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

Vol. 14 No. 3 (2019) P.295~P.329

DOI:	https://doi.org/10.21915/BIMAS.2019302
	10.21915/BIMAS.2019302

ABSTRACT

Let $(\mathfrak g,[p])$ be a finite-dimensional restricted Lie algebra over an algebraically closed field $\mathbb K$ of characteristic $p>0$, and $G$ be the adjoint group of $\mathfrak g$. We say that $\mathfrak g$ satisfies the {\sl generic property} if $\mathfrak g$ admits generic tori introduced in [2]. In this paper, we first prove a generalized conjugacy theorem for Cartan subalgebras by means of the generic property. We then classify the $G$-conjugacy classes of homogeneous Borel subalgebras of the restricted simple Lie algebras $\mathfrak g=W(n)$ when $p>3$, and determine representatives of these classes. Here $W(n)$ is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring $\mathbb K[T_1,\cdots,T_n]/(T_1^p,\cdots,T_n^p)$. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras.

KEYWORDS
Borel subalgebras of restricted Lie algebras, generic elements, generic property, Lie algebras of Cartan type

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 17B50, 17B05, 17B20

MILESTONES

Received: 2018-04-04
Revised :
Accepted: 2018-09-13

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