Varieties of a class of elementary subalgebras

<abstract><p>Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p &gt; 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.</p></abstract>

On the First Hochschild Cohomology of Cocommutative Hopf Algebras of Finite Representation Type

Let $\mathscr{B}_0({\mathcal{G}})\subseteq k\,{\mathcal{G}}$ be the principal block algebra of the group algebra $k\,{\mathcal{G}}$ of an infinitesimal group scheme ${\mathcal{G}}$ over an algebraically closed field $k$ of characteristic ${\operatorname{char}}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology ${\mathcal{L}}:={\operatorname{H}}^1(\mathscr{B}_0({\mathcal{G}}),\mathscr{B}_0({\mathcal{G}}))$ whenever $\mathscr{B}_0({\mathcal{G}})$ has finite representation type. As a consequence, we prove that the complexity of the trivial ${\mathcal{G}}$-module $k$ coincides with the maximal toral rank of ${\mathcal{L}}$.

The nilpotent variety of W(1;n)p is irreducible

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

A Note on the Rank of a Restricted Lie Algebra

In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra.

Examples of uniform exponential growth in algebras

In this paper, we discuss the concept and examples of algebras of uniform exponential growth. We prove that Golod–Shafarevich algebras and group algebras of Golod–Shafarevich groups are of uniform exponential growth. We prove that uniform exponential growth of the universal enveloping algebra of a Lie algebra [Formula: see text] implies uniform exponential growth of [Formula: see text], and conversely should [Formula: see text] be graded by the natural numbers. We prove that a restricted Lie algebra is of uniform exponential growth if and only if its universal enveloping algebra is. We proceed to give several conditions equivalent to the uniform exponential growth of the graded algebra associated to a group algebra filtered by powers of its fundamental ideal.

Vector bundles associated to Lie algebras

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn

In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic

ON CARTAN INVARIANTS FOR GRADED LIE ALGEBRAS OF CARTAN TYPE

Let L = X(m; n), X ∈ {W, S, H, K}, be a graded simple Lie algebra of Cartan type over an algebraically closed field of characteristic p > 3. Then L is a so-called generalized restricted Lie algebra. Let [Formula: see text] be the primitive p-envelope of L, and G = X(m; 1), a subalgebra of [Formula: see text]. In this paper, a close connection between Cartan invariants for [Formula: see text] and U(G, χ) is established, where χ ∈ L* is extended to be a linear function on [Formula: see text] trivially, and 1 ≤ ht (χ) < p-2+δXW. This reduces the study of projective representations of the generalized restricted Lie algebra L to the one of the corresponding restricted Lie algebra G. As a special case, we recover some results in [Shu and Jiang, On Cartan invariants and blocks of Zassenhaus algebras, Comm. Algebra33(10) (2005) 3619–3630].