On some universal algebras associated to the category of Lie bialgebras

Abstract

In our previous work (math/0008128), we studied the set Quant (K) of all universal quantization functors of Lie bialgebras over a field K of characteristic zero, compatible with the operations of taking duals and doubles. We showed that Quant (K) is canonically isomorphic to a product G0(K) x (K), where G0(K) is a universal group and (K) is a quotient set of a set B(K) of families of Lie polynomials by the action of a group G(K). We prove here that G0(K) is equal to the multiplicative group 1 + ℏK[[ℏ]]. So Quant(K) is "as close as it can be" to (K). We also prove that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup generated by the corresponding "square of the antipode." © 2001 Elsevier Science.