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УДК 512.554.3
The Highest Dimension of Commutative Subalgebras in Chevalley Algebras
Galina S. Suleimanova*
Khakas Technical Institute Branch of Siberian Federal University Shchetinkin, 27, Abakan, 665017
Russia
Received 17.12.2018, received in revised form 20.01.2019, accepted 20.02.2019
Let ЬФ(К) denotes a Chevalley algebra with the root system Ф over a field K. In 1945 A. I. Mal'cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra. Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups. The main result of this article allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra ЬФ (К) over an arbitrary field.
Keywords: Chevalley algebra, commutative subalgebra. DOI: 10.17516/1997-1397-2019-12-3-351-354.
Introduction
Let L^(K) denotes a Chevalley algebra with the root system $ over a field K, and let n be a fundamental system of roots. The elements {er, hp | r e £ n} form a basis of L^(K), called a Chevalley basis [1]. Let N$(K) denotes a niltriangular subalgebra in L^(K) with the basis {er I r £ $+}.
In 1945 A. I. Mal'cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups [7]. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra N$(C).
Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups [3-5,8,9]. Given a group-theoretic property P, we recall that every P-subgroup of largest order in a finite group is a large P-subgroup.
The generalization of Mal'cev problem [7] for Chevalley algebras L^(K) over an arbitrary field K was pointed in [6].
The main result of this article is Theorem 1, which allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra L^(K) over an arbitrary field.
* suleymanova@list.ru © Siberian Federal University. All rights reserved
1. Preliminary remarks and notation
Let ) denotes a Chevalley algebra with the root system $ over a field K, and let n be a fundamental system of roots. The elements {er, hp | r £ $,p £ n} form a basis of L$(K), called a Chevalley basis. The elements of this basis multiply together as follows:
hr * hs =0, r, s £ n,
hr * es = Arses, r £ n, s £ $,
er * e-r = hr, r £ $,
er * es =0, r, s £ $, r + s £ $,
er * es Nrser+s:
2(r s)
where the elements Nrs are called the structure constants of L^(K), and Ars = ' . The
(r, r)
elements {hp | p £ n} form a basis for a Cartan subalgebra H [1].
A subset ^ of the root system $ is called a commutative, if r + s £ $ for all r,s £ ^ [7]. A subset ^ of the root system $ is said to be p-commutative, if in the algebra N$(K) over a field K of characteristic p we have er * es =0 for all r,s £ ^ [8].
Further, we use a regular ordering of roots [1, Lemma 5.3.1]. Let x £ L^(K) and
x = aien + a^er2 +-----+ h + c\eSl + c2eS2 + . .., (1)
where ri £ si £ $+, ri < < ..., si < s2 < ..., h = bihp1 + b2hp2 + ..., pi,p2, ■ ■ ■ £ n. We consider the first non-zero term in (1). If this term has the form ter, r £ $, then we denote b(x) = er, else we denote b(x) = h. For M C L we set b(M) = {b(x) | x £ M}.
Lemma 1. If x * y = 0 for x,y £ L^(K), then b(x) * b(y) = 0.
Proof. Let
x = aier1 + a2er2 + ■ ■ ■ + h + cies1 + c2es2 + . ..,
y = a'i er1 + a2er'2 +-----+ h' + c'^ + c2es>2 + ...,
where ri + ri £ ri < r2 < ■ ■ ■ < si < s2 < ..., ri < r'2 < ■ ■ ■ < si < s2 < .... Then in the expression of x * y as a linear combination of er, r £ $, every er (r £ $-) has the form eri+r{, eri+s^, esi+ri, eri or er/. Since ri — ri £ V+, ri — ri £ V+ (i = 1), where V + is a certain positive subspace, then (ri + ri) — (ri + r[) = (ri — ri) + (ri — r[) £ V +, so ri + ri <ri + ri. Since ht(ri + ri) < ht(ri) and ht(ri + r[) < ht(ri), we have ri + ri < ri and ri + ri < ri. Hence b(x * y) = eri+ri, so x * y = 0, that gives a contradiction. Analogously, the remaining cases when b(x), b(y), b(x) * b(y) are not in H give a contradiction.
Let b(x) = h, b(y) = aieri and h * aieri = 0. Then b(x * y) = eri, so again x * y = 0. In the case
x = cies1 + c2es2 + . .., y = a'i er1 + a2er'2 +-----+ h' + c^! + c2es'2 + ...,
where ri = —si, the expression of x * y contains a term hr1. If x * y = 0, then in the expression of x and y there exists another pair of terms of form esi, e-si, respectively. Hence s' < si and — si < —si, a contradiction. This completes the proof of Lemma 1. □
Note that in the case b(M) = {er | r £ $+} the set of corresponding roots r £ $+ coincides with the set L'(M) ( [2,4]).
2. The highest dimension of commutative subalgebras
Theorem 1. Let L^(K) be a Chevalley algebra with the root system $ over an arbitrary field K of characteristic p. Let m be a maximal order of p-commutative subsets of roots in and let k be a dimension of the center of L^(K). The highest dimension of commutative subalgebras of L^(K) equals m + k.
Lemma 2. Let A be a commutative subalgebra of L = L^(K). Then there exists an automorphism of L transforming A to a subalgebra B in L such that for all x € B either b(x) = er, r € $_, or b(x) = h € H, where h is an element of the center of L.
Proof. Let xr (t) denotes an automorphism of a Chevalley algebra, effecting on the elements of a Chevalley basis as follows:
(2)
er er
hs ^ hs — tAsrer (s € n),
e—r ^ e—r \ thr t er,
q
e.s Mr,s,itieir+s (s € $ \ {±r}), MT,sfi := 1
(3)
(4)
(5)
i=0
Mrsi := ± (p(r, s) + i//i) .
Let
x = h + ci es1 + c2es2 +----
Suppose that there exists a fundamental root p such that h * ep = 0 and si = p. Since
h * ep = (bihpi + b2hP2 + ...) * ep = (biAPi + b2AP2 ,p + ... )ep,
then the automorphism x-p (1) transforms h to
h + (biApip + b2Ap2p + ... )ep,
where biApip + b2Ap2,p + ■ ■ ■ = 0. Hence b(x-p(1)(x)) = e-p.
If si = p, then first we obtain ci = 0, up to a certain automorphism xp(t), p € n, and again b(x_p(1)(x)) = e-p.
Suppose that there exists y € A such that b(y) = er, r € $_, and b(x±p(1)(y)) equals h € H or er, r € $+. Taking into account the relations (2)-(5), we deduce that b(y) = e_p. Since h * ep = 0 hence h * e_p = 0. By Lemma 1, this is a contradiction. If
x = cies1 + c2es2 + ...
then b(x_si (1)(x)) = e_si.
Suppose that there exists y € A such that b(y) = er, r € $_, and b(x_si (1)(y) equals h € H or er, r € $+. Taking into account the relations (2)-(5), we deduce that b(y) = e_si. By Lemma 1, this is a contradiction.
The lemma is proved. □
Now the Theorem 1 follows from Lemma 1 and Lemma 2.
References
[1] R.Carter, Simple groups of Lie type, Wiley and Sons, New York, 1972.
[2] E.A.Kirillova, G.S.Suleimanova, Highest dimension commutative ideals of a niltriangular subalgebra of a Chevalley algebra over a field, Trudy of Inst. of Math. & Mech. UrO RAN, 24(2018), no. 3, 98-108.
[3] A.S.Kondratiev, Subgroups of finite Chevalley groups, Russian Math. Surveys, 41(1986), no. 1, 65-118.
[4] V.M.Levchuk, G.S.Suleimanova, Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type, Journal of Algebra, 349(2012), no.1, 98-116.
[5] V.M.Levchuk, G.S.Suleimanova, Thompson subgroups and large abelian unipotent subgroups of Lie-type groups, Journal of Siberian Federal University. Mathematics & Physics, 6(2013), no. 1, 64-74.
[6] V.M.Levchuk, G.S.Suleimanova, The generalized Mal'cev problem on abelian subalgebras of the Chevalley algebras, Lobachevskii Journal of Mathematics, 86(2015), no. 4, 384-388.
[7] A.I.Mal'cev, Commutative subalgebras in semisimple Lie algebras, Izv. Akad. Nauk SSSR. Ser. Mat., 8(1945), 291-300.
[8] E.P.Vdovin, Large Abelian Unipotent Subgroups of Finite Chevalley Groups, Algebra & Logic, 40(2001), no. 5, 292-305.
[9] E.P.Vdovin, Maximal Orders of Abelian Subgroups in Finite Chevalley Groups, Math. Notes, 69(2001), no. 4, 475-498.
Наивысшая размерность коммутативных подалгебр алгебр Шевалле
Галина С. Сулейманова
Хакасский технический институт Филиал Сибирского федерального университета Щетинкина, 27, Абакан, 665017 Россия
Пусть ЬФ(К) — алгебра Шевалле над полем К, ассоциированная с системой корней Ф. В 1945 г. А. И. Мальцев исследовал проблему описания абелевых подгрупп наивысшей размерности в комплексных простых группах Ли. Он решил эту проблему переходом к комплексным алгебрам Ли и редукцией к проблеме описания коммутативных подалгебр наивысшей размерности в нильтре-угольной подалгебре. Позже эти методы модифицировались и применялись для решения проблемы описания больших абелевых подгрупп конечных групп Шевалле. Основной результат данной статьи позволяет вычислить наивысшую размерность коммутативных подалгебр алгебры Шевалле Ьф(К) над произвольным полем.
Ключевые слова: алгебра Шевалле, коммутативная подалгебра.
