Polynomial identities and noncommutative versal torsors

TL;DR

This paper introduces universal algebras associated with cleft Hopf Galois objects, exploring their structures, relationships, and specific cases like the four-dimensional Sweedler algebra, advancing the understanding of polynomial identities in noncommutative algebra.

Contribution

It constructs and analyzes universal algebras U(H,t) and A(H,t) for cleft Hopf Galois objects, establishing their properties, embeddings, and relations, especially in the simple algebra case.

Findings

A(H,t) is a cleft H-Galois extension of a big commutative algebra B(H,t).

U(H,t) is the universal comodule algebra satisfying all identities of H[t].

In the simple case, A(H,t) is isomorphic to a localization of U(H,t).

Abstract

To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra.