A theory of algebraic integration

TL;DR

This paper develops a generalized theory of integration applicable to various algebraic structures, including self-conjugated, matrix, group, and non-associative algebras, extending classical integral properties to these contexts.

Contribution

It introduces a unified integral framework for self-conjugated algebras and demonstrates its consistency with classical properties and examples like Grassmann, matrix, and octonion algebras.

Findings

Integral defined for self-conjugated algebras with properties similar to classical integrals.

Recovery of Berezin's rule for Grassmann algebras.

Representation of integrals as traces over matrices and sums over group representations.

Abstract

In this paper we extend the idea of integration to generic algebras. In particular we concentrate over a class of algebras, that we will call self-conjugated, having the property of possessing equivalent right and left multiplication algebras. In this case it is always possible to define an integral sharing many of the properties of the usual integral. For instance, if the algebra has a continuous group of automorphisms, the corresponding derivations are such that the usual formula of integration by parts holds. We discuss also how to integrate over subalgebras. Many examples are discussed, starting with Grassmann algebras, where we recover the usual Berezin's rule. The paraGrassmann algebras are also considered, as well as the algebra of matrices. Since Grassmann and paraGrassmann algebras can be represented by matrices we show also that their integrals can be seen in terms of traces over the corresponding matrices. An interesting application is to the case of group algebras where we show that our definition of integral is equivalent to a sum over the unitary irreducuble representations of the group. We show also some example of integration over non self-conjugated algebras (the bosonic and the $q$-bosonic oscillators), and over non-associative algebras (the octonions).