Group gradings on the Lie algebra \(\mathfrak{psl}_n\) in positive characteristic
Gradings by abelian groups of the simple Lie algebras of classical type (other that \(D_4\)) over an algebraically closed field of characteristic \(0\) have been described recently in a series of papers. These results were extended to the prime characteristic case (\(p\neq 2\)) by the authors of this paper, together with \textit{S.~Montgomery} [Proc. Am. Math. Soc. 137, No. 4, 1245--1254 (2009; Zbl 1168.17016)], except for the \(A_{pk-1}\) case. The paper under review fills this gap. Given an algebraically closed field \(F\) of characteristic \(p\), let \(R=\text{Mat}_n(F)\) be the algebra of \(n\times n\) matrices over \(F\), with \(n\) a multiple of \(p\): \(n=pk\). Then the simple Lie algebra of type\(A_{pk-1}\) is \(L=\mathfrak{psl}_n(F)=[R,R]/Z(R)\). In characteristic \(0\), any grading over an abelian group \(G\) of a finite dimensional algebra \(A\) is given by an embedding of the character group \(\hat G\) into the algebraic group of automorphisms of \(A\). This is no longer valid in the modular case, but Hopf algebraic techniques can be used instead. This allows the authors to get the gradings on \(L\) from suitable gradings on \(R\) and solve the problem. The results in this paper and the other papers mentioned above have already been simplified and improved by the same authors [``Classification of group gradings on simple Lie algebras of types \(A\), \(B\), \(C\) and \(D\)'', arXiv:0908.0906v1 [math.RA]].
scientific article; zbMATH DE number 5561726
scientific article