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 $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

2011 ◽  
Vol 90 (3) ◽  
pp. 403-430 ◽  
Author(s):  
YU-FENG YAO ◽  
BIN SHU
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Irreducible Representations ◽  
Full Description ◽  
Closed Field ◽  
Maximal Subalgebra ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Restricted Lie Algebra ◽  
Graded Lie Algebra

AbstractLetL=H(2r;n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristicp>2. In the generalized restricted Lie algebra setup, any irreducible representation ofLcorresponds uniquely to a (generalized)p-characterχ. When the height ofχis no more than min {pni−pni−1∣i=1,2,…,2r}−2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebraL0with the aid of an analogy of Skryabin’s category ℭ for the generalized Jacobson–Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations withp-characters of height below this number.

scholarly journals On Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

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.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

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.

A Remark on the Proof of a Theorem of Laufer and Tomber

1971 ◽  
Vol 23 (2) ◽  
pp. 270-270 ◽  
Author(s):  
Hyo Chul Myung
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Cartan Subalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra

In this note we give a correction to the proof of the following theorem [1, Theorem 2].THEOREM. Letbe a flexible, power-associative algebra, over an arbitrary algebraically closed field Ω of characteristic 0. Ifis a simple Lie algebra, thenis a simple Lie algebra isomorphic to.Step (i) of the proof, which proves that the Cartan subalgebra of is a nil subalgebra of , is incomplete. Assuming that is not a nil subalgebra of , there exists an idempotent e ≠ 0 in .

Codimension growth of central polynomials of Lie algebras

2020 ◽  
Vol 32 (1) ◽  
pp. 201-206
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Adjoint Representation ◽  
Closed Field ◽  
Polynomial Identities ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Linearly Independent

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like {(\dim L)^{n}}.

The Loewy series of principal indecomposable modules in the regular nilpotent representations of 𝔰𝔩(3,k)

2019 ◽  
Vol 18 (11) ◽  
pp. 1950205
Author(s):  
Yi-Yang Li ◽  
Yu-Feng Yao
Keyword(s):  
Lie Algebra ◽  
Levi Form ◽  
The Self ◽  
Closed Field ◽  
Prime Characteristic ◽  
Indecomposable Modules ◽  
Algebraically Closed Field ◽  
Enveloping Algebra ◽  
Simple Lie Algebra ◽  
Standard Levi Form

Let [Formula: see text] be a simple Lie algebra of type [Formula: see text] over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text] be the reduced enveloping algebra of [Formula: see text]. In this paper, when the [Formula: see text]-character [Formula: see text] is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable [Formula: see text]-modules and the dimensions for the self-extension of irreducible [Formula: see text]-modules.

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>

Sign in / Sign up

Export Citation Format

Share Document