scholarly journals Varieties of a class of elementary subalgebras

2021 ◽  
Vol 7 (2) ◽  
pp. 2084-2101
Author(s):  
Yang Pan ◽  
◽  
Yanyong Hong ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Positive Characteristic ◽  
Maximal Dimension ◽  
Closed Field ◽  
Characteristic P ◽  
Simple Algebraic Group ◽  
Restricted Lie Algebra ◽  
Irreducible Components ◽  
Type C

<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>

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

Note on Primitive Ideals of Enveloping Algebras in Prime Characteristic

2011 ◽  
Vol 18 (04) ◽  
pp. 701-708
Author(s):  
Yufeng Yao
Keyword(s):  
Maximal Dimension ◽  
Closed Field ◽  
Central Character ◽  
Characteristic P ◽  
Reductive Algebraic Group ◽  
Restricted Lie Algebra ◽  
Primitive Ideals ◽  
Alternative Description ◽  
Stable Ideal ◽  
The Ideal

Let [Formula: see text] be a restricted Lie algebra over an algebraically closed field F of characteristic p > 0, [Formula: see text] the center of the universal enveloping algebra [Formula: see text] of [Formula: see text]. In this note, we study primitive ideals of [Formula: see text]. The following results are included: (1) The ideal of [Formula: see text] generated by the central character ideal associated with any irreducible [Formula: see text]-module has finite co-dimension in [Formula: see text]. Furthermore, the co-dimension is no less than [Formula: see text], where [Formula: see text] is the maximal dimension of irreducible [Formula: see text]-modules. (2) Each annihilator ideal of irreducible [Formula: see text]-modules of maximal dimension is generated by the corresponding central character ideal in [Formula: see text]. (3) Each G-stable ideal in [Formula: see text] for [Formula: see text] contains nonzero fixed points under the action of G, where G is a connected reductive algebraic group. Additionally, the arguments on ideals help us to give an alternative description of the Azumaya locus in the Zassenhaus variety without using the normality of the Zassenhaus variety.

scholarly journals Maximal finite abelian subgroups of E8

2017 ◽  
Vol 147 (5) ◽  
pp. 993-1008 ◽  
Author(s):  
Cristina Draper ◽  
Alberto Elduque
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Homogeneous Component ◽  
Closed Field ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Irreducible Modules ◽  
Abelian Subgroups

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

scholarly journals Derived subalgebras of centralisers and finite -algebras

2014 ◽  
Vol 150 (9) ◽  
pp. 1485-1548 ◽  
Author(s):  
Alexander Premet ◽  
Lewis Topley
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Dimension ◽  
Closed Field ◽  
Natural Action ◽  
Simple Algebraic Group ◽  
Finite Algebras ◽  
Nilpotent Elements ◽  
Primitive Ideals ◽  
Exceptional Lie Algebras

AbstractLet$\mathfrak{g}=\mbox{Lie}(G)$be the Lie algebra of a simple algebraic group$G$over an algebraically closed field of characteristic$0$. Let$e$be a nilpotent element of$\mathfrak{g}$and let$\mathfrak{g}_e=\mbox{Lie}(G_e)$where$G_e$stands for the stabiliser of$e$in$G$. For$\mathfrak{g}$classical, we give an explicit combinatorial formula for the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and use it to determine those$e\in \mathfrak{g}$for which the largest commutative quotient$U(\mathfrak{g},e)^{\mbox{ab}}$of the finite$W$-algebra$U(\mathfrak{g},e)$is isomorphic to a polynomial algebra. It turns out that this happens if and only if$e$lies in a unique sheet of$\mathfrak{g}$. The nilpotent elements with this property are callednon-singularin the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element$e\in \mathfrak{g}$is non-singular if and only if the maximal dimension of the geometric quotients$\mathcal{S}/G$, where$\mathcal{S}$is a sheet of$\mathfrak{g}$containing$e$, coincides with the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element$e$in a classical Lie algebra$\mathfrak{g}$the closed subset of Specm $U(\mathfrak{g},e)^{\mbox{ab}}$consisting of all points fixed by the natural action of the component group of$G_e$is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.

Irreducible Representations of the Generalized Jacobson-Witt Algebras

2012 ◽  
Vol 19 (01) ◽  
pp. 53-72 ◽  
Author(s):  
Bin Shu ◽  
Yufeng Yao
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
Module Structure ◽  
Closed Field ◽  
Witt Algebra ◽  
Divided Power ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Restricted Lie Algebra ◽  
Induced Module

Let L be the generalized Jacobson-Witt algebra W(m;n) over an algebraically closed field F of characteristic p > 3, which consists of special derivations on the divided power algebra R= 𝔄(m;n). Then L is a so-called generalized restricted Lie algebra. In such a setting, we can reformulate the description of simple modules of L with the generalized p-character χ when ht (χ) < min {pni-pni-1| 1 ≤ i ≤ m} for n=(n1,…,nm), which was obtained by Skryabin. This is done by introducing a modified induced module structure and thereby endowing it with a so-called (R,L)-module structure in the generalized χ-reduced module category, which enables us to apply Skryabin's argument to our case. Simple exceptional-weight modules are precisely constructed via a complex of modified induced modules, and their dimensions are also obtained. The results for type W are extended to the ones for types S and H.

scholarly journals On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

2014 ◽  
Vol 58 (1) ◽  
pp. 169-181 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Finite Number ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Irreducible Components

AbstractLet G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

ON CARTAN INVARIANTS FOR GRADED LIE ALGEBRAS OF CARTAN TYPE

2013 ◽  
Vol 13 (03) ◽  
pp. 1350101
Author(s):  
BIN SHU ◽  
YU-FENG YAO
Keyword(s):  
Lie Algebra ◽  
Closed Field ◽  
Graded Lie Algebras ◽  
Characteristic P ◽  
Cartan Type ◽  
Restricted Lie Algebra ◽  
Projective Representations ◽  
Cartan Invariants ◽  
The One ◽  
Special Case

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].

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

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