English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 $\mathfrak{g}$ satisfying the {\sl generic property} if $\mathfrak{g}$ admits generic tori introduced in \cite{BFS}. A Borel subalgebra (or Borel for short) of $\mathfrak{g}$ is by definition a maximal solvable subalgebra containing a maximal torus of $\mathfrak{g}$, which is further called generic if additionally containing a generic torus. In this paper, we first settle a conjecture proposed by Premet in \cite{Pr2} on regular Cartan subalgebras of restricted Lie algebras. We prove that the statement in the conjecture for a given $\mathfrak{g}$ is valid if and only if it is the case when $\mathfrak{g}$ satisfies the generic property. We then classify the conjugay classes of homogeneous Borel subalgebras of the restricted simple Lie algebras $\mathfrak{g}=W(n)$ 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 $\mathbb{K}[T_1,\cdots,T_n]\slash (T_1^p,\cdots,T_n^p)$. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras.

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