On some operator identities and representations of algebras

TL;DR

This paper explores operator identities linked to Lie algebras and q-deformed algebras, providing new insights into their representations using differential and finite-difference operators.

Contribution

It introduces new operator identities and representations for (super) Lie algebras and q-deformed algebras involving differential and finite-difference operators.

Findings

Identified infinite families of operator identities

Established representations of Lie and q-deformed algebras

Connected operator identities with algebraic structures

Abstract

Certain infinite families of operator identities related to powers of positive root generators of (super) Lie algebras of first-order differential operators and $q$-deformed algebras of first-order finite-difference operators are presented.