On the Closedness of Carpets of Additive Subgroups Associated With a Chevalley Group Over a Commutative Ring Текст научной статьи по специальности «Математика»

EDN: BNVWWZ УДК 512.54

On the Closedness of Carpets of Additive Subgroups Associated With a Chevalley Group Over a Commutative Ring

Yakov N. Nuzhin*

Siberian Federal University Krasnoyarsk, Russian Federation

Dedicated to the blessed memory of my teacher Vladimir M. Levchuk

Received 10.06.2023, received in revised form 31.07.2023, accepted 04.09.2023 Abstract. Let A = {Ar | r £ Ф} be a carpet of additive subgroups of type Ф over an arbitrary commutative ring K. A sufficient condition for the carpet A to be closed is established. As a corollary, we obtain a positive answer to question 19.63 from the Kourovka notebook and a confirmation of one conjecture by V. M. Levchuk, provided that the type of Ф is different from Ci, l ^ 5 when the characteristic of the ring K is 0 or 2m for some natural number m > 1. Also, a partial answer to question 19.62 has been obtained.

Keywords: Lie algebra and ring, Chevalley group, commutative ring, carpet of additive subgroups, carpet subgroup.

Citation: Ya.N. Nuzhin, On the Closedness of Carpets of Additive Subgroups Associated With a Chevalley Group Over a Commutative Ring, J. Sib. Fed. Univ. Math. Phys., 2023, 16(6), 732-737. EDN: BNVWWZ.

1. Introduction and preliminaries

Let $ be a reduced indecomposable root system, E($, K) be an elementary Chevalley group of type $ over the commutative ring K with unit 1. The groups E($,K) is generated by the root subgroups xr(K) = {xr (t) | t e K}, r e $. The subgroups xr(K) are Abelian, and

xr (t)xr (u) = xr (t + u), (1)

for any r e $ and t,u e K. We call a carpet of type $ over K a collection of additive subgroups A = {ar I r e $} of the field K with the condition

CijrskaS c air+js, r, s,ir + js e $, i, j > 0, (2)

where Ar = {aO1 | a e ar}, and constants Cij,rs are equal to ±1, ±2 or ±3. Inclusions (2) come from the Chevalley commutator formula

[xs(u),xr (t)] = JJ xir+j^Cij,rS(-t)iuj), r,s,ir + js e $. (3)

i,j>0

* nuzhin2008@rambler.ru © Siberian Federal University. All rights reserved

Every carpet A defines a carpet subgroup E($, A) generated by the subgroups xr(ar), r G $. A carpet A is called closed if its carpet subgroup E($, A) has no new root elements, i.e., if

E($, a) n xr(K) = xr(Ar).

Note that the following fact follows from the condition (2). If t G ar and u G As, then each factor from the right-hand side of (3) lies in E($, A).

The above definition of a carpet was introduced by V. M. Levchuk, and it was first written down in the next question from the Kourovka notebook [1].

Question A). What are the conditions on the carpet a = {ar | r G $} (in terms of ar) over a commutative ring K necessary and sufficient for a to be closed? [1, question 7.28, 1980]

The difficulty of this question lies in the fact that the answer to it, in fact, should be the following statement: the defining relations of the carpet subgroup E($, a) over an arbitrary commutative ring K are exhausted by the relations (1), (3) (which give rise to the carpet conditions (2)), and relations in the subgroup {xr(ar), x-r(a-r), r G $. In the case when the carpet subgroup E($, a) coincides with the entire Chevalley group E($,K) over the field K, this is the well-known result of R. Steinberg on defining relations in Chevalley groups over the field [2, Sec. 6, Theorem 8] In 1983, the author of question A) himself gave the answer to it for a locally finite field K [3] and reformulated it in the case when the main ring K is a field, in the following form.

Question B). Is it true that the carpet a = {ar | r G $} of type $ over a field K is closed if and only if the subcarpets {Ar, A-r}, r G $, of rank 1 are closed? [1, question 15.46, 2002]

The carpet {Ar, A-r}, r G $, of rank 1 corresponds to an elementary matrix carpet of degree 2, and its carpet subgroup is isomorphic to the group generated by opposite elementary transvections t12(u), u G ar and t21(u), u G a-r .

Lemma 14 is noted in [4] without proof, however, after more than 40 years, its proof has not appeared. Therefore, we formulate the assertion of this lemma as a conjecture.

Conjecture C). (V. M. Levchuk, 1982) Inclusions ara-rar C Ar, r G $, are sufficient for the carpet of additive subgroups a = {ar | r G $} over a commutative ring K to be closed.

The next question of the author of this note is a strengthening of conjecture C), and in the case when the characteristic of the main coefficient ring is odd, they are equivalent.

Question D). Are the inclusions A;:A_r C Ar, r G $ , sufficient for the carpet of additive subgroups a = {ar | r G $} over a commutative ring K to be closed? [1, Question 19.63]

In the articles [5,6] give examples of irreducible (if all additive subgroups are nonzero) carpets of any type $ over rings of any even characteristic, for which the inclusions A;:A_r C Ar, r G $, are valid, but the inclusions from hypothesis C) are not satisfied. Question D) was first written in 2012 in [7, p. 199] and this is due to the fact that the inclusions A;;A_r C Ar, r G $, are necessary and sufficient conditions for the carpet subring L($, a) to be invariant under the carpet subgroup E($, a) according to the following definitions.

Let n be the fundamental root system for $. The structure constants in the Chevalley basis {er, r G $; hs, s G n} of the simple complex Lie algebra L($, C) are integers , so one can define a Lie ring (algebra) L($, K) with a Chevalley basis over an arbitrary commutative ring

K (see, for example, [8, p. 62]). By definition, we assume that the subring L($, A) is generated (with respect to both operations) by all the sets Arer, r e $. We will call L($, A) a carpet Lie subring. Note that the basis elements er, hs do not have to lie in L($, A). A carpet A is called L-closed if L($, a) n Ker = arer, r e $. The elementary Chevalley group E($, K) acts on the Lie ring L($, K) as an automorphism group. We will say that the subring R C L($, K) is invariant under G C E($, K) if gr e R for any g e G and r e R. Any carpet subring L($, A) that is invariant under the corresponding carpet subgroup E($, a) is L-closed [7, p. 199].

For any r e $, the automorphism xr (t) of the Lie ring L($,K) acts on the Chevalley basis as follows:

(4)

er er,

e—r e—r 1 thr t er,

hs s t^Asr er, s G n,

es q Cil,rst eir+s, i=0 s G $ \ {±r},

where Ars =

2(r, s) (r, r)

q = q(r, s) is the largest non-negative integer such that s + qr e $ and

by definition C01,rs = 1. Therefore, if s e $ \ {±r} and t e ar, u e As, then under the

q

automorphism xr (t), the vector ues, goes into a linear combination Y1 Cii,rstiueir+s, each term

¿=0

of which, by virtue of the carpet condition, lies in the subring L($, A).

The main result of the article is

Theorem 1. The inclusions A;:A_r C Ar, r e $, are sufficient for closedness of the carpet of additive subgroups a = {ar | r e $} over an arbitrary commutative ring K provided that the type of $ is different from Ci, l ^ 5, when the characteristic of the ring K is 0 or 2m for some natural number m > 1.

2. Proof of Theorem 1

Let xp(t) e E($, a) for fixed p e $ and some t e K. Then, for suitable ri e $ and ti e Ar we have the equality

7(t) = xri (t1)xr2 (t2). ..xrn (tn).

(5)

Our task is to establish the inclusion t G Ap. By virtue of the third equality from (4), for any root q G $ the following equality holds

xp(t)hq hq Aqptep.

(6)

Consider the integer E($, A)-submodule M of the ring L($, K) generated by the element hq and show the inclusion

M < Zhq + L($, a). (7)

For any r G $ we put hr = {u G K | uhr G M}. By virtue of (4), the the following inclusions hold:

hq < Z + AqA—q, (8)

hs asa—s, s = q,

(9)

x

where by definition ar a—r consists of sums of the form u1v1 + • • • + ukvk for ui £ Ar, vi £ a—r, i — 1,...,k, for any natural k. Indeed, the generation of the module M begins with the equality

Xr (Ar ) hq — hq ^Aqr er .

Thus, there is a base of induction. By virtue of (4), we have the equalities:

q

Xr (Ar )ases — J2 Ca}rsar aseir+s, s — -r, (10)

i=0

^c r (Ar) ^a—r e—r ^a—r e—r \ ^ar ^a—r ^hr ^a—r er, (11)

r (Ar )Hs hs — Hshs Asr Ar Hser . (12)

Further, by the definition of the M, these equalities are applied in any order. By Theorem 3.1 in

[7] the subring L($, A) is invariant under the carpet subgroup E($, A) if and only if A;:A—r C Ar, r £ $. Therefore Ker n L($, A) — Arer for all r £ $ [7, p. 199]. This gives us the inclusions (8) and (9), since Ara—rhr £ L($, A), r £ $, due to (11). Thus, the inclusion (7) also holds.

In what follows, we need one lemma, the assertion of which for systems all of whose roots have the same length is obvious. For a system of type F4 it follows from the fact that any of its root lies in a subsystem of type A2. For root systems of types Bl, Cl and G2 it can be verified directly.

Lemma 1. For any root p £ $ there exists a root q £ $ such that

{±2 if $ of type Cl, l ^ 1, and pis a long root, (13)

±1 otherwise.

So now, due to (5), (6) and (7) we have Aqpt £ ap. Hence, by Lemma 1, we obtain the desired inclusion t £ ap if $ is different from Cl, l > 1, in the case of an even characteristic of the ring K. The constraint "$ is different from Cl, l ^ 1" can be relaxed to "$ is different from Ci, l > 5" for an ring K of even characteristic due to the following equalities and inclusions for root systems Ai — Ci C C2 — B2 c C3 c C4 c F4. Indeed, we embed a carpet A of type Cl for l — 1, 2,3,4 into a carpet of type F4 such that Ar — 0 for all r £ F4 \ Cl. Since Lemma 1 is true for a root system of type F4, the above proof also holds in this case. We only note that in this case the root q in Lemma 1 must be taken from the difference F4 \ Cl.

It remains to consider the case when $ is of type Cl, l > 5, and the characteristic of the ring K is equal to 2. In this case, there exists a homomorphism ^ of the group E(Cl ,K) into the group E(Bl, K) such that

( Xr/2 (u) if r is a long root,

f(xr(u)) — < ^ . (14)

I xr (u2) if r is a short root,

for any u £ K (see, for example, [2, Theorem 28] and [9, Sec. 3]). Since the carpet conditions come from the Chevalley commutator formula, y induces a homomorphism of the carpet subgroup E(Cl, A) onto the carpet subgroup E(Bl, A'), where the carpet A' — {A'r | r £ Bl} according to

(14) is defined as

{A2r if r is a short root of root system of type Bl,

2 (15)

A2 if r is a long root of root system of type Bl.

The correspondence (15) between additive subgroups Ar and As is given by the bijection pi ^ qi, i — 1,2,... ,l , between fundamental root systems of type Cl and Bl, where p1 is a long root for type Cl, and q1 is a short root for type Bl, and the sum pi + pi+1 and qi + qi+1 are roots for all i — 1, 2,...,l - 1. The inclusions a';a'—r C a'r, r € Bl, remain valid, since from S2T C S follows (S2)2T2 C S2 for any subsets S and T of the ring K. For $ of type Cl, only the answer to the following question was not known above. Does the inclusion xp(t) £ E(Cl, A), when p is a long root and the characteristic of the ring K is even, imply the inclusion t £ Ap? Since y(xp(t)) — xp/2(t) by definition, then A'p/2 — ap. The inclusion t £ a'p/2 is fulfilled, since we are dealing with a carpet of type Bl. Hence t £ Ap. Which is what needed to be shown.

The Theorem 1 is proved. □

3. Corollaries from Theorem 1

Since the inclusions Ara—rar C ar, r £ $, entails the inclusions A;:A—r C ar, r £ $, then Theorem 1 implies

Corollary 1. Conjecture C) is true if the type of $ is different from Cl, l ^ 5, when the characteristic of the ring K is 0 or 2m for some natural numbers m > 1.

Given an elementary carpet A — {Ar | r £ $} of type $ of rank l > 2, we define a collection of additive subgroups Bp — Cij,rsairajs, p £ $, where the sum is over all natural numbers i, j and roots r,s £ $ for which ir + js — p. The collection B — {Bp | p £ $} is a carpet and is called the derived of A [10]. It is known that for $ — Al the carpet B is closed [11, Proposition 1]. The following question was first written in [10, p. 534, 2011].

Question E). Is every derived carpet of type $ over a commutative ring closed? [1, Question 19.62, 2018]

(r, r)

For each root system $ we define the number m — m($) — max ---. In fact,

r,s£$ (s, s)

( 1 if $ — Al,Dl,El,

m — [ 2 if $ — Bl,Cl,F4, [ 3 if $ — g2.

In [10, Theorem 1] it is proved that any derived carpet B of type $ of rank l > 2 satisfies the inclusions m!BpB—pBp C bp, p £ $. Combining this result with the assertion of Theorem 1, we obtain

Corollary 2. A derived carpet of type $ of rank l ^ 2 over a commutative ring K of characteristic p is closed if gcd(p, 2) — 1 for $ of type Bl, Cl or F4, and if gcd(p, 6) — 1 for $ of type G2.

This work is supported by Russian Science Foundation, project 22-21-00733.

References

[1] The Kourovka notebook: Unsolved problems in group theory, Eds. V. D.Mazurov, E.I.Khukhro, Sobolev Institute of Mathematics, Novosibirsk, no. 20, 2022.

[2] R.Steinberg, Lectures on Chevalley groups, Yale University, 1967.

[3] V.M.Levchuk, On generating sets of root elements of Chevalley groups over a field, Algebra i Logika, 22(1983), no. 5, 504-517 (in Russian).

[4] V.M.Levchuk, Parabolic subgroups of certain ABA-groups, Mathematical Notes, 31(1982), no. 4, 509-525.

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[6] P.S.Badin, Ya.N.Nuzhin, E.N.Troyanskaya, On weakly supplemented carpets of lie type over commutative rings, Vladikavkaz Math. J., 23(2021), no. 4, 28-34 (in Russian).

DOI: 10.46698/g1659-3294-1306-k

[7] Ya.N.Nuzhin, Lie rings defined by the root system and family of additive subgroups of the initial ring, Proc. Steklov Inst. Math. (Suppl 1), 283(2013), no. 1, 119-125.

DOI: 10.1134/S0081543813090125

[8] R.W Carter, Simple groups of lie type, Wiley and Sons, London-New York-Sydney-Toronto, 1972.

[9] Ya.N.Nuzhin, A.V.Stepanov, Subgroups of Chevalley groups of types Bi and Cl containing the group over a subring, and corresponding carpets, St. Petersburg Math. J., 31(2020), no. 4, 719-737. DOI: 10.1090/spmj/1620

[10] Ya.N.Nuzhin, Factorization of carpet subgroups of the Chevalley groups over commutative rings, Journal of Siberian Federal University. Mathematics & Physics, 4(2011), no. 4, 527-535 (in Russian).

[11] V.A.Koibaev, Nets associated with the elementary nets, Vladikavkaz Math. J., 12(2010), no. 4, 39-43 (in Russian).

О замкнутости ковров аддитивных подгрупп, ассоциированных с группой Шевалле над коммутативным кольцом

Яков Н. Нужин

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Установлено достаточное условие замкнутости ковра аддитивных подгрупп А = {Аг | г £ Ф} типа Ф над произвольным коммутативным кольцом К. В качестве следствий получаем положительный ответ на вопрос 19.63 из Коуровской тетради и подтверждение одной гипотезы В. М. Левчука при условии, что тип Ф отличен от Ог, I ^ 5, когда характеристика кольца К есть 0 или 2т для некоторого натурального числа т > 1. Также получен частичный ответ на вопрос 19.62.

Ключевые слова: алгебра и кольцо Ли, группа Шевалле, коммутативное кольцо, ковер аддитивных подгрупп, ковровая подгруппа.