Non-linear generalization of the sl(2) algebra

TL;DR

This paper introduces a non-linear extension of the sl(2) algebra, exploring its finite-dimensional representations through solutions to dynamical and cut condition equations, expanding the algebra's mathematical framework.

Contribution

It generalizes the sl(2) algebra with non-linear functions and constructs finite-dimensional representations using novel methods, including solutions to dynamical and cut condition equations.

Findings

Finite-dimensional representations are constructed for non-linear sl(2) generalizations.

Explicit solutions for quadratic non-linear functions are provided.

Two distinct methods for representation construction are demonstrated.

Abstract

We present a generalization of the sl(2) algebra where the algebraic relations are constructed with the help of a general function of one of the generators. When this function is linear this algebra is a deformed sl(2) algebra. In the non-linear case, the finite dimensional representations are constructed in two different ways. In the first case, which provides finite dimensional representations only for the non-linear case, these representations come from solutions to a dynamical equation and we show how to construct explicitly these representations for a general quadratic non-linear function. The other type of finite dimensional representation comes from solutions to a cut condition equation. We give examples of solutions of this type in the non-linear case as well.