Some topics on the $Z_3$-graded exterior algebras

TL;DR

This paper introduces a $Z_3$-graded exterior algebra with Hopf algebra structure, develops covariant differential calculus, constructs a Grassmann-Heisenberg algebra, finds an R-matrix satisfying graded Yang-Baxter equations, and builds a $Z_3$-graded universal enveloping algebra.

Contribution

It presents the first construction of a $Z_3$-graded Hopf algebra structure and related algebraic frameworks for exterior algebras with two generators.

Findings

A $Z_3$-graded Hopf algebra structure is established.

Two covariant differential calculi are developed.

A $Z_3$-graded universal enveloping algebra is constructed.

Abstract

A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a Grassmann-Heisenberg algebra is constructed. An R-matrix which satisfies graded Yang-Baxter equations is obtained. A $Z_3$-graded universal enveloping algebra $U_q(\widetilde{gl}(2))$ is constructed with the quadratic elements of the Grassmann-Weyl algebra.