Generalization of Superalgebras to Color Superalgebras and Their Representations

TL;DR

This paper introduces two methods to generalize Lie superalgebras into color superalgebras with complex gradings, expanding their structure and representation theory, including a vector field representation derived from boson-fermion systems.

Contribution

It presents two novel constructions of color superalgebras from Lie superalgebras, one based on Clifford algebra properties and the other on boson-fermion operator systems.

Findings

Constructed color superalgebras with ${f Z}_2^{ imes N}$ grading.

Developed a vector field representation from boson-fermion systems.

Extended the algebraic framework for superalgebras to include color superalgebras.

Abstract

For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to any Lie superalgebras and results in color superalgebra of $ {\mathbb Z}_2^{\otimes N} $ grading. The other is discussed with an example, a superalgebra of boson and fermion operators. By treating the boson operators as "second" fermionic sector we obtain a color superalgebra of ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ grading. A vector field representation of the color superalgebra obtaind from the boson-fermion system is also presented.