Differential operators on Hopf algebras and some functorial properties (Q1849597)

Let \(R\) be an associative unital algebra over a field \(\mathbb{K}\). Let \(M\) be an \(R\)-bimodule. Then the centre of \(M\) is defined by \(Z(M)=\{m\in M\mid rm=mr\) for all \(r\in R\}\). The differential part of \(M\) is the \(R\)-bimodule \(M_{\text{diff}}=\bigcup^\infty_{i=0}M_i\), where \(M_i\) is the \(R\)-bimodule generated by \(\{m\in M\mid\overline m\in Z(M/M_{i-1})\) with \(M_{-1}=0\). Then \(M_i\) is called the \(i\)-th differential part of \(M\). \(\Hom_\mathbb{K}(R,R)\) is an \(R\)-bimodule as follows: For \(r,s\in R\) and \(\phi\in\Hom_\mathbb{K}(R,R)\), let \((r\phi)(s)=r\phi(s)\) and \((\phi r)(s)=\phi(rs)\). In this case, the elements of the differential part of the \(R\)-bimodule \(\Hom_\mathbb{K}(R,R)\) are called the differential operators on \(R\). The \(R\)-bimodule \(D^m_\mathbb{K}(R)\) of differential operators on \(R\) of order \(\leq m\) is the \(m\)-th differential part of \(\Hom_\mathbb{K}(R,R)\). \(D(R)=\bigcup^\infty_{i=0}D^i_\mathbb{K}(R)\) is an algebra, called the ring of differential operators. The paper describes some results for Hopf algebras and general rings as follows. Let \(R\) be a Hopf algebra over \(\mathbb{K}\) and let \(M\) be an \(R\)-bimodule. If the first differential part \(M_1\) is a semisimple \(R\)-module under the left adjoint action: \(r\cdot m=\sum r_1mS(r_2)\), then \(M_{\text{diff}}=M_0\). Let \(G\) be a finite group and \(\mathbb{K}\) be a field with characteristic not dividing \(|G|\). Then \(M_{\text{diff}}=M_0\) for any \(\mathbb{K}[G]\)-bimodule \(M\). In particular, \(D(\mathbb{K}[G])=D^0(\mathbb{K}[G])\). Let \(\mathcal G\) be a finite-dimensional semisimple Lie algebra over a field \(\mathbb{K}\) with characteristic 0 and let \(U:=U({\mathcal G})\) be the universal enveloping algebra. Then \(M_{\text{diff}}=M_0\) for any \(U\)-bimodule \(M\). In particular, \(D(U)=D^0(U)\), and the ring \(D^0(U)\) is Noetherian. Let \(Z\) be the centre of \(U\). Then \(U\otimes_ZU^o\cong D^0(U)\) as rings, and every derivation of \(U\) is inner, where \(U^o\) denotes the opposite ring of \(U\). Let \(R\) and \(S\) be two \(\mathbb{K}\)-algebras. Let \(M\) be an \(R\)-bimodule and \(N\) an \(S\)-bimodule. Then \(M\otimes N\) is an \(R\otimes S\)-bimodule in a natural way. There is an isomorphism of bimodules \((M\otimes N)_{\text{diff}}\cong M_{\text{diff}}\otimes N_{\text{diff}}\). If \(R\) and \(S\) are finitely generated \(\mathbb{K}\)-algebras, then \(D(R\otimes S)\cong D(R)\otimes D(S)\). Let \(\Gamma\) be an Abelian group. If \(R\) is \(\Gamma\)-graded, finitely generated as an algebra over \(\mathbb{K}\), then \(D(R)\subset\text{grHom}_\mathbb{K}(R,R)\), i.e. \(D(R)\) is a \(\Gamma\)-graded \(R\)-bimodule. Let \(A=\bigoplus_{\alpha\in\Lambda}A_\alpha\) be a direct sum of \(\mathbb{K}\)-algebras. For any \(\alpha\in\Lambda\), let \(M_\alpha\) be an \(A_\alpha\)-bimodule. Then \(M=\bigoplus_{\alpha\in\Lambda}M_\alpha\) is given the obvious \(A\)-bimodule structure and \(M_{\text{diff}}=\bigoplus_{\alpha\in\Lambda}(M_\alpha)_{\text{diff}}\). The ring of differential operators on \(A\) is a direct sum \(D(A)=\bigoplus_{\alpha\in\Lambda}D(A_\alpha)\). Note that \(\Gamma\) has to be a finite set, otherwise, \(A\) is not unital.