ALGEBRAS, DERIVATIONS AND INTEGRALS

Abstract

In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include, for instance, the para-Grassmann algebras of order p, the quaternionic algebra and the toroidal algebras. We study the relation between derivations and integration, proving a generalization of the standard result for the Riemann integral about the translational invariance of the measure and the vanishing of the integral of a total derivative (for convenient boundary conditions). We consider also the possibility, given the integration over an algebra, to define from it the integral over a subalgebra, in a way similar to the usual integration over manifolds. That is by projecting out the submanifold in the integration measure. We prove that this is possible for para-Grassmann algebras of order p, once we consider them as subalgebras of the algebra of the (p + 1) × (p + 1) matrices. We find also that the integration over the subalgebra coincides with the integral defined in the direct way. As a by-product we can define the integration over a one-dimensional Grassmann algebra as a trace over 2 × 2 matrices.