Quantization of \(\Gamma \)-Lie bialgebras (Q933371)

A cocommutative bialgebra \(P\) with multiplication \(m\) and comultiplication \(c\) is said to be co-Poisson if it also has a cobracket \(d\) of \(P\) to the exterior product of \(P\) with itself which is a derivation for \(m\) and a coderivation for \(c\). A quantization of \((P,m,c,d)\) is a bialgebra structure \((P[[h]],m_h,c_h)\) such that \(m_h=m+O(h)\), \(c_h=c+O(h)\) and the skew-symmetrization of \((c_h)(p)\) is \(d(p)+O(h^2)\) for all \(p\) in \(P\). For \(L\) a Lie algebra, co-Poisson bialgebra structures on \(U(L)\) correspond to Lie bialgebra structures on \(L\). A quantization of such co-Poisson bialgebras was given by \textit{P. Etingof} and \textit{D. Kazhdan} [Sel. Math., New Ser. 2, No. 1, 1--41 (1996; Zbl 0863.17008)], and the first author of the paper under review gave an alternative version [Adv. Math. 197, No. 2, 430--479 (2005; Zbl 1127.17013)]. If \(L\) is a Lie algebra, \(G\) a group acting by automorphisms on \(L\), the semi-direct product of \(U(L)\) and \(G\) is a \(G\)-graded co-Poisson bialgebra. The \(G\)-graded co-Poisson bialgebra structures on this semi-direct product correspond to pairs \((d,f)\), where \(d\) is a cobracket on \(L\) giving a Lie bialgebra structure, and \(f\) is a map from \(G\) to to the exterior product of \(L\) with itself satisfying certain conditions. The Lie bialgebra \(L\), together with the action of \(G\) on \(L\), and \(f\), is called a \(G\)-Lie bialgebra. The paper under review gives a quantization of the corresponding \(G\)-graded co-Poisson bialgebra structure. The main ingredient is to use some results of an article by the authors [Quantization of coboundary Lie bialgebras, to appear in Ann. Math. (2), Preprint \url{mathQA/0603740}] to show that the quantization of Etingof and Kazhdan is compatible with twists of Lie algebras. This is based on the alternative quantization of Lie bialgebras, mentioned earlier, due to the first author.