On Graded Quasi-Cartan Pairs and Twisted Steinberg Algebras

TL;DR

This paper extends the theory of quasi-Cartan and Cartan subalgebras to graded settings, establishing a correspondence with graded twisted Steinberg algebras and proving the uniqueness of associated graded twists.

Contribution

It introduces graded versions of quasi-Cartan, Cartan, and diagonal pairs, generalizing previous ungraded results and including all discrete group algebras.

Findings

Established a correspondence between graded algebraic pairs and graded twisted Steinberg algebras.

Proved the uniqueness of the associated graded discrete twist.

Included all discrete group algebras in the generalized framework.

Abstract

We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted Steinberg algebras and that the associated graded discrete twist is unique. Our results include all discrete group algebras, and so are more general than the ungraded version.