Quasi-triangular Hopf algebras and invariant Jacobians

We show that two module homomorphisms for groups and Lie algebras established by Xi can be generalized to the setting of quasi-triangular Hopf algebras. These module homomorphisms played a key role in his proof of a conjecture of Yau (1998). They will also be useful in the problem of decomposition of tensor products of modules. Additionally, we give another generalization of result of Xi in terms of Chevalley-Eilenberg complex.