Local automorphisms of nil-triangular subalgebras of classical Lie type Chevalley algebras Текст научной статьи по специальности «Математика»

УДК 512.54 + 512.55

Local Automorphisms of Nil-triangular Subalgebras of Classical Lie Type Chevalley Algebras

Igor N. Zotov*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny 79, Krasnoyarsk, 660041

Russia

Received 10.05.2019, received in revised form 10.07.2019, accepted 20.08.2019 We study the problem of describing local automorphisms of nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring with identity.

Keywords: automorphism, local automorphism, standard central series, characteristic ideal, Chevalley

algebra, nil-triangular subalgebra.

DOI: 10.17516/1997-1397-2019-12-5-598-605.

Introduction

A local automorphism of an algebra A is arbitrary modular automorphism which acts on each element a G A as suitable automorphism of this algebra. The local automorphisms of an algebra A form a group under the composition of mappings (Lemma 1 in Section 1). Automorphisms of an algebra are its trivial local automorphisms.

Local automorphisms and local derivations of an algebra are systematically studied since the 1990s. According to [1], local automorphisms of the algebra M(n, C) of complex n x n matrices exhausted by automorphisms and anti-automorphisms. See also [2] for local automorphisms of the simple Lie algebra sln over a field of characteristic zero. R. Crist [3] constructed first example of a nontrivial local automorphism for subalgebra of triangular matrices in M(3, C) with pairwise coincide elements on each diagonal.

Let K be an associative commutative ring with identity. In [4] and [5] local automorphisms of the algebra NT(n, K) of nil-triangular n x n matrices over K and associated Lie algebra are investigated; they are described for n = 3 and, when K is a field, for n = 4. In this article we study the more general problem of describing local automorphisms of nil-triangular subalgebra N$(K) of the Chevalley algebra over K associated with a root system The main result is a reduction Theorem 1 in Section 1. The proof of the theorem is devoted to Section 2. See also the remarks in Section 3.

1. Remarks and the main theorem

Further K is an arbitrary associative commutative ring with identity, unless specified otherwise.

* [email protected] © Siberian Federal University. All rights reserved

According to [5] and [6], a local automorphism of an arbitrary K-algebra A is an automorphism of the K-module A which acts on each element a G A as some automorphism depending, in general, on the choice of a. (Definition in [1] has certain difference.) Local ring automorphisms are defined analogously. Denote by Laut A, the set of all local automorphisms of algebra A.

Lemma 1. The local automorphisms of any algebra A (similarly for the ring) form a group under the composition of mappings.

Proof. Choose an arbitrary G Laut A. They acts on each element x G A as some automorphism "x of an algebra A. It is necessary to show that

(^")x = = x G A-

It's evident that, x = 0(z) = (z) with uniquely z G A. Hence

r\x) = z = (z)) = (4>z )-1(x) = (rl)x(x),

(№)(x) = $($x(x)) = ^x(x)("^x(x)) = (^(x)^x)(x) = (№)x(x).

This completes the proof for local ring automorphisms. From here lemma follows easily. □

We investigate local automorphisms of a nil-triangular subalgebra in Chevalley K-algebras.

A Chevalley algebra over a field K is associated with each indecomposable root system $ in the Euclidean space and characterized by Chevalley base consisting of generating elements er (r G $) [7, Sec. 4.4]. We fix a base n in $. Positive system of roots $+ D n in $ is unique [7]. The subalgebra N$(K) with the base {er | r G $+} is said to be a niltriangular subalgebra. According to Chevalley's theorem on base [7, Sec. 4.2], if r,s G $+, then

er * es = Nr,ser+s = -es * er (r + s G $), er * es = 0 (r + s G $),

where either Nr,s = ±1 or |r| = |s| < \r + si and Nr,s = ±2 or $ is of type G2 and Nr,s = ±2 or ±3. The signs of the structure constants Nr,s may be chosen arbitrarily (up to isomorphisms N$(K)) for extraspecial pairs (r,s) G $+, [7, Proposition 4.2.2].

The height of the root r is the sum ht(r) of the coefficients in the expansion of r in the base n in $. The Coxeter number h = h($) of the system $ equals ht(p) + 1, where p is a maximal root in $+, [7,8]. Subalgebras Lm with base {er | r G $+, ht(r) > m} form in the algebra Li = N$(K) the standard central series

Li D L2 Lh-i = Kep D Lh = 0, h = ht(p) + 1. (1)

We now may to formulate our main theorem.

Theorem 1. The ideal L2 of Lie algebra N$(K) of classical type of rank > 4 is characteristic and any local automorphism of N$(K) acts as its suitable automorphism, modulo L2.

2. Proof of the main theorem

It is well known that the standard central series (1) of the algebra N$(K) is an upper central (or hypercentral) and lower central, except the cases 2K = K, when the system $ has no roots of different lengths and, also, the case 6K = K for type G2. Thus, all ideals Lm in the Lie algebra

N$(K) are characteristic when all roots of the system $ have the same length or 2K = K for the types Bn, Cn and F4.

The Lie algebra N$(K) of type An-1 is associated to the algebra NT(n,K) of all lower nil-triangular (with zeros on and above the main diagonal) n x n matrices over K. Usual matrix units eij (1 < j < i < n) gives Chevalley base {er | r G er = ej} after the corresponding numbering of the roots.

The Lie algebras N$(K) of type Bn, Cn and Dn are given in [9] similarly in the base of matrix units eiv, respectively

—i < v < i ^ n, —i ^ v < i ^ n, v = 0, 1 ^ |v| < i ^ n.

We assume that r = riv as er = eiv. The sums of two roots that are the root, in addition to the standard rj + rjv = riv, as for the type An, here also rkv + rm-v = rk,-m (k > m > |v|) and for type Cn, moreover, rkv + rk,-v = rk,-k (k > |v|). Any element of the Lie algebra N$(K) here is represented by a $+-matrix Haiv | = ^ aiveiv for corresponding type. Thus, the B+-matrix has the form

a10

a2,- 1 a20 a2i

an,-n+i • • • an,-i an0 ani • • • an,n-1'

If we cancel zeros column, then we obtain ^+-matrix.

Let Tim be the ideal of all $+-matrices of \\auv|| with the condition auv =0 for u < i or v > m. We assume that Tim := Tim if for the selected $ and m the number i is the smallest.

n

In the Lie algebra N$(K) of type Bn (or NBn(K)), we select submodules Rj := Kei0,

i=j

1 ^ j ^ n, and also select the submodule Lj with base {euv \ 0 < v < u < n, u — v > j}. We need the following two lemmas from [10].

Lemma 2. Let 2K = K and n > 2. Then the Lie rings NBn(K) and NCn(K) generate

{Keii-i (1 < i < n); Ke2-i}, {Keii-i (2 < i < n); KeH- (1 < i < n)}, respectively, and no Keiv can be dropped in them.

Denote by A2, the annihilator of the element 2. Lemma 3. Hypercenters of the Lie algebra NCn(K) (n ^ 2) are written as

Zi = L2n-i + A2L2n-i-i (1 < i < 2n — 1), Z2n-i = Li. For the algebra NBn(K) (n ^ 2) we have

Zi = L2n-i + A2Rn+i-i (1 ^ i ^ n — 2), Zn-i = Ln+i + A2R2 + A2eni, Zn+i = Ln-i + A2Ri + A2L^n_^2 (0 < i < n — 3), Z2n-2 = L2 + A2Li .

The diagonal automorphisms h(x) ■ er ^ x(r)er (r G Ф+) of the Lie algebra NФ(К) correspond to each K-character x of the root lattices to the multiplicative group К" of invertible elements of the ring К [7, Sec. 7.1].

For any root r the mapping t ^ xr(t) ■= exp (t ■ ad.er) (t G К) generate an isomorphism of the additive group К + ■= (К, +) to the automorphism group of the algebra Chevalley. Root subgroups Xr = xr (К) generate a Chevalley group, [7,11]. The restrictions of automorphisms of its unipotent subgroup UФ(К) = (Xr (r G Ф+)} generate the subgroup J of inner automorphisms of the Lie algebra NФ(К).

The standard automorphisms of the Lie algebra NФ(K) include inner, diagonal, graph [7, Ch. 12] and central automorphisms, that is, the identity automorphism modulo the center.

According to [9], if the Lie ring (or group) does not coincide with its mth hypercenter, then its automorphism is said to be hypercentral of height m, or simply hypercentral, if it is the identity automorphism modulo the mth hypercenter and an outer automorphism modulo the (m — 1)th hypercenter.

The main hypercentral automorphisms of height > 1 of Lie algebras of NФ(K) of classical types are revealed previously, [12-14]. Let V(Ф,К) denote the subgroup generated by them.

It is well known that the adjoint group of the ring R = NT(n, К) under the adjoint multiplication a о b = a + b + ab is isomorphic to the unitriangular group UT(n, К). The automorphism group of the associated Lie algebra Л(R) (that is, NФ(K) of type An-i) is found in [12]:

Aut Л(R) = Z ■ J ■ V ■D■ W (n> 4), (2)

where Z, D and W are subgroups of central, diagonal and idempotent automorphisms, respectively. The subgroup V is generated by the main hypercentral automorphisms of height 2 and for A2 = 0 - of height 3. For the Lie ring Л(R) the subgroup ~ Aut К of induced automorphisms is added, [12, Theorem 1]. The description of automorphisms in [12] also for n = 3,4 is certain difference.

In the Lie algebra NФ(K) of type Bn, the ideal L2 is larger than the commutant as 2K = К, by Lemma 2 and Lemma 3. When A2 = 0, it less than hypercenter Z2n-2. The Lie algebra NBn(K) admits hypercentral automorphisms, whose height depends linearly on the rank n. To any pair t,d G A2 there corresponds such automorphism

n-1

Xt,d ■ a ^ a + ak,-i(tek0 + den,-k), k=2

which translates the ( — 1)th column of the B+-matrix to 0th column.

The subgroup in Aut NBn(K) isomorphic to the adjoint group in A2 form semi-diagonal automorphisms

d(-1) ■ ekv ^ (1 + c)ekv (0 < —v<k < n), ekv ^ ekv (0 < v < k < n),

with invertible 1 + c G 1 + A2.

According to [13], for simple symmetric roots r and r = r (r = r) of a system roots Ф of type Dn (n > 4), an isomorphic embedding~of the subgroup

S = {a = \\aUv||G SL(2,K) ■ 2anai2 = 2a2ia22 = 0} of the group SL(2, К) into the group Aut NDn(K) is defined by the rule

a ■ er ^ aiier + ai2ef, ef ^ a2ier + a22ef, es ^ es (s G П \ {r, r}).

The description of automorphisms of Lie rings N$(K) of classical types is completed in [13]. It is summarized by the following theorem.

Theorem 2. Any automorphism of the Lie ring NCn(K) (n> 4) is a product of the standard and hypercentral of V($, K) automorphisms. For the Lie ring NBn(K) (n > 4), a semi-diagonal automorphism is added as a factor, and for the Lie ring NDn(K) (n > 4) an automorphism of S is added as a factor.

Remark that the ideal L3 is not even invariant under the hypercentral automorphism \t,d. On the other hand, we have the following lemma.

Lemma 4. The ideal L2 in the Lie algebra N$(K) of rank > 4 of the classical type is always characteristic.

Proof. Obviously, the ideal L2 in the Lie algebra NBn(K) is xt,d-invariant. Its V($, K)-invariance follows directly from the definitions of the other main hypercentral automorphisms of height > 1 in all cases with regard to the restriction on Lie rank, [13,14].

Graph automorphisms of Lie algebras N$(K) of rank > 4 are defined only for root systems of the same length (more precisely, for An, Dn and En types, n = 6,7,8). As noted above, in these cases all ideals of Lm are characteristic. With respect to diagonal (and ring) automorphisms, all one-dimensional subalgebras Ker are invariant, and therefore any Lm ideals are invariant. This is also true for semi-diagonal automorphisms of the Lie algebra NBn(K).

The inner automorphisms act on Lm identically modulo Lm+1, which also implies J-invariance of all Lm. Taking into account that under the conditions of the lemma the center Z1 always is in L2, we obtain the characteristic of the ideal L2 with respect to any standard automorphism. This completes the proof of the lemma. □

The first statement of main theorem 1 is given by Lemma 4. Let us prove the second statement.

The Lie algebra N$(K) of type An-1 is represented, as above, by the algebra A(R) for R = NT(n, K) (n > 4). The group automorphisms Aut A(R) is factorized by the product (2), and its normal subgroup Z ■ J ■ V acts identically modulo L2.

Let i' := n +1 — i. According to [12], any idempotent g of the ring K defines a Lie automorphism

Tg : eij ^ geij + ( — 1)i-j-1(1 — g)eji, 1 < j <i < n,

called idempotent and W =< Tg | g G K, g2 = g >. In particular, for the case i' = i + 1 we obtain the characteristic ideal

Ti' i'-1 = T0(Ti+1i) = Ti+1i

of the Lie algebra A(R).

Any local automorphism p of the Lie algebra A(R) acts on an arbitrary element modulo L2 as a suitable automorphism of D ■ W. Taking into account that p is K-module automorphism, we have, up to its multiplication by an automorphism of D ■ W,

p(e21) = e21, p(xe21 ) = xp(e21 ) = xe21 mod L2 (x G K).

We need the following lemma (cf. [15, Lemma 1.3.5]). Lemma 5. If p(e21) = e21 for a local automorphism p of Lie algebra A(R) (n > 4), then

p(ei+u) G Ti+U + L2, 1 < i <n. (3)

Proof. Assume that the inclusion (3) is violated for some number i, 1 < i < n and i' = i + 1. Taking into account the condition n > 4, for some idempotent f = 1 we obtain, up to a multiplication of y by a diagonal automorphism, the following equalities:

y(e2i) = e2i, <p(ei+ii) = Tf (ei+ii) mod R2, 1 — f = 0.

Since there exists automorphism " G Aut A(R) acting on e2i + ei+ii similarly to y, so

e2i + fei+ii + (1 - f )ei'i'-i = "(e2i) + "(ei+ii) mod L2. (4)

We can assume that " G D ■ W and hence " = Sts for a suitable idempotent g and diagonal automorphism S. If i < n — 1, then (n,n — 1)-projection of the element (4) on the right is in (1 — g)Kand on the left is zero. Hence g = 1 and " = S. Comparing now (i', i' — 1)-projections of matrices in (4) on the right and left, we obtain the equality 1—f = 0, that gives a contradiction.

It remains to investigate the case i = n — 1. Using (4), we find invertible elements c,d G K such that

(2 — f )e2i + fenn-i = (ge2i + (1 — g)enn-i)S + (genn-i + (1 — g)e2i)S,

(2 — f )e2i + fenn-i = (cg + c(1 — g))e2i + (d(1 — g) + dg)enn-i.

The last equality gives d = f. It is easily follows f = 1. This contradicts to the condition f = 1.

□

As corollary of the proved lemma we obtain the equalities for some elements ci G K

V(ei+ii) = ciei+ii (mod L2).

Now the equality (2) shows that ^ acts on each element of ei+ii modulo L2 as a suitable automorphism of D. It follows that all elements of ci are invertible in K. This completes the proof of the theorem for the type An.

Any local automorphism of the y Lie algebra NDn(K) acts on arbitrary element, as a suitable automorphism. Up to multiplication by an automorphism, one can even assume that the equality y(e2-i) = e2--i is satisfied.

Then y acts modulo L2 on each element ei+ii as a suitable automorphism from the product D ■ S. Therefore

y(e2i) = a2ie2,-i + cie2i mod L2,

y(ei+ii) = ciei+ii mod L2, 2 < i < n,

where all elements of ci (1 ^ i < n) are invertible in K and 2a2i = 0. Up to multiplication of y by a diagonal automorphism, we can assume that ci = 1(1 < i <n). Then y acts modulo L2 as an automorphism a with matrix

1 01

( 1 0) .

\a2i 1J

Further we note that if an ideal of an algebra is characteristic, then it is invariant with respect to any local automorphism of the algebra. From the Theorem 2 easily implies the following lemma

Lemma 6. In the Lie ring NCn(K) (n > 4), the ideals Tij for i < n and the ideals Tiv for v < 0 are characteristic.

It follows from lemma that the ideal T2,-2 is characteristic in the Lie algebra NCn(K) (n > 4) and this shows that any local automorphism of p induces a local automorphism of the factor algebra

NCn(K)/T2,-2 - NAn(K) - NT(n + 1, K),

moreover, p(T1--1) = T1--1. Applying the proved case for the type An, we obtain the statement of the theorem for the type Cn.

In the Lie ring NBn(K) (n > 5) the ideals T10 and T10 + T21 are always characteristic and

NBn(K)/T10 - NAn-1(K) - NT(n,K).

Applying the proved case for the type An, we obtain the statement of theorem for the type Bn. The theorem is proved. □

3. Some remarks

The algebra R is called enveloping for the Lie algebra L if replacing the multiplication in R with new a * b := ab — ba gives algebra R(-) isomorphic to L. It's obvious that Aut R C Aut R(-), [16]. Unlike the Lie algebras N$(K) the enveloping algebras (in general, non-associative) that are constructed for them in [16,17] depend on the choice of signs of the constants Nr,s.

When the choice of signs of the constants of the Lie algebra N$(K) of the classical type corresponds to its representation in [9], the enveloping algebra is denoted by R$(K). The developed methods are applicable for transferring the main theorem to algebras R$(K).

The restrictions in the main theorem on the rank of n are related to the fact that some basic hypercentral automorphisms of the Lie algebra N$(K) of classical Lie type for small n can remain automorphisms that are not hypercentral automorphisms. In these cases, the action of Aut N$(K), modulo L2, becomes exceptional. See [4,12] for the type An and the description of Aut ND4(K) in [9].

The author thanks professor V. M. Levchuk for statement of a problem and attention to the work.

References

[1] D.R.Larson, A.R.Sourour, Local derivations and local automorphisms of B(H), Proc. Sym-pos. Pure Math., 51(1990), 187-194.

[2] T.Becker, J.Escobar Salsedo, C.Salas, R.Turdibaev, On local automorphisms of stn, arXiv:1711.11297, 2018.

[3] R.Crist, Local automorphisms, Proc. Amer. Math. Soc., 128(2000), 1409-1414.

[4] A.P.Elisova, Local automorphisms of nilpotent algebras of matrices of small orders, Russian Mathematics (Iz. VUZ), 57(2013), no. 2, 40-48 (in Russian).

[5] A.P.Elisova, I.N.Zotov, V.M.Levchuk, G.S.Suleimanova, Local automorphisms and local derivations of nilpotent matrix algebras, The Bulletin of Irkutsk State University. Series Mathematics, 4(2011), no. 1, 9-19 (in Russian).

[6] I.N.Zotov, Local Automorphisms of Nilpotent Algebras of Matrices of Small Orders, Conference thesis of XLII regional student scientific conference of mathematics and computer science, Krasnoyarsk, SibFU, 2009, 24-25 (in Russian).

[7] R.W.Carter, Simple groups of Lie type, New York, Wiley and Sons, 1972.

[8] N.Bourbaki, Groupes et algebraes de Lie (Chapt. IV-VI), Hermann, Paris, 1968.

[9] V.M.Levchuk, Automorphisms of unipotent subgroups of Chevalley groups, Algebra And Logic, 29(1990), no. 3, 315-338 (in Russian).

10] I.N.Zotov, V.M.Levchuk, The Mal'tsev correspondence and isomorphisms of niltriangular subrings of Chevalley algebras, Trudy Inst. Mat. i Mekh. UrO RAN, 24(2011), no. 4, 135-145 (in Russian).

11] C.Chevalley, On Some Simple Groups, Matematika, Periodic Collection of Translations of Foreign Papers, 2(1958), no. 1, 3-53 (in Russian).

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17] V.M.Levchuk, Niltriangular subalgebra of Chevalley algebra and the enveloping algebras, Group Theory in Ankara, Middle East Technical University, 2019, 13-14.

Локальные автоморфизмы нильтреугольных подалгебр алгебр Шевалле классических типов

Игорь Н. Зотов

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Исследуется задача описания локальных автоморфизмов нильтреугольной подалгебры алгебры Шевалле над ассоциативно-коммутативным кольцом с единицей.

Ключевые слова: автоморфизм, локальный автоморфизм, стандартный центральный ряд, характеристический идеал, алгебра Шевалле, нильтреугольная подалгебра.