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 et al. (Adv Math 126:52–92, 1997) 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 $\mathfrak {d}$.