# $\mathbb{Z}_2 \times \mathbb{Z}_2$ generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions

## TL;DR

This paper introduces a new class of infinite-dimensional $Z_2 imes Z_2$-graded color superalgebras, classifies their central extensions, and explores their algebraic properties, extending the understanding of conformal Lie superalgebras.

## Contribution

It provides the first complete classification of central extensions for these novel $Z_2 imes Z_2$-graded color superalgebras and demonstrates their algebraic structures.

## Key findings

- Many members have non-trivial central extensions
- Color superalgebras possess adjoint and superadjoint operations
- Complete classification of central extensions achieved

## Abstract

We introduce a class of novel $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras is presented. It turns out that infinitely many members of the class have non-trivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and superadjoint operations.