Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras

Abstract

Abstract Let W ⁢ ( n ) {W(n)} be the Jacobson–Witt algebra over algebraic closed field 𝕂 {\mathbb{K}} with characteristic p > 2 {p>2} . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ ( n ) {W(n)} , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ {\mathcal{B}} consisting of all B-subalgebras of W ⁢ ( n ) {W(n)} in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ {\mathcal{B}} is related to the variety of all full flags in 𝕂 n + 1 {\mathbb{K}^{n+1}} . Additionally, we provide a detailed description of the varieties for W ⁢ ( 1 ) {W(1)} as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras.