Note on the cohomology of color Hopf and Lie algebras (Q2497463)

Let \(G\) be an abelian group, \(K\) a field of characteristic zero, and \(X\) an antisymmetric bicharacter from \(GxG\) to \(K\). \(X\) can be used to give a new \(G\)-graded algebra structure \(A^X\) to a a \(G\)-graded algebra \(A\), a new \(G\)-graded module structure \(M^X\) to a \(G\)-graded module \(M\), and a new \(G\)-graded coalgebra structure \(C^X\) to a \(G\)-graded coalgebra \(C\). This leads to the notion of a \((G,X)\)-Hopf algebra, one feature of which is that the antipode is a graded map. If the antipode is bijective, a \((G, X)\)-Hopf algebra is called a color Hopf algebra. The first main theorem is that if \(A\) is a color Hopf algebra and \(M\) is a graded \(A\)-bimodule, then the graded Hochschild cohomology \(H(H_{gr}^n(A, M))\) is isomorphic to \(\text{Ext}_{A-gr}^n(K, {}^{ad}M)\), where \(K\) is the trivial graded \(A\)-module (via the counit of \(A\)), and \({}^{ad}M\) is the adjoint A-module of \(M\). This result is applied to color Lie algebras \(L\), i.e., \((G,X)\)-Lie algebras (there is a graded Jacobi identity with scalar coefficients involving \(X\)), to obtain that if \(M\) is a graded \(U(L)\)-module, then the graded Hochschild cohomology \(H(H_{gr}^n(U(L), M))\) is isomorphic to the graded cohomology \(H_{gr}^n(L, {}^{ad}M)\). The proof also uses information coming from the color Koszul resolution of the trivial module \(K\) of \(L\). This answers a question of \textit{M. Scheunert} in the case of degree zero [Recent advances in Lie theory. Selected contributions to the 1st colloquium on Lie theory and applications, Vigo, Spain, July 17-22, 2000. Lemgo: Heldermann Verlag. Res. Expo. Math. 25, 77--107 (2002; Zbl 1022.17013)].