стр. 104 |

AO(2, 1) ? S : 0, (1,2,3);

ASL(3, R): 0, (1,2,3);

Подалгебры псевдоортогональной алгебры AO(3, 3) 449

AGL(3, R): 0, (1,2,3);

A2 + A3 + ?D + ?S (? ? 0): 0, (3), (1,2), (1,2,3);

A2 + A3 + ?D, S (? ? 0): 0, (3), (1,2), (1,2,3);

A2 + A3 + ?S, D + ?S (? ? 0): 0, (3), (1,2), (1,2,3);

A2 + A3 , D, S : 0, (3), (1,2), (1,2,3);

ASL(2, R): 0, (3), (1,2), (1,2,3);

ASL(2, R) ? S : 0, (3), (1,2), (1,2,3);

ASL(2, R) ? D + ?S : 0, (3), (1,2), (1,2,2);

AGL(2, R) ? S : 0, (3), (1,2), (1,2,3);

A2 + A3 + ?D + T3 : 0, (1,2);

A2 + A3 + T3 , D + ?T3 (? ? 0): 0, (1,2);

A2 + A3 , D + T3 : 0, (1,2);

ASL(2, R) ? D + T3 : 0, (1,2).

1. Фущич В.И., Никитин А.Г., Симметрия уравнений Максвелла, Киев, Наук. думка, 1983, 200 с.

2. Баранник Л.Ф., Фущич В.И., О непрерывных подгруппах конформной группы пространства

Минковского R1,n , Препринт 88.34, Киев, Ин-т математики, 1988, 48 c.

3. Баранник А.Ф., Фущич В.И., О непрерывных подгруппах псевдоортоганальных и псевдоунитар-

ных групп, Препринт 86.67, Киев, Ин-т математики, 1986, 48 c.

4. Фущич В.И., Баранник А.Ф., Москаленко Ю.Д., Подалгебры афинной алгебры AIGL(3, R),

Препринт 89.65, Киев, Ин-т математики, 1989, 32 с.

5. Patera J., Winternitz P., Zassenhaus H., Continous subgroups of the fundamental groups of physics.

General method and the Poincar? group, J. Math. Phys., 1975, 16, № 8, 1597–1614.

e

6. Patera J., Sharp R., Winternitz P., Zassenhaus H., Continous subgroups of the fundamental groups

of physics. The de Sitter groups, J. Math. Phys., 1977, 18, № 12, 2259–2288.

7. Тауфик М.С., О полупростых подалгебрах псевдоунитарных алгебр Ли, в Геометрические мето-

ды в задачах алгебры и анализа, Ярославль, Яросл. гос. ун-т, 1960, 86–115.

W.I. Fushchych, Scientific Works 2001, Vol. 3, 450–467.

Continuous subgroups of the generalized

?

Schrodinger groups

L.F. BARANNIK, W.I. FUSHCHYCH

Some general results on the subalgebras of the Lie algebra ASch(n) of the generalized

Schr?dinger group Sch(n) and on the subalgebras of the Lie algebra ASch(n) of the

o

generalized extended Schr?dinger group Sch(n) have been obtained. The subalgebra

o

structure of the algebras ASch(n) and ASch(n) are studied with respect to inner

automorphisms of the groups Sch(n) and Sch(u), respectively. The maximal Abeli-

an subalgebras and the one-dimensional subalgebras of the algebras ASch(n) and

ASch(n) have been explicitly found. The full classification of the subalgebras of the

algebras ASch(3) and ASch(n), which are nonconjugate to the subalgebras of ASch(2),

ASch(2), respectively, has been carried out.

1. Introduction

To construct exact solutions of both linear and nonlinear Schr?dinger and heat

o

equations it is important to know the subgroup structure of the extended Schr?dinger

o

group Sch(3) (see [1]). Other important applications of subgroup structure of this

group were discussed in [2, 3]. It is natural to generalize the notions of the three-

dimensional Schr?dinger group for the case of arbitrary n-dimensional Euclidean space

o

and to solve the problem of subgroup classification for these generalized groups. If

we restrict ourselves by continuous subgroups, then the problem will be reduced to

classification of subalgebras of correspondent Lie algebras. This classification was

realized for n = 1 in [4] and for n = 2 in [2].

In the present paper we study subalgebra structure of both the Lie algebra ASch(n)

of the Schr?dinger group Sch(n) and the Lie algebra ASch(n) of the extended Schr?-

o o

dinger group Sch(n) with respect to inner automorphisms of the group Sch(n) and

the group Sch(n), respectively. This paper is a continuation of investigations that

were carried out in [5–9]. The applied general method of Patera, Winternitz, and

Zassenhaus [10] gets further development for classes of groups under consideration.

In Sec. 2 we give definitions of the generalized Schr?dinger groups and algebras

o

and introduce some other concepts and basis notation used in the whole paper. In

Sec. 3, completely reducible subalgebras of the algebra AO(n)?ASL(2, R) are derived,

and all subalgebras of this algebra are described for n = 3. In Sec. 4 a number

of general results about splitting subalgebras of the algebra ASch(n) are obtained.

Abelian subalgebras of the extended Schr?dinger algebra ASch(n) are described in

o

Sec. 5. Classification of subalgebras of the algebras ASch(3) and ASch(3) is carried

out in Sec. 6. The conclusions are summarized in Sec. 7.

J. Math. Phys., 1989, 30, № 2, 280–290.

Continuous subgroups of the generalized Schr?dinger groups

o 451

2. Definitions of Schr?dinger groups and algebras. Main notation

o

Let R be the real number field, R an arithmetical n-dimensional Euclidean space,

and AG the Lie algebra of the Lie group G. The Schr?dinger group Sch(n) is the

o

multiplicative group of matrices

? ?

Wva

? 0 ? ? ?,

0??

where W ? O(n), a, v ? Rn , and ?? ? ?? = 1 (?, ?, ?, ? ? R). If ? = ? = 1,

? = 0, we obtain matrices that are elements of the Galilei group G(n). If at the same

time ? = 0, we have elements of the isochronous Galilei group G0 (n). Besides, the

Schr?dinger group Sch(n) can be realized as the transformation group

o

W x + tv + a ?t + ?

x> t>

, ,

?t + ? ?t + ?

where t is time and x is a variable vector of the space Rn .

The Lie algebra ASch(n) of the group Sch(n) consists of real matrices

? ?

Xv a

? 0 ? ? ?,

0 ? ??

where X ? AO(n), ?, ?, ? ? R, and a, v ? Rn . Let Iab be a matrix of degree n + 2

having unity at the intersection of the ath line and the bth column and zeros at the

other places (a, b = 1, . . . , n + 2). Then the basis of the algebra ASch(n) is formed by

the matrices

Jab = Iab ? Iba , Ga = Ia,n+1 , Pa = Ia,n+2 ,

D = ?In+1,n+1 + In+2,n+2 , S = ?In+2,n+1 , T = In+1,n+2

(a < b, a, b = 1, . . . , n). They satisfy the following commutation relations:

[Jab , Jcd ] = ?ad Jbc + ?bc Jad ? ?ac Jbd ? ?bd Jac , [Pa , Jbc ] = ?ab Pc ? ?ac Pb ,

[Pa , Pb ] = 0, [Ga , Jbc ] = ?ab Gc ? ?ac Gb , [Ga , Gb ] = 0, [Ga , Pb ] = 0,

[D, Jab ] = [S, Jab ] = [T, Jab ] = 0, [D, Pa ] = ?Pa , [D, Ga ] = Ga ,

[S, Pa ] = Ga , [S, Ga ] = 0, [T, Pa ] = 0, [T, Ga ] = ?Pa ,

[D, S] = 2S, [D, T ] = ?2T, [T, S] = D, (a, b, c, d = 1, 2, . . . , n).

The extended Schr?dinger algebra ASch(n) is obtained from the algebra ASch(n)

o

by adding the central element M , and, moreover, [Ga , Pb ] = ?ab M and other com-

mutation relations do not change. The factor algebra ASch(n)/ M is identified with

ASch(n). We shall denote the generators of algebras ASch(n) and ASch(n) by the

same symbols.

?

The algebra AG0 (n) = AO(n) ? M, P1 , . . . , Pn , G1 , . . . , Gn is called the exten-

+

?

ded isochronous Galilei algebra, and the algebra AG0 (n) = AG0 (n)/ M is called the

isochronous Galilei algebra.

Since the Lie algebra L = M, P1 , . . . , Pn , G1 , . . . , Gn is nilpotent, L is a Lie

algebra of some connected and simply connected nilpotent Lie group H. As H is an

452 L.F. Barannik, W.I. Fushchych

exponential group, any of its elements can be denoted as exp(?M ) exp(vG + aP ),

where ? ? R, vG = v1 G1 + · · · + vn Gn , and aP = a1 P1 + · · · + an , Pn (ai , vi ? R,

i = 1, . . . , n). The multiplication law is derived by the Campbell–Hausdorf formula.

Let

? ?

W00

?? 0 ? ? ?

0 ? ??

be an element of O(n) ? SL(2, R). It is not difficult to show that in Sch(n) we have

? · exp(vG + aP ) = exp((?W v ? ?W a)G + (??W v + ?W a)P ) · ?. (1)

An arbitrary element of the group Sch(n) has the form

exp(?M ) · exp(vG + aP ) · ?.

By definition, exp(?M ) · ? = ? · exp(?M ), and the equality (1) holds true for

? · exp(vG + aP ). Using these equalities andmultiplication laws in H and O(n) ?

SL(2, R) we shall establish multiplication in Sch(n) in the usual way. Evidently,

Sch(n) = H?(O(n)) ? SL(2, R)).

Subalgebras L1 and L2 of the algebra ASch(n) are called Sch(n) conjugated

if gL1 g ?1 = L2 for some element g ? Sch(n). Mapping: ?g : X > gXg ?1 ,

X ? ASch(n), is called an automorphism corresponding to the element g. If g =

diag[W, 1, 1], where W ? O(n), then ?g is called an O(n) automorphism correspondi-

ng to the matrix W . We shall identify the automorphism ?g with the element g.

Henceforth we shall use the following notations: X1 , . . . , Xs is a vector space

or Lie algebra over R with the generators X1 , . . . , Xs ; V [k, l] = Gk , . . . , Gl (k ?

l) is a Euclidean space having the orthonormal basis Gk , . . . , Gl , V [k] = V [k, k];

W [k, l] = Pk , . . . , Pl (k ? l), W [k] = W [k, k]; M[r, t] = M, Pr , . . . , Pt , Gr , . . . , Gt

(r ? t), M[r] = M[r, r], M[r, t] = M[r, t]/ M ; ?, ?, ? , , and ? are projections of

the algebras ASch(n) and ASch(n) onto AO(n) ? ASL(2, R), AO(n), V [1, n], and

W [1, n], respectively.

?

Let U be a subspace of M[1, n] and F be a subalgebra of ASch(n) such that

? ? ?

?(F ) = F . The notation F + U means that [F, U ] ? U and F ? M[1, n] ? U .

? ? ?

Considering algebras (F + U1 ), . . . , (F + Us ) we shall use the notation F : U1 , . . . , Us .

In the case of the algebra ASch(n) this notation has the same meaning.

Let L be the direct sum of Lie algebras L1 , . . . , Ls , K a Lie subalgebra of L,

and ?i the projection of L onto Li . If ?i (K) = Li , for all i = 1, . . . , s, then K is

called the subdirect sum of algebras L1 , . . . , Ls . In this case we shall use the notation

K = L1 + · · · + Ls . The subdirect sum of modules over a Lie algebra is defined in a

? ?

similar way.

3. On the subalgebras of the algebra AO(n) ? ASL(2, R)

In this section a number of auxiliary results to be used in following sections are

obtained.

Lemma 3.1. Subalgebras of the algebra ASL(2, R) are exhausted with respect to

SL(2, R) conjugation by the following algebras: O, D , T , S + T , D, T ,

ASL(2, R). The written algebras are not conjugated mutually.

Continuous subgroups of the generalized Schr?dinger groups

o 453

Later on, when we speak about subalgebras of the algebra ASL(2, R) we shall

mean the subalgebras given by Lemma 3.1.

By direct calculations we are convinced that the normalizer of D in the group

SL(2, R) consists of matrices

0 ? ? 0

,

???1 ??1

0 0

where ? ? R, ? = 0. The normalizer of T and the normalizer of D, T in the group

SL(2R) consist of matrices ± exp(?1 D) · exp(?2 T ), where ?1 , ?2 ? R. The normalizer

of S + T coincides with the group

cos ? sin ?

??R .

SO(2) =

? sin ? cos ?

Proposition 3.1. Let AH(n) be the Cartan subalgebra of the algebra AO(n). Up to

conjugacy under O(n) ? SL(2, R) the algebra AO(n) ? ASL(2, R) has two maximal

solvable subalgebras AH(n) ? S + T , AH(n) ? D, T .

Proposition 3.1 follows immediately from Lemma 3.1 and the fact that AO(n) has,

with respect to O(n) conjugation, the only maximal solvable subalgebra AH(n).

Proposition 3.2. Up to conjugacy under O(n) ? AL(2, R) the algebra AO(n) ?

ASL(2, R) has the following subalgebras: (i) F ? K, where F ? AO(n), K ?

ASL(2, R); (ii) F ? X + Y , where F ? X ? AO(n), Y ? ASL(2, R); and (iii)

X + D ? (F ? T ), where F ? X ? AO(n).

+

Proposition 3.2 is proved by the Goursat twist method [11].

Corollary. Subalgebras of the algebra AO(3)?ASL(2, R) are exhausted with respect

to O(3) ? SL(2, R) conjugation by the following algebras:

O; J12 ; D ; T ; S + T ; J12 + ?D (? > 0);

J12 + T ; J12 + ?(S + T ) (? > 0); J12 + ?D, T (? > 0);

D, T ; J12 , D ; J12 , T ; J12 , S + T ; J12 , D, T ;

AO(3); ASL(2, R); J12 ? ASL(2, R); AO(3) ? D ;

AO(3) ? T ; AO(3) ? S + T ; AO(3) ? D, T ; AO(3) ? ASL(2, R).

The written algebras are not conjugated mutually.

The space can M[1, n] be considered as an exact module the Lie algebra AO(n) ?

ASL(2, R). Let L be a subalgebra of this algebra. If M[1, n] is a completely reducible

L module, then the algebra L will be called completely reducible.

Theorem 3.1. A subalgebra L of the algebra AO(n) ? ASL(2, R) is completely

reducible if and only if ? (L) does not coincide with T and D, T .

Proof. If ? (L) = 0, then L is a completely reducible algebra. If ? (L) = D, T , then

L = L1 ? L2 , where L1 ? AO(n), L2 = X + D, T , X ? AO(n). Since the algebra

L2 is solvable and non-Abelian, then L is not a completely reducible algebra [12]. Let

? (L) = ASL(2, R). Since direct decomposition of F ? AO(n) can be realized through

every ideal, and since every subalgebra of the algebra AO(n) is not compact, then

L = ?(L) + ? (L). That is why [12] L is completely reducible.

Let us assume that ? (L) = D . Since [D, Pa ] = ?Pa , [D, Ga ] = Ga , then M[1, n]

can be decomposed into a direct sum of L-irreducible spaces. Consequently L is a

completely reducible algebra.

454 L.F. Barannik, W.I. Fushchych

As [S + T, Pa ] = Ga and [S + T, Ga ] = ?Pa , then the skew-symmetric matrix

?E

0

E 0

corresponds to the operator S + T in a basis P1 , . . . , Pn , G1 , . . . , Gn of the space

M[1, n]. Hence it follows that if ? (L) = S + T , then in the basis mentioned above

every element of an algebra L is represented by a skew-symmetric matrix of degree

2n, and that is why L is a completely reducible algebra.

Let ? (L) = T , and V [k, l] be an irreducible ?(L) module. Evidently V [k, l] +

W [k, l] is an L module. Since by Lemma 4.2 of [9] this module can not be decomposed

into a direct sum of irreducible L modules, an algebra L is not completely reducible.

The theorem is proved.

4. The structure of splitting subalgebras of the Schr?dinger algebra

o

The aim of this section is to study up to conjugation the subspaces of the space

M[1, n] invariant under subalgebras of the algebra AO(n) ? ASL(2, R). The main

results are Theorems 4.1 and 4.2.

?

Let F be a subalgebra of AO(n) ? ASL(2, R), and F be a subalgebra of the

? ?

algebra ASch(n) such that ?(F ) = F . If algebra F is Sch(n) conjugated to the

?

стр. 104 |