Generalized exterior algebras

TL;DR

This paper introduces a new class of generalized exterior algebras called N-metric exterior algebras, extending classical structures like Grassmann and Clifford algebras, with potential applications in gravitational theories involving multiple metrics.

Contribution

The paper defines N-metric exterior algebras depending on N matrices of structure constants, generalizing existing algebraic frameworks for potential use in gravity models.

Findings

Defined N-metric exterior algebras for N≥2

Connected 0-metric and 1-metric algebras to classical structures

Proposed applications in gravitation and fermion description

Abstract

Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\geq2$ can be considered as generalizations of the Grassmann algebra and Clifford algebra. Specialists consider models of gravity that based on a mathematical formalism with two metric tensors. We hope that the considered in this paper 2-metric exterior algebra can be useful for development of this model in gravitation theory. Especially in description of fermions in presence of a gravity field.