GROWTH OF SUBALGEBRAS FOR RESTRICTED LIE ALGEBRAS AND TRANSITIVE ACTIONS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1151-1168 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Group Theory ◽  
Finite Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finitely Generated ◽  
Subgroup Growth ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Transitive Actions

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then an(L) is the number of restricted subalgebras H ⊂ L such that dim 𝔽q L/H = n, n ≥ 0. We compute the numbers an(Fd) explicitly and find asymptotics, where Fd is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

2008 ◽  
Vol 18 (02) ◽  
pp. 271-283 ◽  
Author(s):  
HAMID USEFI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Groups ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

scholarly journals Subalgebras of free restricted Lie algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 147-156 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Main Step ◽  
Free Lie Algebra ◽  
Generating Set ◽  
Restricted Lie Algebra ◽  
Corrected Proof ◽  
Restricted Lie Algebras ◽  
Remarkable Result ◽  
Algebra Over A Field

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

Restricted Lie Algebras and Their Envelopes

1995 ◽  
Vol 47 (1) ◽  
pp. 146-164 ◽  
Author(s):  
D. M. Riley ◽  
A. Shalev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Codimension ◽  
Augmentation Ideal ◽  
Nilpotent Ideal ◽  
Enveloping Algebra ◽  
Group Of Units ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Algebra Over A Field

AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

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

2019 ◽  
Vol 18 (03) ◽  
pp. 1950056
Author(s):  
Cong Chen
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Closed Field ◽  
Nilpotent Variety ◽  
Restricted Lie Algebra ◽  
Finite Dimensional ◽  
Restricted Lie Algebras ◽  
Minimal Formula

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].

scholarly journals On powerful and p-central restricted Lie algebras

2007 ◽  
Vol 75 (1) ◽  
pp. 27-44 ◽  
Author(s):  
S. Siciliano ◽  
Th. Weigel
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Ring ◽  
Cup Product ◽  
Restricted Lie Algebra ◽  
Finite Dimensional ◽  
Restricted Lie Algebras ◽  
Product Map

In this note we analyse the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups (Theorem A). Every finite-dimensional restricted Lie algebra  is isomorphic to for some finite-dimensional p-central restricted Lie algebra (Proposition B). In particular, for restricted Lie algebras there does not hold an analogue of J.Buckley's theorem. For p odd one can characterise powerful restricted Lie algebras in terms of the cup product map in the same way as for finite p-groups (Theorem C). Moreover, the p-centrality of the finite-dimensional restricted Lie algebra  has a strong implication on the structure of the cohomology ring H•(,) (Theorem D).

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

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

scholarly journals The Nowicki conjecture for free metabelian Lie algebras

2019 ◽  
Vol 19 (05) ◽  
pp. 2050095
Author(s):  
Vesselin Drensky ◽  
Şehmus Fındık
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Algebra ◽  
Vector Subspace ◽  
Classical Theorem ◽  
Finitely Generated ◽  
Commutator Ideal ◽  
Ideal Formula ◽  
Locally Nilpotent ◽  
Derivation Formula

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

scholarly journals Restricted Lie algebras of maximal class

1999 ◽  
Vol 59 (2) ◽  
pp. 217-223
Author(s):  
D.M. Riley
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Class ◽  
Infinite Dimensional ◽  
Restricted Lie Algebras ◽  
Infinite Dimensional Lie Algebra

Let L be a possibly infinite-dimensional Lie algebra of maximal class. We show that if L admits the structure of a Lie p-algebra then the dimension of L can be at most p + 1. Furthermore, this bound is best possible.

scholarly journals On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras

2021 ◽  
Vol 52 ◽  
Author(s):  
Farshid Saeedi ◽  
Nafiseh Akbarossadat
Keyword(s):  
Group Theory ◽  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Short Exact Sequence ◽  
Upper Bound ◽  
Tensor Square ◽  
Algebra Over A Field

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

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