Symmetric bilinear form on a Lie algebra

TL;DR

This paper constructs a symmetric bilinear form on a Lie algebra $rak d$, a generalization of a simple Lie algebra $rak g$, related to quantum groups, expanding understanding of invariant forms in algebraic deformations.

Contribution

It introduces a $rak d$-invariant symmetric bilinear form on a generalized Lie algebra related to quantum groups, extending classical invariant form concepts.

Findings

$rak d$ is a generalization of $rak g$

A $rak d$-invariant symmetric bilinear form is constructed

The form extends classical invariant forms to quantum group related algebras

Abstract

Let $\frak g$ be the finite dimensional simple Lie algebra associated to an indecomposable and symmetrizable generalized Cartan matrix $C=(a_{ij})_{n\times n}$ of finite type and let $\frak d$ be a finite dimensional Lie algebra related to a quantum group $D_{q,p^{-1}}(\frak g)$ obtained by Hodges, Levasseur and Toro \cite{HoLeT} by deforming the quantum group $U_q(\frak g)$. Here we see that $\frak d$ is a generalization of $\frak g$ and give a $\frak d$-invariant symmetric bilinear form on $\frak d$.