PBW deformations of quantum symmetric algebras and their group extensions. (Q2797004)

Let \(V\) be a vector space with a basis \(v_1,\ldots,v_n\) over a field \(k\) of characteristic zero. Fix a group \(G\) of invertible linear operators in \(V\) and \(\kappa\) a bilinear map \(V\otimes V\to(k\oplus V)\otimes kG\) such that \(\kappa(v_i,v_j)=q_{ij}\kappa(v_j,v_i)\) for all \(i,j=1,\ldots,n\). Here \(q_{ij}\in k^*\) and \(q_{ii}=q_{ij}q_{ji}=1\). Denote by \(\mathcal H_{\mathbf q,\kappa}\) the algebra \(T(V)\rtimes G\) factorized by relations \(v_iv_j-q_{ij}v_jv_i-\kappa(v_i,v_j)\) for all \(i,j\).NEWLINENEWLINE The object of study is the quantum Drinfeld orbifold algebra \(\mathcal H_{\mathbf q,\kappa}\) whose associated graded algebra is isomorphic to \(\mathcal O_{\mathbf q}\rtimes G\), where \(\mathcal O_{\mathbf q}\) is a quantum polynomial algebra with defining relations \(v_iv_j=q_{ij}v_jv_i\). There are found conditions under which \(\mathcal H_{\mathbf q,\kappa}\) is a quantum Drinfeld orbifold algebra. It is shown that in this case the action of \(G\) is compatible with passing to the associated graded algebra. It is shown that generalized universal enveloping algebra \(U_\omega(L)\) of a color Lie algebra \(L\) is isomorphic to \(\mathcal H_{\mathbf q,\kappa}\) with \(G=1\). In section 4 there is given an interpretation of condition being quantum Drinfeld orbifold algebra in terms of Gerstenhaber brackets and Hochschild cohomology.