Noncommutative Differential Calculus: Quantum Groups, Stochastic Processes, and the Antibracket

TL;DR

This paper develops a noncommutative differential calculus framework connecting quantum groups, stochastic processes, and the Batalin-Vilkovisky formalism, revealing new algebraic structures and relations.

Contribution

It introduces a noncommutative differential calculus on smooth functions, linking quantum groups, stochastic calculus, and superspace formalisms in a unified approach.

Findings

Established relations with bicovariant differential calculus on quantum groups

Connected noncommutative calculus to stochastic processes

Linked superspace calculus to Batalin-Vilkovisky formalism

Abstract

We explore a differential calculus on the algebra of smooth functions on a manifold. The former is `noncommutative' in the sense that functions and differentials do not commute, in general. Relations with bicovariant differential calculus on certain quantum groups and stochastic calculus are discussed. A similar differential calculus on a superspace is shown to be related to the Batalin-Vilkovisky antifield formalism.