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 \(\mathfrak {d}\).