On Dixmier-Duflo isomorphism in positive characteristic -- the classical nilpotent case (Q2437447)

Let \(\mathfrak g\) be a finite dimensional Lie algebra over a field \(F\). The set consisting of the elements of \(S(\mathfrak g)\) which are annihilated by \(\mathrm{ad}\,\mathfrak g\), is denoted by \(S(\mathfrak g)^g\). In zero characteristic, there is a well-known algebra isomorphism of \(Z(\mathfrak g)\) onto \(S(\mathfrak g)^{\mathfrak g}\). In this paper, the author studies the structure of \(Z(\mathfrak g)\) and \(S(\mathfrak g)^{\mathfrak g}\) in positive characteristic. Let \(\mathfrak g\) be the nil radical of the Borel subalgebra of one of the classical simple Lie algebras (of type \(A_n\), \(B_n\), \(C_n\), \(D_n\)) over a field \(F\) of positive characteristic \(p\). The author proves that there is a well-known algebra isomorphism of \(Z(\mathfrak g)\) onto \(S(\mathfrak g)^{\mathfrak g}\) under a certain condition. Moreover, he presents two examples of non-nilpotent Lie algebras \(\mathfrak g\) with \(Z(\mathfrak g)\) not isomorphic to \(S(\mathfrak g)^{\mathfrak g}\).