Commuting variety of Witt algebra
Let \(\mathfrak{g}\) be a Lie algebra over an algebraically closed field. Then the commuting variety \(\mathcal{C}(\mathfrak{g})=\{(x,y)\in\mathfrak{g} \times\mathfrak{g}\mid[x,y]=0\}\) of \(\mathfrak{g}\) is a Zariski closed subset of \(\mathfrak{g}\times\mathfrak{g}\). \textit{P. Levy} [J. Algebra 250, No. 2, 473--484 (2002; Zbl 1034.17012)] proved under mild conditions on the characteristic of the ground field that the commuting variety of the Lie algebra of a reductive algebraic group is an irreducible algebraic variety. In the paper under review the authors investigate the commuting variety of the smallest simple Lie algebra that is not the Lie algebra of a reductive algebraic group, namely, the \(p\)-dimensional Witt algebra \(W_1\) over a field of characteristic \(p >3\). It is shown that the commuting variety of \(W_1\) is the union of \(\frac{p-1}{2}\) irreducible components, one of which is of dimension \(p+1\), and the others are all of dimension \(p\). In particular, this variety is never irreducible, and as the origin belongs to any irreducible component, it follows that this variety is also never normal.
scientific article; zbMATH DE number 6985520
scientific article