Nilpotent commuting varieties of the Witt algebra

Abstract

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathcal{N}=x\in ⋙\mid x^{[p]}=0$ be the nilpotent variety of $\mathfrak{g}$, and $\mathcal{C}(\mathcal{N}):=\{(x,y)\in \mathcal{N}\times \mathcal{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathcal{C}(\mathcal{N})$ is reducible and equidimensional. Irreducible components of $\mathcal{C}(\mathcal{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.

10 pages. Comments are welcome

References in corpus (2)

• Nilpotent commuting varieties of reductive Lie algebras
• Borel subalgebras of the Witt algebra $W_1$

Cited by in corpus (1)

• Commuting varieties for nilpotent radicals