Symmetric bilinear form on a Lie algebra

Abstract

Let $\mathfrak{g}$ 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 $\mathfrak{d}$ be a finite dimensional Lie algebra related to a quantum group $D_{q,p^{-1}}(\mathfrak{g})$ obtained by Hodges, Levasseur and Toro \cite{HoLeT} by deforming the quantum group $U_q(\mathfrak{g})$. Here we see that $\mathfrak{d}$ is a generalization of $\mathfrak{g}$ and give a $\mathfrak{d}$-invariant symmetric bilinear form on $\mathfrak{d}$.