Borel subalgebras of the Witt algebra

Yao, Yu Feng; Chang, Hao

doi:10.1007/s10114-015-4425-z

Cited by 4 publications

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, Gr(B) contains Kx∂. This implies that B contains nonzero semi-simple elements, and then contains some maximal torus of g. Therefore, the above proposition covers the main result of [23].…”

Section: Recallsupporting

confidence: 64%

Generic Property and Conjugacy Classes of Homogeneous Borel Subalgebras of Restricted Lie Algebras of Cartan Type (I): Type W

Shu¹

2019

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

View full text Add to dashboard Cite

Let (g, [p]) be a finite-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0, and G be the adjoint group of g. We say that g satisfying the generic property if g admits generic tori introduced in [2]. A Borel subalgebra (or Borel for short) of g is by definition a maximal solvable subalgebra containing a maximal torus of g, which is further called generic if additionally containing a generic torus. In this paper, we first settle a conjecture proposed by Premet in [15] on regular Cartan subalgebras of restricted Lie algebras. We prove that the statement in the conjecture for a given g is valid if and only if it is the case when g satisfies the generic property. We then classify the conjugay classes of homogeneous Borel subalgebras of the restricted simple Lie algebras g = W (n) under G-conjugation when p > 3, and present the representatives of these classes. Here W (n) is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring K[T 1 , · · · , T n ]/(T p 1 , · · · , T p n ). We also describe the closed connected solvable subgroups of G associated with those representative Borel subalgebras. BIN SHUclass of Cartan subalgebras, by definition the ones containing a maximal torus of maximal dimension, are proved to be conjugate by means of a finite number of so-called elementary switchings (some invertible linear transformations provided by root vectors). He further proposed a conjecture on the conjugation situation of regular Cartan subalgebras under the adjoint group G of g (see Conjecture 1.5). As to Borel subalgebras, there is less study on them so far. Especially, for nonclassical restricted simple Lie algebras, we have no answer to the questions how many and what the conjugacy classes of Borel subalgebras are, and what kind of role the Borel subalgebras could play in the representations of the Lie algebras (for those Lie algebras, Cartan subalgebras have been well-known (cf. [20]). Our motivation is to understand more on Borel subalgebras of non-classical restricted simple Lie algebras, and to exploit some possible connection among the theory of Borel subalgebras, representations and others.The purpose of the present paper is twofold. One is to settle Premet's conjecture mentioned above, the other one is to study Borel subalgebras of restricted Lie algebras.According to Block-Wilson-Strade-Premet classification of restricted simple Lie algebras over an algebraically closed field of characteristic p > 3 (cf. [17]), it is known that aside from the analogues of the complex simple Lie algebras (called classical Lie algebras) there are usually four additional classes of restricted simple Lie algebras 1 , the so-called restricted Lie algebras of Cartan type, among of which the Jacobson-Witt algebras W (n) will be a main topic of the present paper. In his article [16], Premet studied analogy of Weyl groups and of the Chevalley restriction theorem in the complex simple Lie algebras, and the variety of nilpotent elements for W (...

show abstract