Varieties of Borel subalgebras for the Jacobson-Witt Lie algebras

摘要

Let W(n) be the Jacobson-Witt algebra over algebraic closed field K with characteristic p > 2. In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 (2022), no. 11, Paper No. 2250210], we introduced the so-called B-subalgebra of 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 consisting of all B-subalgebras of W(n) in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that B is related to the variety of all full flags in Kn+1. Additionally, we provide a detailed description of the varieties for 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.

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