Projective Group Algebras

TL;DR

This paper extends algebraic integration techniques to projective group algebras, enabling harmonic analysis on noncommutative spaces relevant for M-theory compactification on noncommutative tori.

Contribution

It introduces a generalized algebraic integration framework for projective group algebras, applicable even when classical representation spaces are noncommutative or ill-defined.

Findings

Algebraic integration applies to projective group algebras.

Extension of harmonic analysis to noncommutative spaces.

Simplified calculus for abelian groups in noncommutative setting.

Abstract

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the $M$ theory on noncommutative tori. This turns out to be an interesting field of applications, since the space $\hat G$ of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over $\hat G$. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space $\hat G$ has no classical meaning. This allows a generalization of the usual group harmonic analysis. A particular attention is given to abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.