Commuting varieties of Lie algebras over fields of prime characteristic
Let \(\mathfrak{g}\) be a Lie algebra over an algebraically closed field \(K\). The commuting variety \(\mathcal{C}(\mathfrak{g})\) of \(\mathfrak{g}\) is defined as \(\{(x,y)\in \mathfrak{g\times g}\mid [x,y]=0\}\), and \(\mathcal{C}(\mathfrak{g})\) is clearly a Zariski closed subset of \(\mathfrak{g\times g}\). In [\textit{P. Tauvel}, Bull. Sci. Math. (2) 113, 51--83 (1989; Zbl 0681.17008)], it was proved that if \(K\) is of characteristic zero and \(\mathfrak{g}\) is a reductive Lie algebra over \(K\), then \(\mathcal{C}(\mathfrak{g})\) is an irreducible variety. In the article under review, the author extends this result to the case where \(K\) is of characteristic \(p > 0\) under certain mild restrictions on \(\mathfrak{g}\).
scientific article; zbMATH DE number 1763542