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УДК 519.11+512.5
Enumerations of Ideals in Niltriangular Subalgebra of Chevalley Algebras
Nikolay D. Hodyunya*
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 18.11.2017, received in revised form 20.12.2017, accepted 20.02.2018 Let NФ(К) be the niltriangular subalgebra of Chevalley algebra over a field К associated with a root system Ф. We consider certain non-associative enveloping algebras for some Lie algebra NФ(К). We also study the problem of enumeration of standard ideals in algebra N Ф(К) over any finite field К ; for classical Lie types this is the problem which was written earlier (2001).
Keywords: Chevalley algebra, niltriangular subalgebra, enveloping algebra, ideal. DOI: 10.17516/1997-1397-2018-11-3-271-277.
Introduction
Any Chevalley algebra over a field K is characterized by a root system $ and a Chevalley basis consisting of elements er (r e $) and a base of suitable Cartan subalgebra [1, Sec. 4.2]. We fix a positive root system $+ C $. The subalgebra N$(K) with the basis {er | r e $+} is said to be a niltriangular subalgebra. In the present paper we consider the following problem.
(A) Find the number of standard ideals of Lie algebra N$(K) over any finite field K.
For classical Lie types this problem has arisen earlier as Problem 1 in [2]. In these cases Problem (A) had been solved recently by G. P. Egorychev, V. M.Levchuk, and the author. Standard ideals of a Lie ring N$(K) are distinguished in Sec. 1.
Main Theorem 2.1 in Sec. 2 solves Problem (A) for exceptional Lie types.
Also, we study enveloping algebras of Lie algebras N$(K). According to [3], an algebra R = (R, +, ■) (possibly, non-associative) is called an enveloping algebra of a Lie algebra L if L is isomorphic to the algebra R(— := (R, +, [ , ]), [a,b] := ab — ba. (See also Lie-admissible algebras [4,5].) The well-known enveloping algebra R of Lie algebra N$(K) [3, Proposition 1] has also base {er | r e $+} and its choice depends on signs of structural constants of Chevalley basis.
The representation [6] of Lie algebra N$(K) of classical Lie types determines uniquely their enveloping algebra R. If $ = Dn, then all ideals of such enveloping ring R are exactly standard ideals of Lie ring N$(K). By [3], it is not true for Lie type Dn (n > 4) and, as a corollary, for Lie types En (n = 6,7,8).
As it is shown in the following section, there exist enveloping algebras of type F4 both having nonstandard ideals and not having them.
* [email protected] © Siberian Federal University. All rights reserved
We use standard notation from [1]. Let ht(r) be the height of r £ $. The highest root in $+ is denoted by p. The Coxeter number h of $ is defined by ht(p) + 1 = h($) = h.
1. Ideals and enveloping algebras of Lie algebra N$(K)
The definition of an enveloping algebra R of an arbitrary finite dimensional Lie algebra L (see Introduction) shows that both algebras can be constructed on the same vector space. Also, every ideal of the enveloping ring R is an ideal of the Lie ring L.
In this section we study certain enveloping algebras of a Lie algebra N$(K) and their ideals. According to [1, Proposition 4.2.2], we have
[er,es] = Nrser+s = -\es,er] (r + s £ $), [er,es] = 0 (r + s £ $ \ {0}),
where Nrs = ±1 or |r| = |s| < \r + s\ and Nrs = ±2 or $ is of type G2 and Nrs = ±2 or ±3.
Proposition 1.1 ( [3, Proposition 1]). A K-algebra with the basis {er | r £ $+} is an enveloping algebra of N$(K) if the product is defined as follows: eres = 0 when r + s £ $, and if r + s £ $+
and Nrs ^ 1, then eres = er+s and eser = (1 — Nrs)er+s.
We distinguish the following ideals in a Lie algebra N$(K) putting on r < s (r,s £ $+) if s — r is a linear combination of simple roots with nonnegative coefficients:
T(r):=J2 Kes, Q(r):=J2 Kes.
r^s r<s
Roots r and s are called incident ones if T(r) C T(s) or T(s) C T(r) (i.e., s ^ r or r ^ s). Any set L of pairwise non-incident roots in $+ is called a set of corners in $+.
If H C T(r) and the inclusion fails under every substitution of T(r) by Q(r), then
rec
L = L(H) is said to be a set of corners in H. By [7], a set F(H) is said to be a frame of H if F(H) CJ2Ker, F(H) = H mod Q(L) (Q(L) = J2Q(r)).
rec rec
An ideal H of a Lie ring N$(K) is said to be standard if H = F(H) + Q(L). Evidently, all standard ideals of Lie ring N$(K) are ideals of any enveloping ring R from Proposition 1.1
The representation [6] of Lie algebras N$(K) of classical Lie types determines uniquely their enveloping algebra R. All ideals of such enveloping ring R for $ = Dn are exactly standard ideals of Lie ring N$(K). By [3], it is not true for Lie type Dn (n > 4) and also, as a corollary, for Lie types En (n = 6,7,8). We now show that both cases are possible for Lie type F4.
Theorem 1.1. For Lie type F4 Proposition 1.1 allows to construct enveloping algebras Ri having nonstandard ideals, and R2 in which all ideals are standard.
Proof. Note that the enveloping algebra R from Proposition 1.1 depends on choice of signs of structural constants Nrs.
Similarly to [1, Lemma 5.3.1], we use an ordering ^ on the space containing roots $ such that r ~< s implies h(r) < h(s). An ordered pair (r, s) of roots is called a special pair if r + s £ $ and 0 ^ r ~< s. An ordered pair (r, s) is called extraspecial if (r, s) is a special pair and if for all special pairs (r', s') with r + s = r' + s' we have r ^ r'.
Proposition 1.2. The signs of the structure constants Nrs may be chosen arbitrarily for extraspecial pairs (r, s), and then the structure constants for all pairs are uniquely determined.
Proof. See [1, Proposition 4.2.2].
□
For the root system $ of type F4, we need notation from [8]. The positive root systems of types Bn and Cn [9, Tables I-IV] can be written, respectively, as
C+ = {piv I 0 < H < i < n,v = i}, Pi,mj = — mej, 1 < j < i < n, m = 0,1, —1;
B+ = {qij I 0 < Ij I <i < n}, qi,mj = ei — mej. Then the positive system F+ is represented as the union C+ U B+ with the given intersection
B+ n C+ = {qio,Pi— (1 < i < 4)}.
Also, we use the following diagram from [8]. (The roots are accompanied by the notation (abcd) from [9, Table VIII].)
Fig. 1. The positive roots of the system F4
The relation q32 — q2i — qio — P32 of simple roots determines uniquely the ordering — (Fig. 1). Using Proposition 1.1 choose an arbitrary enveloping algebra R for Lie algebra N$(K)
of type F4. Recall that an ideal H of R is standard iff Q(r) C H for all r £ L(H).
It is clear that if M is a subset in an ideal H of R and F(M) = Ker, then T(r) C H. Let Li = T(r), 1 < i < h. It is not difficult to prove the following lemma.
i^h(r)
Lemma 1.1. Every ideal H C L4 in the enveloping ring R is standard.
Now we construct algebras Ri from Theorem 1.1. Assume N(r, s) := Nrs, and also,
N(q2i, q3,-i) = 1, N(q2i,P3i) = —1, N(q32,P3,-i) = 1, N(qio,P3,-i) = —1, (1) N(q32, q2,-i) = —1, N(q32, q2i) = —1, N(qw,p32) = —1, N(q2i, qw) = —1.
For algebra Ri we additionally set N(q32, q2o) = —1, N(qio, q2o) = 2, and
N(P32, q2,-i) = —1, N(P32,q30) = —1, N(qio, q3i) = 1.
By choosing arbitrarily the remaining structural constants Nrs we obtain the algebra Ri. One can see that ideals in the algebra Ri of the form
K (eq30 + Ce 92,-1 ) + K (eP42 + CeP3,-2
) + T(q3,-i) + T(q43) (c £ K*) (2)
are nonstandard. Moreover, It can easily be checked that all other ideals in the algebra Ri are standard.
Lemma 1.2. The algebra Ri has nonstandrd ideals and they are exhausted by ideals (2).
Further, we use the following lemma to construct algebra R2 which is not isomorphic to Ri .
Lemma 1.3. All ideals in the ring R are standard if the following equalities are satisfied:
N (q2i,q3,-i) = —N (q2i,P3i), N (q32,P3,-i) = —N (q32,q2,-i),
N(qio,P3,-i) = —N(qio, q3i), N(P32,q3o) = —N(P32,q2,-i),
N (q2i,P3i) = N (q32,q2i), N (qio,q3i) = —N (qio ,P32),
N(q32,q2i) = N(q2i, qio), N(q2i, qio) = N(qio,P32).
Proof. The proof is by direct calculation. □
To construct algebra R2, as before, assume (1). Also set N(q32, q2o) = —1 and N(qio, q2o) = —2. Then, by the Jacobi identity, N(p32,q2o) = 1 and
N (P32, q2,-i) = 1, N (P32, q3o) = —1, N (qio, q3i) = 1. By choosing arbitrarily the remaining structural constants Nrs we obtain the algebra R2. Lemma 1.4. All ideals in the algebra R2 are standard.
Finally, by combining Lemmas 1.2 and 1.4 we prove Theorem 1.1. □
2. The completion of problem's (A) solution
Denote by N$(q) the algebra N$(K) over finite field K = GF(q). Problem (A) of enumeration of standard ideals in Lie algebras N$(q) had been recently solved for classical Lie types (as Problem 1 in [2]) by G. P. Egorychev, V. M.Levchuk, and the author. In these section we complete the solution of Problem (A).
The following theorem gives the solution of Problem (A) for exceptional Lie types.
Theorem 2.1. The number of standard ideals of a Lie algebra N$(q) of exceptional Lie type is equal to
G2
F4 :
Eq
Er
Es
q + 7;
q4 + 3 q3 + 44 q2 +32 q + 25;
q9 +3 qs + 4 qr + 67 q6 + 69 q5 + 230 q4 + 306 q3 + 94 q2 +22 q + 37; 2 (q
+ 708 q4 + 300 q3 - 79 q2 + 31 q + 32);
16
q
+ 4283 qr + 5829 q6 + 7055 q5 + 3773 q4 - 2361 q3 - 244 q2 + 239 q + 121.
-12 + q11 + 3 q10 + 32 q9 + 90 qs + 118 qr + 394 q6 + 449 q5 +
16 + 3 q15 +4 q14 + 7 q13 + 237 q12 + 239 q11 + 693 q10 + 1647 q9 + 3554 qs +
Proof. We need the following definition. A subspace S of the space Km is called m-proper if for all i, 1 ^ i ^ m, there exists an element (ai,..., am) £ S such that ai = 0.
Similarly to Section 1, every standard ideal H of Lie algebra N$(q) is characterized by a set of corners L(H) = {r1,r2,... ,rm} and a frame F(H). So, to each standard ideal H there corresponds a unique pair (L, S) such that H is equal to the ideal
H(L, S) = Q(L) + {a,1ßri + a2er2 +----+ ameTm | (a1, a2,. .., am) G S}.
(3)
The second term in (3) is a frame of the ideal H(L, S). This yields that the enumeration of standard ideals coincides with the enumeration of ideals of the form (3). Denote by Vm,t the number of all m-proper t-dimensional subspaces in Km and by m) denote the number of sets of corners L in with \L\ = m. From the established one-to-one correspondence between standard ideals and pairs (L, S), we obtain the following
Lemma 2.1. The number of standard ideals in the algebra N$(q) of Lie rank n is
n m
Q($,q) = 1+^ B($,m) ]T V„ht.
m=1 t=1
Besides the solution of Problem 1 for type An, [10] provides the formula
(4)
Vm,t =
E
(qt _ 1)m-jt 1—1 qk _ 1
(q 1)t-Jt ^(q—1r+i-jk-1 (1 < t < m).
(q-1)1 jt Lo q-1
k=2
1=ji<j2<...<jt^m
In his paper [11], G. P. Egorychev has found a simpler form of this formula.
Lemma 2.2 ( [11, Lemma 4]). The number of m-proper t-dimensional subspaces of the space Km over the finite field K = GF(q) is
Vm
m-t
E
fc=0
(_1)m-t-k qk
m-1 \t + k - \)
t + k - 1
(5)
k
q
By using Lemma 2.1 we immediately obtain q) = q + 7 for type G2. In the remaining cases, we obtain the numbers B(§, m) by using the representations of of type F4 in [8] and of types En (n = 6,7, 8) in [12]. Tab. 1 represents the results of computations. (See also [13, Remark 5.2].)
Substituting the corresponding values of Tab. 1 and (5) for B(§, m) and Vm,t in (4), we prove Theorem 2.1. ' □
Table 1. The values of B(§, m) for types F4 and En
$/m 0 1 2 3 4 5 6 7 8
F4 1 24 55 24 1
Eq 1 36 204 351 204 36 1
E7 1 63 546 1470 1470 546 63 1
Es 1 120 1540 6120 9518 6120 1540 120 1
The author expresses his gratitude to V. M. Levchuk for guidance and important corrections.
The author was supported by RFBR (project no. 16-01-00707).
References
[1] R.Carter, Simple Groups of Lie type, Wiley and Sons, New York, 1972.
[2] G.P.Egorychev, V.M.Levchuk, Enumeration in the Chevalley algebras, ACM SIGSAM Bulletin, 35(2001), no. 2, 20-34.
[3] V.M. Levchuk, Niltriangular subalgebra of Chevalley algebra: the enveloping algebra, ideals and automorphisms, Dokl. Math., 478(2018), no. 2.
[4] A.Albert, Power-Associative Rings, Trans. Amer. Math. Soc., 64(1948), no. 3, 552-593.
[5] H.C.Myung, Some Classes of Flexible Lie-Admissible Algebras, Trans. Amer. Math. Soc., 167(1972), 79-88.
[6] V.M.Levchuk, Automorphisms of unipotent subgroups of Chevalley groups, Algebra and Logic, 29(1990), no. 3, 211-224.
[7] V.M.Levchuk, G.S.Suleimanova, Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type, J. Algebra, 349(2012), no. 1, 98-116.
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[9] N.Bourbaki, Groupes et algebres de Lie, Chapt. IV-VI, Hermann, Paris, 1968.
[10] V.P.Krivokolesko, V.M.Levchuk, Enumeration of ideals in exceptional nilpotent matrix algebras, Trudy IMM UrO RAN, 21(2015), no. 1, 166-171 (in Russian).
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Перечисления идеалов в нильтреугольной подалгебре алгебры Шевалле
Николай Д. Ходюня
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
В работе Г. П. Егорычева и В. М. Левчука 2001 г. была записана проблема 1, заключающаяся в перечислении стандартных идеалов нильтреугольных подалгебр NФ(GF(q)) алгебр Шевалле классических типов. Мы решаем аналог проблемы 1 для исключительных типов. С помощью недавно введенной конструкции В. М. Левчука обертывающих алгебр для NФ(К) исключительного типа F4 найдены обертывающие алгебры как с нестандартными иделами, так и без них.
Ключевые слова: алгебра Шевалле, нильтреугольная подалгебра, обертывающая алгебра, идеал.
