Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Generation of relative commutator subgroups in Chevalley groups

2015 ◽  
Vol 59 (2) ◽  
pp. 393-410 ◽  
Author(s):  
R. Hazrat ◽  
N. Vavilov ◽  
Z. Zhang
Keyword(s):  
Normal Subgroup ◽  
Chevalley Group ◽  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Elementary Subgroup ◽  
Elementary Group ◽  
Commutator Width ◽  
Relative Commutator ◽  
Elementary Generator ◽  
Elementary Chevalley Group

AbstractLet Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

scholarly journals Generation of relative commutator subgroups in Chevalley groups. II

2020 ◽  
Vol 63 (2) ◽  
pp. 497-511 ◽  
Author(s):  
Nikolai Vavilov ◽  
Zuhong Zhang
Keyword(s):  
Root System ◽  
Commutative Ring ◽  
Chevalley Group ◽  
Congruence Subgroup ◽  
Unexpected Result ◽  
Chevalley Groups ◽  
Elementary Subgroup ◽  
Commutator Formula ◽  
Rank 2 ◽  
Relative Commutator

AbstractIn the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

On ϕ-ideals and the structure of Lie algebras

2020 ◽  
pp. 2150193
Author(s):  
Leila Goudarzi ◽  
Zahra Riyahi
Keyword(s):  
Normal Subgroup ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Normal Subgroups ◽  
Basic Properties ◽  
Finite Dimensional

This paper aims to study the concept of [Formula: see text]-ideals of a finite-dimensional Lie algebra which is analogous to the concept of [Formula: see text]-ideal and [Formula: see text]-normal subgroup. We compile some basic properties of [Formula: see text]-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for [Formula: see text]-ideals of Lie algebras and [Formula: see text]-normal subgroups are not valid here.

Normal Structure of Maximal Parabolic Subgroups in Chevalley Groups over Rings

2009 ◽  
Vol 16 (04) ◽  
pp. 631-648 ◽  
Author(s):  
Anastasia Stavrova
Keyword(s):  
Commutative Ring ◽  
Parabolic Subgroup ◽  
Chevalley Group ◽  
Normal Structure ◽  
Levi Subgroup ◽  
Parabolic Subgroups ◽  
Chevalley Groups ◽  
Maximal Parabolic Subgroup ◽  
Elementary Subgroup ◽  
Maximal Parabolic Subgroups

We study the normal structure of maximal parabolic subgroups of a Chevalley group over a commutative ring. More precisely, we describe the subgroups of a maximal parabolic subgroup P normalized by the elementary part of its Levi subgroup. As a corollary, we obtain a description of the subgroups in P normalized by its elementary subgroup EP.

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

scholarly journals A combinatorial identity for the derivative of a theta series of a finite type root lattice

2003 ◽  
Vol 172 ◽  
pp. 1-30
Author(s):  
Satoshi Naito
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Finite Type ◽  
Cartan Subalgebra ◽  
Theta Series ◽  
Combinatorial Identity ◽  
Root Lattice ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.

On basic Cycles of An, Bn, Cn and Dn

1985 ◽  
Vol 37 (1) ◽  
pp. 122-140 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Dual Space ◽  
Cartan Subalgebra ◽  
Closed Field ◽  
Generating Set ◽  
Enveloping Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Positive Roots

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type

1991 ◽  
Vol 119 (3-4) ◽  
pp. 191-212 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Number Field ◽  
Function Field ◽  
Normal Subgroups ◽  
Dedekind Ring ◽  
Ring Of Integers

SynopsisLet R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.

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