheme automorphisms, providing a geometric classification of algebra twists.

Findings

Dimension 1 noncommutative projective spaces are all isomorphic to commutative ones.

Automorphisms in higher dimensions relate to point scheme automorphisms.

Complete description of Zhang twists for quantum projective spaces and Sklyanin algebras.

Abstract

We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras. We analyse this automorphism groupoid, showing that in dimension 1 it is connected, so that every noncommutative $\mathbb{P}^{1}$ is isomorphic to commutative $\mathbb{P}^{1}$. For dimension 2 and above, we use the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme. We apply our results to two important examples, quantum projective spaces and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.