Generalized Reduced Mal’tsev Problem on Commutative Subalgebras of E6 Type Chevalley Algebras over a Field Текст научной статьи по специальности «Математика»

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Серия «Математика»

2019. Т. 29. С. 31-38

УДК 512.554.3 MSG 17В30

DOI https://doi.org/10.26516/1997-7670.2019.29.31

Generalized Reduced Mal'tsev Problem on Commutative Subalgebras of Eq Type Chevalley Algebras over a Field

E. A. Kirillova

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract. In 1905 I. Shur pointed out the largest dimension of commutative subgroups in the groups SL(n, C) and proved that for n > 3 such the subgroups are automorphic to each other. In 1945 A.I. Mal'tsev investigated the problem of description of the largest dimension commutative subgroups in the simple complex Lie groups. He solved the problem by the transition to the complex Lie algebras and by the reduction to the same problem for the maximal nilpotent subalgebra. Let N be a niltriangular subalgebra of a Chevalley algebra. The article deals with the problem of describing the largest dimension commutative subalgebras of N over an arbitrary field. The solution of this problem is obtained for the subalgebra N of Es type Chevalley algebra.

Keywords: Chevalley algebra, niltriangular subalgebra, largest dimension commutative subalgebra.

In 1905 I. Shur [9] pointed out the largest dimension of commutative subgroups in the group SL(n, C) and proved that for n > 3 such the subgroups are automorphic to each other. In 1945 A. I. Mal'tsev investigated the problem of description of the largest dimension commutative subgroups in all finite dimension simple Lie groups using the transition to complex Lie algebras.

Let L$(K) be a Chevalley algebra over an arbitrary field K associated with the root system [2]. The Chevalley base in this algebra consists of the elements er (r € and the appropriate base of Cartan subalgebra. The elements er (r € form the base on niltriangular subalgebra N§(K).

1. Introduction

A. I. Mal'tsev solved his problem by reduction to analogous problem for the Lie algebras C). Later his methods were modified and applied to description of maximal order commutative subgroups of a finite Chevalley group and to reduced problem for an unipotent radical IJ of its Borel subgroup [4]- [10], [1] - [7].

The following problems were stated in [6]:

Generalized Mal'tsev's problem: Describe the largest dimension commutative subalgebras of Chevalley algebra over arbitrary field.

Generalized reduced problem: Describe the largest dimension commutative subalgebras in the subalgebra N$(K) of Chevalley algebra over arbitrary field.

The generalized reduced problem was studied in [3] and such the hypotheses was confirmed: any largest dimension commutative ideal of the algebra N$(K) is its largest dimension commutative subalgebra. We proved (Theorem 1) that the algebra N&(K) of Eq type has no another largest dimension commutative subalgebras.

2. Preliminary remarks

The structure constants of the algebra N&(K) are determined by Chevalley theorem on base:

if r + s € then er * es = Nrser+s, Nsr = —Nrs;

if r + s ^ <J>+, then er * es = 0.

The coefficients Nrs in the algebra N&(K) of Eq type are equal only ±1.

The sum ht(r) of coefficients in the root r base decomposition is called the root height. The subset \I> of the root system $ is called commutative if for any two roots r, s € ^ holds r + s ^ <J>. Let {r}+ be the set of roots s € such that the base decomposition of the root s — r contains only non-negative coefficients. Let, further, T(r) and Q(r) be the subalgebras of N$(K) with the bases {es|s € {r}+} and {es | s € M+\{y}}, respectively.

If the condition H C T(r\) + T(r2) + • • • + T(rm) holds and any substitution of T(ri) for Q(ri) leads to wrong inclusion then the set

{n,r2,.. .,rm} = C{H)

is called the corner set for H.

Let the regular ordering is determined for the positive root system $+. Any element a € N$(K) may be written as a sum a = a\eri +a2er2 +... + anern (cn / 0), where r\, r2, ■ ■ ■, rn are the roots from <J>+ which are ordered on increase. Then the root r\ is called the first corner of the element a. If

Известия Иркутского государственного университета. 2019. Т. 29. Серия «Математика». С. 31-38

M C N®(K) is arbitrary subalgebra then C\(M) is the set of first corners of all its elements.

Further we will use the A. I. Mal'tsev's notation [8] for the base roots cti (i = 1,... ,6) in the root system of E§ type:

a\ a2 a3 04 a5 a6

ÜJ i - w2 W4 + W5 + W6 w2 - w3 LÜ3 - U4 W4 — W5 cos ~ coe

The positive root system of Eq type consists only of the roots u)i — u)j (1 < i < j < 6), u)i + ujj + ujk (i < j < k; i, j, k = 1,...,6) and wo = wi + ... + ujq. The height of roots is calculated as j — i for m — ujj, as 16 — (i + j + k) for u)i + ujj + Wfc. The root ujq is maximal and it has the height 11. See Pic. 1 for positive root system of E§ type.

Further we denote and eWfc+W;+Wm as e^ and ekim, respectively

It is easy to note that:

the product of e^ and e« (i < j < k < 1) is non-zero only if the case j = k; the product of e^ and e^m is non-zero only if i £ {k,l,m} and j € {k, I, m}; the product e^ and eimn is non-zero only in the case {i, j, k, I, m, n} = {1,2,3,4,5,6}.

3. Largest dimension commutative subalgebras

Theorem 1. A largest dimension commutative subalgebra of the algebra N$(K) of Es type over the field K coincides with either T(a\) orT(a§).

Proof It is proved [3] that a subalgebra M of the algebra N$(K) over the field K is a largest dimension commutative subalgebra if and only if the set of roots C\(M) is a commutative root set of a maximal order in

According A. I. Mal'tsev, the maximal order commutative subsets in Eq are only {cki}+ and {o;6}+- Now we consider the largest dimension commutative subalgebras with such the sets as the sets of first corners for their elements.

Lemma 1. Let M be a largest dimension commutative subalgebra of the algebra N$(K) over the field K. Then there exists a base in M consisting of the elements

r>s

js = es + A, (s € C\(M)), where e ^ Ker. (3.1)

r€$+\£i(M)

Proof. For any s € C\(M) there exists an element in M with the first corner s. Choose such the element as with the s-coordinate equal to one. It is of a form

Известия Иркутского государственного университета . 2019. Т. 29. Серия «Математика». С. 31-38

r>s r>s

as = es + Ps + As, where fjs € ^ Ker, As € ^ Ke,..

re£i(M) re$+\£i(M)

We can map to zero all items from fis decomposition by adding the elements from for all possible s' > s. □

Let's show that for $ of Eq type all As are zero. The elements As may be written as

p>q(s)

-4s = Cseq(s) + 7s, where 7^ € ^ Kep.

p€$+\£i(M)

If As / 0 then we suppose that cs / 0 and so q(s) is a first corner in As. Let's choose such s and arbitrary r € C\(M). The product 7^ * 7r is equal to zero and is equal to

cr(es * eq(r)) + es * + cs(eg(s) * er) + cscr{eg* eg(r))+ (3.2) +cs(eg(s) * 7^) + 7s * er + cr(7^ * eg(r)) + 7s * 7r Lemma 2. If r + g(s) and s + g(r) are i/ie equal roots then

Na,q{r)Cr + Ng(s),rCs = 0.

If r + g(s) is a root which is not equal to s + q{r) then r + g(s) — s is a root and corresponding coordinate of is equal to — (Nq^>rNs>r+q^_s)cs.

Proof If r + g(s) is a root then the third item in (3.2) is cs(eq^ * er) = ±cser+q(s) / 0. r + g(s)-coordinate in the product 7^ * 7r is non-zero, so the sum (3.2) contains, except the third item, another items with non-zero r + g(s)-coordinate. It is easy to prove that it may be either first or second items, but not simultaneously. Then r + g(s)-coordinate in (3.2) is a sum of corresponding coordinates of either first and third items or second and third ones. The first case is possible only if r + q{s) = s + q(r), and in this case NStg(r}Cr + Ng(s}trcs = 0. The second case contains such the element aet that aet * es = —Nq^s^rcser+qs. So aet * es = NttSaet+s, and we have t = r + q(s) - s. Then a = -(Nq{s)ir/Nr+q{s)_SiS)cs. □

This lemma leads to

Corollary 1. If r + q(s) is a root then r + q{s) — s is a root too.

In the case $ = E§ and C\(M) = {cki}+ for arbitrary s and q{s) there exists such the root r that r + q{s) is a root but r + q{s) — s is not a root. We will not consider q{s) which does not commute with the roots t of the height > 8 (because At in (3.1) is equal to zero).

1) For the roots s = uj 1 — uji and q(s) = uji — ujj we choose the root r = (jj\ + ujj + Uk, where k / i.

2) For the roots s = uj\ — UJi and q(s) = ujj — ujk (i / j) we choose the root r = (jJ\ + UJi + Wfc.

3) For the roots s = uo\ — UJi and q(s) = uji + ujj + ujk we choose the root

r = (jj\ — ujj.

4) For the roots s = uj i—loî and q(s) = ujj +ujk + uji (i / j, k, I) we choose the root r = wo — q(s).

5) For the roots s = wi + W5 + W6 and q(s) = 002 —ujq we choose the root

r = (jj\— uj2-

6) For the roots s = wi + uii + ujj and q(s) = UJi + ujk + uji we choose the root r = (jj\ — uji.

7) For the roots s = wi + uii + ujj and g(s) = + wi + (s + q(s) = wo) we choose the root r = u\ — Uk-

The possibility of such a choice contradicts to the corollary and, so, contradicts to the proposition As / 0. So, the largest dimension commutative subalgebra of the algebra NEq(K) with the first corners set {cki}+ is only T(ot\). Acting to it by the graph automorphism we obtain the analogous result for the first corners set {a:6}+. So, the largest commutative subalgebras of NEq(K) are only T(a{) and T(a§). □

4. Conclusion

We proved that the list of largest dimension commutative subalgebras of the algebra NEq(K) completely coincides with the list of its largest dimension commutative ideals, which was obtained earlier. So, the generalized reduced Mal'tsev's problem is completely solved in this case. The problem of description of all maximal commutative ideals of the algebra N&(K) is written in [3] and now this problem is solved not for all root system types.

References

1. Barry M.J.J. Large Abelian subgroups of Chevalley groups. J. Austral. Math. Soc. Ser. A., 1979, vol. 27, no. 1, pp. 59-87. https://doi.org/10.1017/S1446788700016645

2. Carter R. Simple groups of Lie type. Wiley and Sons, New York, 1972, 331 p.

Известия Иркутского государственного университета. 2019. T. 29. Серия «Математика». С. 31-38

3. Kirillova E.A., Suleimanova G.S. Highest dimension commutative ideals of a niltriangular subalgebra of a Chevalley algebra over a field. Trudy Inst. Mat. i Mekh. UrO RAN, 2018, vol. 24, no. 3, pp. 98-108. (in Russian) https://doi.org/10.21538/0134-4889-2018-24-3-98-108

4. Kondrat'ev A.S. Subgroups of finite Chevalley groups. Uspekhi Mat. Nauk, 1986, vol. 41, no. 1 (247), pp. 57-96. (in Russian) https://doi.org/10.1070/RM1986v041n0lABEH003203

5. Levchuk V.M., Suleimanova G.S. Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type. J. Algebra, 2012, vol. 349, iss. 1, no. 1, pp. 98-116.

6. Levchuk V.M., Suleimanova G.S. The generalized Mal'cev problem on abelian subalgebras of the Chevalley algebras. Lobachevskii Journal of Ma,them,a,tics, 2015, vol. 86, no. 4, pp. 384-388.

7. Levchuk V.M., Suleimanova G.S. Thompson subgroups and large abelian unipotent subgroups of Lie-type groups. J. Siberian Federal University. Math. & Physics, 2013, vol. 6, no. 1, pp. 64-74.

8. Malcev A.I. Commutative subalgebras of semi-simple Lie algebras. Izv. Akad. Nauk SSSR Ser. Mat., 1945, vol. 9, no. 4, pp. 291-300. (in Russian)

9. Schur I. Zur theorie der vertauschbaren matrizen. J. reine und angew. Math., 1905, vol. 130, pp. 66-76. https://doi.org/10.1515/crll.1905.130.66

10. Vdovin E.P. Large abelian unipotent subgroups of finite Chevalley groups. Algebra, and, Logic, 2001, vol. 40, no. 5, pp. 292-305. (in Russian) https://doi.Org/10.1023/A:1012549701336

11. Vdovin E.P. Maximal Orders of Abelian Subgroups in Finite Chevalley Groups. Mat. Zametki, 2001, vol. 69, no. 4, pp. 524-549. (in Russian)

Evgeniya Kirillova, Postgraduate, Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, tel.: (391)2062148 (e-mail: [email protected]) Received 06.05.19

Обобщённая редукционная задача Мальцева о коммутативных подалгебрах алгебр Шевалле типа Е6 над полем

Е. А. Кириллова

Сибирский федеральный университет, Красноярск, Российская Федерация

Аннотация. В 1905 г. И. Шур указал наивысшую размерность коммутативных подгрупп группы SL(n, С) и доказал, что коммутативные подгруппы этой размерности при п > 3 автоморфны. В 1945 г. А. И. Мальцев исследовал задачу описания коммутативных подгрупп наивысшей размерности в комплексных простых группах Ли. Он получил решение, применив переход к комплексным алгебрам Ли и редуцирование к аналогичной задаче для максимальной нильпотентной подалгебры. Пусть N — нильтреугольная подалгебра алгебры Шевалле. Исследуется задача описания коммутативных подалгебр наибольшей размерности подалгебры N алгебры Шевалле, ассоциированной с системой корней типа Ее, над произвольным полем. Ранее при работе над этой задачей было получен полный список коммутативных идеалов наибольшей размерности подалгебры N типа Ее- В настоящей статье показа-

но, что коммутативные подалгебры наивысшей размерности также исчерпываются этим списком; таким образом решена обобщённая редукционная задача Мальцева для алгебр Шевалле типа Ее-

Ключевые слова: алгебра Шевалле, нильтреугольная подалгебра, коммутативная подалгебра наибольшей размерности.

Список литературы

1. Barry М. J. J. Large Abelian subgroups of Chevalley groups. // J. Austral. Math. Soc. Ser. A. 1979. Vol. 27, N 1. P. 59-87. https://doi.org/10.1017/S1446788700016645

2. Carter R. Simple groups of Lie type. New York : Wiley and Sons, 1972. 331 p.

3. Кириллова E. А., Сулейманова Г. С. Коммутативные идеалы наибольшей размерности нильтреугольной подалгебры алгебры Шевалле над полем // Тр. ИММ УрО РАН. 2018. Т. 24, № 3. С. 98-108. https://doi.org/10.21538/0134-4889-2018-24-3-98-108

4. Кондратьев А. С. Подгруппы конечных групп Шевалле // Успехи мат. наук. 1986. Т. 41, № 1 (247). С. 57-96. https://doi.org/10.1070/RM1986v041n01ABEH003203

5. Levchuk V. М., Suleimanova G. S. Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type //J. Algebra. 2012. Vol. 349, Iss. 1, N 1. P. 98-116.

6. Levchuk V. M., Suleimanova G. S. The generalized Mal'cev problem on abelian subalgebras of the Chevalley algebras // Lobachevskii Journal of Mathematics. 2015. Vol. 86, N 4. P. 384-388.

7. Levchuk V. M., Suleimanova G. S. Thompson subgroups and large abelian unipotent subgroups of Lie-type groups //J. Siberian Federal University. Math. & Physics. 2013. Vol. 6, N 1. P. 64-74.

8. Мальцев А. И. Коммутативные подалгебры полупростых алгебр Ли // Изв. АН СССР. Сер. матем. 1945. Т. 9, № 4. С. 291-300.

9. Schur I. Zur theorie der vertauschbaren matrizen //J. reine und angew. Math. 1905. Vol. 130. P. 66-76. https://doi.org/10.1515/crll.1905.130.66

10. Вдовин E. П. Большие абелевы унипотентные подгруппы конечных групп Шевалле // Алгебра и логика. 2001. Т. 40, № 5. С. 523-544. https://doi.Org/10.1023/A:1012549701336

11. Вдовин Е. П. Максимальные порядки абелевых подгрупп в конечных группах Шевалле // Мат. заметки. 2000. Т. 68, № 1. С. 53-76.

Евгения Алексеевна Кириллова, аспирант, Институт математики и фундаментальной информатики, Сибирский федеральный университет, Российская Федерация, 660041, г. Красноярск, пр. Свободный, 79 тел.: (391)2062148 (e-mail: kea92bk.ru)

Поступила в редакцию 06.05.19

Известия Иркутского государственного университета. 2019. Т. 29. Серия «Математика». С. 31-38