ON PAIRS OF ADDITIVE SUBGROUPS ASSOCIATED WITH INTERMEDIATE SUBGROUPS OF GROUPS OF LIE TYPE OVER NONPERFECT FIELDS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-5-604-610 УДК 512.54

On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields

Yakov N. Nuzhin*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 06.03.2021, received in revised form 20.04.2021, accepted 24.6.2021 Abstract. The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Ai, 2DU 2E6, D in the case when the larger field F is an algebraic extension of the smaller nonperfect field K of exceptional characteristic for the group G(F) (characteristics 2 and 3 for the type 3D4 and only 2 for other types). It turned out that apart from, perhaps, the type 2Di, such intermediate subgroups are standard, that is, they are exhausted by the groups G(P)H for some intermediate subfield P, K С P С F, and of the diagonal subgroup H normalizing the group G(P). In this note, it is established that intermediate subgroups are also standard for the type 2Di.

Keywords: groups of Lie type, nonperfect field, intermediate subgroups, carpet of additive subgroups. Citation: Ya.N. Nuzhin, On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields, J. Sib. Fed. Univ. Math. Phys., 2021, 14(5), 604-610. DOI: 10.17516/1997-1397-2021-14-5-604-610.

1. Introduction and preliminaries

Groups of Lie type G(F) over the field F consist of Chevalley groups of type $ = Al, Bl, Cl, Dl, E6, E7, E8, F4, G2 and twisted Chevalley groups of type = 2Al, 2Dl, 3D4, 2E6, 2B2, 2G2, 2F4. The number of fundamental reflections generating the Weyl group associated with the group G(F) is called its Lie rank. Groups of type Ai, 2A2, 2B2, 2G2 constitute all groups of Lie rank 1. Groups of type 2Al, 2Dl, 3D4, 2E6 are also called by Steinberg groups, groups of type 2B2 are called by Suzuki groups, groups of type 2G2, 2F4 are called by Ree groups, in honor of their discoverers.

The exceptional characteristics of the ground field F for the group G(F) are usually:

- characteristic 2 for types Bl, Cl, F4, 2Al, 2Dl and 2E6;

- characteristics 2 and 3 for types G2 and 3D4.

This is due to the fact that the Coxeter graph associated with the group G(F) has edges of multiplicity 2 or 3.

In what follows, everywhere the field F is an algebraic extension of the field K. The intermediate subgroups between the groups G(K) and G(F) are described in the author's papers [1-3]. For exceptional characteristics, the description depends on whether the field K is perfect. By definition, a field K of characteristic p > 0 is called perfect if Kp = K.

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

In 1983, the following result was obtained in [1]. If the Lie rank of the group G(F) is greater than one and it is different from the group Re of type 2F4, and in the exceptional characteristics for G(F) the field K is perfect, then the groups lying between the groups G(K) and G(F), are exhausted by the groups G(P)H for some intermediate subfield P and a diagonal subgroup H normalizing the group G(P). We call such intermediate subgroups standard.

In [2], the groups that lie between the Chevalley groups G(K) and G(F) of type Bi, Cl, F4, G2 are described in the case of an nonperfect field K that is exceptional characteristics for the group G(F). It turned out that in each of these cases, except type G2 in characteristic 2, nonstandard subgroups appear and they are parameterized by two additive subgroups of the field F. Moreover, if G(K) is of type F4 or G2, then both additive subgroups are fields, and if G(K) is of type Bl (l > 3) or Cl (l > 3), then one additive subgroup is a field. The paper [4] contains examples of non-standard intermediate subgroups for types Bl (l > 3) and Cl (l > 3), which are parameterized by two additive subgroups, one of which is not a field, and for the type B2 = C2 both such additive subgroups may not be fields.

In [3], the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Al, 2Dl, 2Ee, 3 D4 are described in the case of nonperfect fields K of exceptional characteristic for the group G(F). It turned out that except, perhaps, the type 2Dl, the intermediate subgroups are standard.

In this note, we classify pairs of additive subgroups that parameterize non-standard subgroups between the groups G(K) and G(F) (Section 2) and prove the standardness of such intermediate subgroups for the type 2Dl (Section 3). Thus, non-standard groups lying between the groups G(K) and G(F) appear only for Chevalley groups of normal type Bl, Cl, F4 and G2 over the nonperfect field F of characteristic 2 and, respectively 3. Note also that if we remove the condition of algebraicity of the extension of a larger field over a smaller one, then the description of intermediate subgroups becomes immeasurable for Chevalley groups associated with Coxeter graphs without multiple connections [5,6].

2. Pairs of additive subgroups associated with intermediate subgroups of Chevalley groups of type Bl, Cl, F4 h G2

Let $ be a reduced indecomposable root system, $(F) be a Chevalley group of type $ over the field F generated by the root subgroups

xr(F) = {xr(t) 1t e F}, r e $.

Following V. M. Levchuk [7], by a carpet of type $ over F, we mean a family of additive subgroups A = {Ar | r e $} of the field F with the condition

CijrsXAS c Air+js, r,s,ir + js e $, i,j> 0, (1)

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

[xs(u),xr (t)] = JJ Xir+js(Cij,rs(-t)iUj), r,s,ir + js e $. (2)

i,j>0

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

$(A) n xr (F) = xr (Ar).

Summing Theorems 3.1 and 4.1 from [2], we obtain the following result.

Theorem 1 ([2]). Let F be an algebraic extension of an nonperfect field K of characteristic p and M be a group lying between Chevalley groups $(K) and $(F) of type $ = Bl (l ^ 2), Cl (l > 2), F4, G2. Let p = 2 for $ = Bl, Cl, F4 and p = 3 for $ = G2. Then M is the product of the carpet subgroup $(A) and some diagonal subgroup HM normalizing $(A). The carpet A = {Ar | r € $} is closed and

{P, if r is a short root, Q, if r is a long root,

for some additive subgroups P and Q of the field F with the conditions

R < Pp < Q < P < K.

Moreover, depending on the type of the Chevalley group $(K), the following refinements hold for the additive subgroups P and Q of the field F and the diagonal subgroup HM ■

a) if $ = Bl and l ^ 3, then Q is a field;

b) if $ = Cl and l ^ 3, then P is a field;

c) if $ = F4,G2, then both additive subgroups P and Q are fields and HM is the unit subgroup. Here, for any additive subgroup A of some field, by definition

Ap = {tp I t € A},

A-1 = {0}U{t € A I t€ A}.

For $ = F4, G2, the structure of the additive subgroups P and Q is clear, they are fields. The next proposition clarifies their structure for $ = Bl,Cl. For any root r € $ and any t from the multiplicative group F* of the fields F by definition

nr (t) = xr (t)x-r (-t-1)xr (t),

hr (t) = nr (t)nr ( — 1).

Proposition 1. Let M, P and Q be the same as in Theorem 1 and p = 2. Then the additive subgroups P and Q satisfy the following conditions:

A1) 1 G P n Q;

A2) PQ < P;

A3) P2 Q < Q;

A4) P2 P < P ;

A5) Q2Q < Q;

A6) P-1 = P ;

A7) Q-1 = Q.

Moreover, P2 and Q2 are fields, P and Q are P2-modules, and the subgroup M contains all diagonal elements of the form hr (tu), t,u € P \ {0} (respectively, t,u € Q \ {0}), if r is a short root (respectively, if r is a long root).

Proof. Condition A1) follows from definition of the subgroup M. Conditions A2) and A3) follow from the commutator formula (2) and the carpet condition for the subgroup M. In [1, p. 535] it was established that for any t e P/{0} (respectively t e Q/{0}) the polynomial ring K[t] (respectively K2[t]) lies in P (respectively, in Q). Hence, since the extension F/K is algebraic, we obtain equalities A6) and A7). For any short root r and any t,u e P/{0}, equality A6) implies that hr(t)hr(u) = hr(tu) e M. Similarly, for any long root r and any t,u e Q/{0} from A7), we obtain the inclusion hr(tu) e M. Conjugating the subgroup xr(Ar), r e $, by these diagonal elements, we obtain the inclusions A4) and A5). It follows from A4) and A5) that P2 and Q2 are fields. Finally, from A3) and A4) we obtain that P and Q are P2-modules. The proposition is proved. □

In [4, Sec. 7] for types Bl (l > 2) and Cl (l > 2), examples of subgroups P and Q from Theorem 1, one of which is not a field, and for the type B2 = C2 both of which are not fields, are given. Therefore, the inclusion of diagonal elements of the form hr(tu), t,u e P \ {0} (respectively, t, u e Q\ {0}) if r is a short root (respectively, if r is a long root) into the subgroup M, despite the fact that the product tu may not lie in the subgroup P (respectively, in Q).

Any algebraic extension of a perfect field is perfectly [8, p. 217] and any finite field is perfect, so the results of the paper [1] say that there are no finite additive subgroups that are not fields that parameterize intermediate subgroups in groups of Lie type. The next proposition asserts that they do not exist even under weaker constraints.

Proposition 2. If the characteristic of the field F is equal to 2 and its finite additive subgroup P satisfies the conditions A1) and A4), then P is finite field.

Proof. The inclusions A1) and A4) imply the inclusion P2 ^ P, and since squaring is an isomorphism of any field of characteristic 2, taking into account the finiteness of P, we obtain the equality P2 = P. Hence and again in view of A4), P is a ring and, therefore, a field, since any finite integral domain is a field. The proposition is proved. □

3. Groups lying between twisted Chevalley groups

Let A be a subset of the field F. The sets An and A-1 have the same meaning as in Section 2. The Steinberg group G(F) of type nXl is associated with an automorphism a of order n of the fields F. By Fa we denote the subfield of fixed elements of the automorphism a. By definition, we set a(u) = u, A = {u | u e A} and Aa = A n Fa. The groups lying between the Steinberg groups G(K) and G(F), where F is an algebraic extension of a nonperfect field K of exceptional characteristic p, are described by the author in [3].

Theorem 2 ([3]). Let M be a group lying between the Steinberg groups G(K) and G(F) of type 2Al, l ^ 4, 2Di, l ^ 3, 2E6 or 3D4, where F is an algebraic extension of an nonperfect field K of characteristic p, and p = 2 or 3 if G(F) is of type 3D4, and p = 2 otherwise. Then:

1) If G(F) is of type 2Al, l > 4, 2E6 or 3D4, then M = G(P)HM for some intermediate, subfield P, K C P C F, and some diagonal subgroup HM normalizing the group G(P).

2) If G(F) is of type 2Di, l ^ 3, then M = G(P,Q)HM for some diagonal subgroup HM normalizing the group G(P, Q) which is generated by intersections

M n xR(K)= xr (Ar), r e 2Dl, where {

Ar = P, if r short root, Ar = Q, if r long root,

P and Q are subgroups of the additive group of the field F containing the subfield K and respectively Ka, and they satisfy the following conditions: PQ Ç P, P2P Ç P, P-1 = P = P, uu, u + u G Q for all u G P, and if l ^ 4, then Q is a field.

The next proposition axiomatizes the properties of the additive subgroups P and Q from

i,

Theorem 2 for G(F) of type 2Dh l > 3.

Proposition 3. Let M, P and Q be the same as in Theorem 2 for G(F) of type 2 A, l > 3. Then the additive subgroups P and Q satisfy the following conditions:

B1)1 € P n Q, P £ Fa u Q £ Fa;

B2) PQ £ P;

B3) uut, uv + uv € Q for any u,v € P and t € Q;

B4) P2P £ P;

B5) Q2Q £ Q;

B6) P-1 = P;

B7) Q-1 = Q.

Next, we need the following technical lemma on algebraic extensions fields.

Lemma 1. Let F be an algebraic extension of the field K, the field F has an automorphism a, and Fa and Ka be centralizers of the automorphism a in the fields F and K, respectively. Then the extension Fa/Ka is also algebraic.

Proof. Let f be an arbitrary nonzero element from Fa. Since the extension F/K is algebraic, there exists a smallest natural number m such that

fm + km-1fm-1 + ... + kf + k0 =0

for some simultaneously non-zero elements ki from the field K. But then

fm + a(km-1)fm-1 + ... + a(k1)f + a(ko) = 0.

Subtracting the second equality from the first, we obtain

(km-1 - a(km-1))fm-1 + ... + (k1 - a(k1))f + (ko - a(ko)) = 0.

Hence, either for some i ^ 1 the difference (ki — a(ki)) is nonzero, which is impossible due to the minimality of m, or all the differences (ki — a(ki)) are zero, and then the element f is algebraic over the field Ka, as required. The lemma is proved. □

Proposition 4. Suppose that a field F of characteristic 2 has an automorphism a of order 2, P and Q are its additive subgroups satisfying conditions B1)-B7). Then the subgroups P and Q are fields, and Q = Pa.

Proof. Since Q £ Fa, the inclusion Q £ Pa follows from B1) and B2). Since P £ Fa, then there is an element t € P such that the sum t +1 is nonzero and due to B3) lies in Q, and in force B7)

— € Q. Hence and by virtue of B2) the element u =-= lies in P, and u + u = 1. Let v € Pa.

t +1 t +1 _ _ _ _ Then by virtue of B3) the subgroup Q contains the element uv + uv = uv + uv = (u + u)v = v. Therefore, Pa £ Q. So Q = Pa. Now B2) implies the inclusion QQ £ Q. Therefore, Q is a

ring. By virtue of Lemma 1, the extension Fa/Ka is algebraic, and since the ring Q is enclosed between Ka and Fa, it is a field.

Let us show that P is a field. Since the extension F/K is algebraic, it suffices to show that for any two elements of P their product lies in P. So, let u,v G P. If one of the elements u or v lies in Pa, then by condition B2) and the equality Q = Pa proved above, we obtain the inclusion uv G P. Let both u and v not lie in Q. Then they are the roots of the irreducible polynomials x2 + (u + u)x + uu and, accordingly, x2 + (v + v)x + vv of degree 2 over the field Q. The polynomial ring Q[u] is a field and, by B2) and B4), lies in P. If v G Q[u], then uv G P. If v G Q[u], then the polynomial ring Q[u, v] is an algebraic extension of degree 4 of the field Q and again, by B2) and B4), lies in P. Therefore, in any case, uv G P. The proposition is proved. □

Combining Theorem 2 and Proposition 4, we obtain the following theorem, which gives a uniform and standard description of intermediate subgroups for Steinberg groups over nonperfect fields in exceptional characteristics.

Theorem 3. Let M be a group lying between the Steinberg groups G(K) and G(F) of type 2Ai, l ^ 4, 2Di, l ^ 3, 2E6 or 3D4, where F is an algebraic extension of an nonperfect field K of characteristic p, and p = 2 or 3 if G(F) is of type 3D4, and p = 2 otherwise. Then M = G(P)HM for some intermediate subfield P, K С P С F, and some diagonal subgroup HM normalizing the group G(P).

This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement no. 075-02-2020-1534/1) and RFBR (project 19-01-00566).

References

[1] Ya.N.Nuzhin, Groups contained between groups of Lie type over different fields, Algebra i Logika, 22(1983), no. 5, 526-541 (in Russian).

[2] Ya.N.Nuzhin, Intermediate groups of Chevalley groups of type Bl, Cl, F4, G2 over nonperfect fields of characteristic 2 and 3, Siberian Math. J., 54(2013), 119-123.

DOI: 10.1134/S0037446613010151

[3] Ya.N.Nuzhin, Groups lying between Steinberg groups over non-perfect fields of characteristics 2 and 3, Trudy Inst. Mat. Mekh. UrO RAN, 19(2013), no. 3, 245-250 (in Russian).

[4] Ya.N.Nuzhin, A.V.Stepanov, Subgroups of Chevalley groups of types Bl and Cl containing the group over a subring, and corresponding carpets, St. Petersburg Math. J., 28(2020), no. 4. Translated from: Algebra i Analiz, 31(2019), no. 4, 198-224 (in Russian).

[5] A.V.Stepanov, Nonstandard subgroups between Е„(Д) and GL„(A), Algebra Colloq., 11(2004), no. 3, 321-334.

[6] A.V.Stepanov, Free product subgroups between Chevalley groups G^,F) and G^,F[i]), J. Algebra, 324(2010), no. 7, 1549-1557.

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

[8] S.Lang, Algebra, Mir, Moscow, 1968 (translation into Russian).

О парах аддитивных подгрупп, ассоциированных с промежуточными подгруппами групп лиева типа над несовершенными полями

Яков Н. Нужин

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Ранее (Труды ИММ УрО РАН, 19(2013), № 3) автор описал группы, лежащие между скрученными группами Шевалле С(К) и С(Е) типа 2Л;, 2В;, 2Е6, 3В4 в случае, когда большее поле Е является алгебраическим расширением меньшего несовершенного поля К исключительной характеристики для группы 0(Е) (характеристики 2 и 3 для типа 3В4 и только 2 для остальных типов). Оказалось, что кроме, быть может, типа 2В1, такие промежуточные подгруппы стандартны, то есть они исчерпываются группами 0(Р)Н для некоторого промежуточного подполя Р, К С Р С Е и диагональной подгруппы Н, нормализующей группу 0(Р). В данной заметка установлено, что промежуточные подгруппы являются стандартными и для типа 2В;.

Ключевые слова: группы лиева типа, несовершенное поле, промежуточные подгруппы, ковер аддитивных подгрупп.