Quantizations and classical non-commutative non-associative algebras

Hilja Lisa HURU and Valenti

Abstract

In this paper we study quantizations, associativity constraints and braidings in the monoidal category of monoid graded modules over a commutative ring. All of them can be described in terms of the cohomology of underlying monoid. The case when the monoid is a finite topology has the main interest for us. The cohomology classes which are invariant with respect to homeomorphism group produce remarkable algebraic constructions. We study in details the Sierpinski and discrete topology and show the relations with the Clifford algebras, the Cayley algebra and their quantizations. All of them are ®-associative and ¾-commutative for suitable associativity constraints ® and braidings ¾.