Normal forms of elements of classical real and ...

Normal forms of elements of classical real and complex Lie and Jordan algebras D. Ž. Djoković, J. Patera, P. Winternitz, and H. Zassenhaus Citation: Journal of Mathematical Physics 24, 1363 (1983); doi: 10.1063/1.525868 View online: http://dx.doi.org/10.1063/1.525868 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/24/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in From vector fields to normal forms: A basic approach to lie algebras in classical mechanics AIP Conf. Proc. 255, 308 (1992); 10.1063/1.42322 Prolongation structures and Lie algebra real forms J. Math. Phys. 27, 1266 (1986); 10.1063/1.527131 Versal deformations of elements of classical Jordan algebras J. Math. Phys. 24, 1375 (1983); 10.1063/1.525869 Dimensions of orbits and strata in complex and real classical Lie algebras J. Math. Phys. 23, 490 (1982); 10.1063/1.525407 The maximal solvable subalgebras of the real classical Lie algebras J. Math. Phys. 17, 1028 (1976); 10.1063/1.523011

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Normal forms of elements of classical real and complex Lie and Jordan algebras8 ) D.

Z.

Djokovic

Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada

J. Patera and P. Winternitz Centre de recherche de mathematiques appliquees, Universite de Montreal, Montreal, Quebec, Canada

H. Zassenhaus Department of Mathematics, Ohio State University, Columbus, Ohio 43210

(Received 21 April 1982; accepted for publication 1 July 1982) Elements of the classical complex and real Lie and Jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical Lie group. Normal forms of representatives of each conjugacy class are chosen so as to resemble the Jordan normal forms of n X n complex matrices. For completeness similar results are given for gl (n,q,gl (n,R), andgl (n,H). PACS numbers: 02.1O.Sp, 02.20.Hj, 02.20.Sv

1. INTRODUCTION The purpose of this paper is to provide a comprehensive list of normal forms of matrices of two kinds. The first are matrices belonging to the defining representations of real and complex classical algebras. The normal forms are suitably chosen representatives of the conjugacy classes of the elements under the action of the corresponding classical Lie group. The solution of this problem has been known 1 for a long time for the algebras gl (N,D ) and sl (N,D ), where D stands either for the real numbers R, the complex numbers C, or the quaternions H. The normal forms are the familiar Jordan normal forms of the corresponding N X N matrices. For the remaining classical Lie algebras, i.e., those with an involution a, the problem has been solved relatively recently by Burgoyne and Cushman, 2 although for Lie algebras of several types it has been considered earlier by a number of authors. 3.4 Our solution of the problem is independent of that of Ref. 2; the results are naturally equivalent. In addition to a "basis independent" description of the normal forms, we summarize the results in an explicit ready-to-use matrix form. The set (J'_(aIDNXN)

=

(XIXEgl(N,D) and a(X)

=

-Xl

(Ll) of a-skew-symmetric matrices over D is closed under commutation and thus forms the classical Lie algebra (J' _(aiD N XN) with the involution a. In many applications, however, one also encounters matrices which are a-symmetric. The set (J'+(aID NXN ) = (XIXEg/(N,D) and a(X)=Xl (1.2)

of such matrices is closed under anticommutation and thus forms a Jordan algebra. It is defined in terms of the same involution a which determines the Lie algebra (1.1). The

a)

Supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Ministere de l'Education du Quebec. Published with the help of an FCAC grant for support of research.

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J. Math. Phys. 24 (6), June 1983

main result of this paper is a unified method of treatment of a-symmetric and a-skew-symmetric matrices. The new results are just for the case D = Hand p = ± 1. For the connection to some related classification tasks see the discussion at the very end of the paper. This article contains a classification ofthe elements X of (J'p(aIDNXN),p = ± 1 up to equivalence transformation gXg- 1, gEO (aiD NXN), and in each conjugacy class a particularly simple representative is chosen as the normal form. In applications the suggested normal forms make it particularly easy to recognize to which orbit an arbitrarily chosen element belongs and they simplify the task of transferring results obtained for a particular Lie algebra, say sl (n,q, to all other classical Lie algebras. Indeed, our normal forms have proven to be particularly suitable for the study of versaI deformations. 5 They can be used whenever the problem of orbit analysis on Lie groups and Lie algebras or specific sets of matrices arises, which is becoming more and more frequent in theoretical physics and applied mathematics. As examples let us mention: (i) The relationship between normal forms in phase space. 4 (ii) The study of the separation of variables in partial differential equations6 (an "ignorable variable" corresponds to each different normal form and a classification of "essential" separable variables is related to a classification of a-symmetric matrices). (iii) The problem of finding systems of ordinary nonlinear differential equations with nonlinear superposition principles. 7 Our own interest in the problem arose in connection with the problem of classifying all maximal abelian subalgebras of the classical Lie algebras 8 for which a convenient normal form of the corresponding Lie algebra elements was indispensable. The article is organized as follows. Section 2 contains notations, definitions and some conventions. The results of the paper are found in Sec. 3. There are four classification theorems, two for the algebras of the linear type and two for the quadratic ones, as well as Table III containing the indecomposable normal forms in the quadratic case. Section 4 contains a discussion of the results, some examples, as well as the discussion of the classification of a-normal matrices.

0022-2488/83/061363-12$02.50

@ 1983 American Institute of Physics

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2. DEFINITIONS AND NOTATIONS

(2.6) and a' by

1. LetDbe one of the three division algebras R, C, and H of dimension I, 2, and 4, respectively, over the real field, and Fbe the center of D. Thus one has

a'(X)=LXtL- I ,

detL=I=O,

then a and a' are equivalent iff there exists S such that SED NXN, 4. The set of matrices

detS=I=O,

L=).SKSt,

DI

RClHl F RCR 2. The general linear classical Lie algebra

(2.1)

gl(N,D) = {XIXEDNXNj

(2.2)

is the Lie algebra associated with the N X N matrices over D. The special linear classical Lie algebra sl(N,D) = {XIXED NXN and TrX=Oj

(2.3)

is the Lie algebra of traceless matrices of D N x N. Thus there are three Lie algebras of the general linear type: gl (N,R), gl (N,C), andgl (N,lHl). Since lHl has a nontrivial center F = R, the Lie algebra gl (N,lHl) of quaternionic N X N matrices is a real Lie algebra of dimension d = 4N 2 • The general linear group G L(N,D ) consists of all invertible matrices of D N X N: GL(N,D) = {YIYED NXN and yy- I =IN },

SL(N,D) = {YIYED NXN and det Y= lj

(2.5)

of GL(N,D ) elements with determinant equal to I as a normal subgroup. 3. The involution a is an involutory antiautomorphism a:D N XN_D N xN acting on XED N xN according to I,

det K =1= 0,

K

t

=

±K =

EK.

(2.6) The symbol X

O=l=)'EF.

(2.9)

ap(aIDNXN) = {XIXED

NXN

and a(X)=pX}, p=

± 1, (2.10)

which is either a-symmetric (p = + 1) or a-skew-symmetric (p = - 1) over the field of reference (2.1 I) is the main subject of interest in this paper. The classification of ap(alD N XN) hinges on the classification of none qui valent involutions a. The results are well known. 9 They are summarized in Table I, where also the Lie algebra a _(aiD N XN) corresponding to each a is shown. In particular one has a(K) = EK or equivalently Kt = EK.

(2.12)

(2.4)

where IN is the N XN identity matrix. The group GL(N,D) hasN 2 parametersoverD,sothatithasd = N 2dim D IRreal parameters. It contains the special linear group

a(X) = KX t K -

(2.8)

Lt= ±L,

t, defined by

5. The largest of the locally isomorphic Lie groups of matrices corresponding to the Lie algebra a _ (aiD N X N) is defined by O(aIDNXN) = {XIXED NXN and a(X)=X-lj. (2.13) Notice that from (2.13) one has XKXt=K. (2.14) 6. It is often convenient to represent complex numbers and quaternions by 2 X 2 matrices. One has the 1-1 corre-

spondence

a=a+ib~(_: q= a

xt=(xl/)t = (ao(xI/W = (ao(xjI))'

+ ib + jc + kd~ (

(2.7)

is the transposition of X combined with the action of an involution ao:D-D fixing the real number field. Two involutions a, a' are equivalent if one is obtained from the other by conjugation by an automorphism of D N x N, i.e., Sa(X)S - I = a'(SXS -I). Thus, if a is given by

!),

a+id + ic

= ( - b

(2.15)

_

~.

; .)

b+iC)

(2.16)

a - id '

where a,b,c,d E R, a,/3,y EC, and P = f = k 2 = - 1, ij = k (cyclic). Then, in particular, the conjugation turns out to be the transposition in the case of C and Hermitian conjugation for lHl:

TABLE I. Involutions defining the classical quadratic Lie algebras.

a

E

D

F

F

Lie algebras

N

K

\

C C R

C C IR

C C R

o(N,q sp(2n,q o(p,q)

N 2n p+q

sgn K

= (p,q)

-\ \

IR C

IR C

IR R

sp(2n,lR) u(p,q)

2n p+q

K _Kl K= _KT K=K T K= _KT Kt =K

sgnK

= (p,q)

IR

R

IR

p+q n

sgnK = (p,qj

R

sp(p,q) o·(2n)

Kt =K

-\

IHI IHI

~

a()(x) x x x x x· x· x·

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-\ \

J. Math. Phys., Vol. 24, No.6, June 1983

Kt = -K

Djokovi6 et a/.

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V that is invariant under right multiplication by elements of If[. A matrix which is not decomposable is said to be indecomposable. A decomposable X is GL(N,D )-conjugate to a block diagonal sum of indecomposables X;,

(2.17)

q* ......(13 -y*

y)T(O

13*

s

1

X =

Here x* denotes complex (quaternionic) conjugation if x E C (x E If[) and Hermitian conjugation if x is an n X n matrix, n> 1. 7. It is useful to distinguish two types of elements X of gl (N,D ): decomposable and indecomposable ones. A matrix X is decomposable if there exist two invariant nontrivial subspaces VI and V2 of the space Von which X acts such that ffi V2, XVI~ VI' XV2~ V2.

IN; =N.

(2.20)

i= 1

Two elements X and X of gl (N,D ) which differ only by permutation of the blocks X; are GL(N,D )-conjugate. They transform into each other by means of a suitable permutation matrix. For the purpose of our classification X and X should not be distinguished. By the general decomposition theorem (McLagan-Wedderburn-Remak-Krull-SchmidtFitting lO ) the block diagonal components are unique up to conjugacy. 8. The classification of elements into decomposable and indecomposable ones has to be refined for X E a p (a/D N x N). Indeed, a decomposition mayor may not be compatible with the involution a. Consequently, one has the following hierarchy of elements: I

I

(2.19)

In case D = If[ the "linear" space Von which gl (N,If[) acts is the column space If[N x I of all N-columns over If[ with componentwise addition and multiplication with elements of If[ on the right. A linear subspace then is any submodule of

X;EgI (N;,D),

i= 1

(2.18)

D NXI = V= VI

ffi Xi>

I

a decomposable

decomposable but not a decomposable

a indecomposable

indecomposable

More precisely X E ap(a/DNXN) is a-decomposable ifin addition to the properties (2.19) one has also the a orthogonality: N

VI 1V2 , i.e.,

I

;J~

a o (J:)Kij/; = 0

(2.21)

I

for any / E VI'!' E V2. Here a o and the matrix K define the involution a [cf. (2.6) and Table I]. 9. The minimal polynomial m(A) over Fofa matrix X ED N xN is the monic polynomial (i.e., its highest degree coefficient equals 1) of the lowest degree over F such that m(X)

= O.

(2.22)

The minimal polynomial always divides the characteristic 1365

J. Math. Phys., Vol. 24, No.6, June 1983

I polynomial X(A ), X(A) =det(X -AI),

(2.23)

where in the case D = If[ the matrix X is of degree 2N over C. The characteristic polynomial X (A ) shares all its irreducible divisors with the minimal polynomial m(A ). The minimal polynomial for an indecomposable matrix is a power of a polynomial morA ), which is irreducible over F and called the irreducible polynomial of X. For D = R or If[ the irreducible polynomials over Fare of degree 1 or 2, for D = C they are of degree 1. As examples, let us consider several matrices which will reappear as ingredients of normal forms. Let a, b, E R; then one has the following table: Djokovic et a/.

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x

m(A}

A -a

X(A}

A -a

A -a

A-a

m(A)

X(A)

(A - a)2

+ b2

Similarly let a, p, E C, then one has for the generic case (a#a*, P #O)

x

( _a *

:*)

(-:'

P a* 0

(-~'

:.

p

m(A}

X(A}

(a - A )(a* - A ) +

a -P*

:J

[(a -A )(a* -A)

IPl2

X(A)

X(A)

+ IPI 2f

X(A)

(a - A )(a* - A)

(a - A )(a* - A)

+ IPI 2

+ IPI 2

m(A)

a -P*

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J. Math. Phys., Vol. 24, No.6, June 1983

DjokoviC et al.

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a normal, though not necessarily a-symmetric or a-skew symmetric. Our results will be extended in Sec, 4 to a classification of the a-normal elements under 0 (aiD N XN) conjugacy.

10. We make use ofthe matrix operations

AffiB=(~ ~), A®B=A®(b rs

where B = (b rs

),

)

= (Ab rs

),

(2.24) 3. CLASSIFICATION THEOREMS

and the n X n matrices

(2.25)

J

In this section we present four theorems. The first two deal with classical Lie algebras oflinear typegl (N,D ), D = H, C, and lHl. Their content is well known and they are listed here just for completeness. No proofs are given. The last two theorems apply to quadratic classical Lie algebras a _(aIDNXN) as well as to a+(aIDNXN). Their content is known only for a_(aIDNXN) (cf. Ref. 2) and a+(aIC NXN ) (cf. Ref. 1). For a + (aiD N XN), where D = H or lHl, they appear here for the first time. Theorem 1: The normal form of an indecomposable element X of a linear classical Lie algebragl (N,D ) can be chosen to be (3.1)

where we allow x to be also a matrix. Then, for instance, one has

8

o o

o

a

f3

(2.26)

r 11. The most often used involution a of complex matrix calculus is the involution which sends any matrix of degree N over C on its transpose complex conjugate. The corresponding classifying group is the group U(N) of unitary matrices of degree N over C. Symmetry is Hermitian symmetry aD. Antisymmetry of X is equivalent to Hermitian symmetry of iX. It is well known that the principal axis theorem asserting the diagonalizability of such matrices under conjugation by unitary matrices extends precisely to the normal matrices which are characterized by the condition a(X)X = Xa(X)

(2.28a)

of the a-symmetric matrix X + = (X + a(X))/2,

= (g~.),;EC,Im;>O.

Comment: For D = lHl we interpret X as an N X N matrix of 2 X 2 complex matrices, cf. (2.16). The indecomposable normal forms of elements of gl (N,D ) are summarized in Table II. Theorem 2: The normal form of a decomposable element of a classical linear Lie algebragl (N,D ) can be chosen to be the block diagonal sum of indecomposable matrices X; of the form (3.1);

X=

and the a normality of X is tantamount to the commutativity rule

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J. Math. Phys., Vol. 24, No.6, June 1983

s>2,

(3.2)

s

IN; =N.

;= I

The sum (3.2) is defined uniquely up to permutations of the blocks X;. TABLE II. Indecomposable normal forms of elements of the linear classical Lie algebras. For notations see subsec. 2.10.

gl(n,q gl(2n I,R)

+

gl(2n,R)

(2.28d) Both a-summetric and a-skew-symmetric elements are a normal. But, for example, the elements of 0 (aiD N XN)are

X;,

where

Algebra

(2.28c)

ffi ;=1

(2.28b)

and the a-skew-symmetric matrix X - = (X - a(X))l2,

X"'o

(2.27)

which is easy to test. Generalizing the concept of normal matrices of complex matrix calculus we define a matrix X of D N xN to be a normal for an arbitrary involution a of D N x N if it satisfies (2.27). Equivalently, the a-invariant F-algebra generated by IN' X, a(X) is commutative. Each matrix X of degree N over D is uniquely decomposed into the sum X=X++X-

where X"'o is an irreducible matrix of degree g over D with minimal polynomial molt ) over F, N = gj-l, Ip. and Jp. are given in (2.25). For D = C we haveg = 1 andX",o =f3E C; for D = H we have either g = 1 and X", = a E H, or g = 2 and X "'0 = ( _ ~ ~), a E H, b> 0; "for D = lHl we have g = 1 and

Indecomposable normal form In(a) J 2 n+ 1 (a) J 2n la)

I

n

gl(n,lHI)

(

a -b

aEC aER aER

b)a

In(~ ~*)

a,hER,b> 0

;= a + ib, a,hER,b;;.O

Djokovic et a/.

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TABLE III. List of a-indecomposable normal forms of elements of classical Lie and Jordan algebras. The parameter K = nonequivalent normal forms (X,K) and (X, - K). Normal form

t

D

E

P

+ +

X

Restrictions

Signature p-q

K

I n(;)

Hn

J 2n +, (0)

F 2n +

±

I indicates the existence of two

Classification group O(n,Cl

O(2n

1

No. I

+ I,Cl

2

-

I n(;)

Ell (-In(;n T

I n(;)

Ell In(;)T

®(~

n even if;= 0 In

~)

O(2n,Cl

3

Sp(2n,Cl

4

Sp(2n,Cl

5

Sp(2n,Cl

6

T

+

In

J 2n (0)

C

-

~)

F 2n

-

J~(;)EIl( - In(;)(

noddif;=O In

In(a)

~)

®(_ ~

~(I- (- I)") 2

U(p,q),

p+q=n

7

0

U(n,n),

p=q=n

8

~(I-(-I)") 2

U(p,q),

p+q=n

9

0

U(n,n),

p=q=n

10

KHn

~(I_(_I)n) 2

O(p,q),

p+q=n

II

H 2n

0

O(n,n),

p=q=n

12

K( - I)n F 2n + ,

K

O(p,q),

p+q=2n+1

13

®(~ ~)

0

O(n,n), p=q=n

14

K(I-(-I)")

O(p,q),

15

0

O(2n,2n), p=q=2n

16

Sp(2n,R)

17

Sp(4n,R)

18

Sp(2n,R)

19

Sp(2n,R)

20

Sp(2n,R)

21

Sp(4n,R)

22

KHn

+

Ell I n(;)·

I n(;)

•

®(_ ~

Im;>O

In

®(~ ~)

+ Ki" - I Fn

In(ib) -

In(;)EIl( -In(;n·

Re;>O

In(a)

In

®(~

~)

+ I n( - :

!)

b>O

J 2n + ,(0)

+ I n(a)

Ell (- I n(a)(

a;;.O

In

nevenifa=( -

R

T

I n( _

~ ~)

I n(

a b) b a

I n(a)

b>O ( a Ell (_ J n_b !)f a>O,b>O

Ell I n(a)T

K( _

!) Ell (In( _: !)f

b>O

1)"-'

o

I 2n

®(~

In

®( _ ~ ~)

+ I n( - :

~

~)

®(- ~

I 2n

®Fn

~)

KF 2n

J 2n (0)

p+q=2n

-

In(a)

I n(

Ell (-In(an T o

b) b 0

I n( - :

1368

a;;.O n odd if a = 0 In

b>O

!)EIl( - In(_: !)f a>O,b>O

J. Math. Phys., Vol. 24, No.6, June 1983

®( _ ~ ~)

K(_ I 2n

~

I)" ®Fn o

®(- ~ ~)

DjokoviO

/2n

Sp(p,q), p + q = n

23

0

Sp(n,n)

24

®Fn

~(I-(-I)")

Sp(p,q), p+q=n

25

~(1-( -I)")

Sp(p,q), p+q=n

26

0

Sp(n,n)

27

O*(2n)

28

O*(2n)

29

O*(2n)

30

O*(4n)

31

2

®(~ ~) ,

K"(_~ ~)"

In(~ ~)

No.

~(I-(-l)n)

KH 2n

In(~ ~*)$(Jn(~ ~*)r

Classification group

In(_~ ~)

b>O

K(_ ~ o1)"-' ®Fn

In(~ ~*)$( -In(~ ~*)r

Ret>O Imt>O

/2n

In(~ ~*)

Imt>O

(

2 2

-

H

*

+

In(~ ~)

®(~ ~)

0 I Kn- '(

-

o b)

I n(

b 0

In(~ ~*)$( -In(~ ~*)r

b>O Re~>O Im~>O

K( /2n

Let us now turn to elements of up(a/DN XN). In order that X E up (a/D N X N), it is necessary and sufficient to have a(X) =pX = KXtK

-I

(3.3)

as follows from (2.6) and (2.10). Equivalently, KXt - pXK = 0

(3.4)

or

®

U ~-n F _ IOn

~r

o ®Fn

®( _~ ~)

composable. A list of a-indecomposable normal forms of pairs (X,K) is given in Table III. Theorem 4 deals with adecomposable elements. Theorem 3: A. Let XED N X N be an indecomposable matrix such that a(X) = pX, P = ± 1. The hormal form of the pair (X,K) can be chosen as follows. The matrix X=JI'(Xm.l =Xmo ®II' +Ig ® JI' (0)

(XK)t - EpXK = O.

(3.5)

In order to fix a particular up (a/D N X N), the values of E andp have to be fixed (cf. Table I). If in addition a particular form ofthe matrix K is chosen, the requirement (3.4) is invariant with respect to the transformation X_YXy- l ,

YEO(a/DNXN).

as a consequence of and

(Y,L)~(Y',L').

The problem is then to classify simultaneously the pairs of matrices (X,K) under the action S:(X,K )-(SXS - 1,SKS t)

= SXS -

I,

K'

= SKS t

(3.8)

for some S E GL (N,D ). In Theorem 3 we treat simultaneously a-indecomposable elements that are decomposable and those that are indeJ. Math. Phys., Vol. 24, No.6, June 1983

+ 1,

for

p=

for

p = - 1,

(3.10)

where K is a nonsingular matrix of degreeg over D satisfying the symmetry condition (3.11)

with E = - 1 for sp(2n,R), sp(2n,q, o*(2n), and E = the remaining algebras. In addition X m" satisfies -

t

-_

+ 1 for (3.12)

B 1. Let X be an a-indecomposable decomposable matrix such that a(X) = pX, P = ± 1. The normal form of the pair (X,K) can be chosen to be

INI2)

o '

(3.7)

of S E GL(N,D). It is straightforward to verify that (3.4) is invariant with respect to (3.7). We say that two pairs (X,K) and (X',K') are equivalent and write (X,K)~(X',K') iff X'

is the same as in Theorem 1 and

KX m" -pXm"K - O.

(XE9 Y, KE9L )~(X' E9 Y',K' E9L') (X,K)~(X',K')

(3.9)

(3.6)

It is, however, desirable for the purpose of the universal classification to let K vary within the range permitted by the equivalence relation (2.9) of the involutions. The full range should not be used because we need relations

1369

0 I

~)®Hn

(3.13)

where Xm is as in Theorem 1, i.e., m = m~ and Xm = JI'(Xmu),N = 2gfl·

B2. Conversely, a pair of matrices (X,K) as given by (3.13) is a-indecomposable precisely if either (i)Xm andpX~ are not conjugate under GL(N /2,D ), i.e., X m andpX ~ have distinct minimal polynomials over F, or, equivalently, Xm andpX~ have distinct eigenvalues, or (ii) Xm andpX~ are conjugate under GL(N /2,D) and one of the cases 3 (with Djokovic et al.

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; = 0), 4, 6 (with; = 0),

14 (with a

= 0),

17, 18,20 (with

a = 0), and 24 of Table III occur.

The result of the classification of a-indecomposable pairs may be summarized as follows: Let D, p, ~, t be fixed (one choice from Table III). The pair (X,K) ofTable III corresponding to that choice is a-indecomposable and is nonequivalent to any other choice. Every a-indecomposable pair is equivalent precisely to one pair in Table III. Pro%/Theorem 3: First the reader must work his way through Table III in order to verify that each entry (X,K) satisfies the required conditions. This work consists (1) of verifying that each case is the appropriate particular case of the generic form of the a-indecomposable pair (X,K) of Theorem 3; (2) one must verify that the generic form really meets the requirements stated in Theorem 3. (I) Thus, for example, for line 12 of Table III one must verify that J" ( _ % ~), with a, b real, b > 0, and K = H 2", actually is an indecomposable element of o( p, p), where p = n, K = Kt as claimed in the ~ column, KXtK - I = X as claimed in the p column, and finally that all matrices J" ( _ % ~), a, bE JR, b =1= 0, are equivalent to the normal form of Table III. Because of the invariance of the minimal polynomial, it follows indeed that the pair (J" ( _ % ~ ),H2n) is equivalent to the pair (In( _ I~ II~ I ),H2n)' (2) A. It follows from Theorem 1 thatX defined by (3.9) is indecomposable. A simple computation shows that K defined by (3.10) meets the conditions K t = ~K, KXtK - I = pX, provided (3.11) and (3.12) are satisfied. (3)B. It follows from Theorems 1 and 2, that X defined by (3.13) is fully decomposed so that the pair (X,K ) of(3.13) is a-indecomposable. Conversely, let us assume that (X,Ko) is an a-indecomposable pair. We have to show that it is equivalent to the normal form provided by Theorem 3. A. IfX is indecomposable then according to Theorem 1, after a suitable equivalence transformation, it is as in (3.9). Regarding Ko we merely know that K"6 = EKo, KoXtK 0- I = pX with stated sign of p and E. Writing Ko = (K ik ), i,k = 1, ... ,11, as a matrix of matrices of degree g of D, one verifies that KoX t = pXKo implies Kik = if i + k0, 1m ;;>0, stands for aquaternion represen ted by a 2 X 2 complex matrix and transformed without loss of generality to the diagonal form. One could have equally well transformed it, for instance, into the form (_~ :) with a >0 and b;>O. Note that in lines 25 and 29 if n is even we have just one matrix K because K n = 1, but for n odd we obtain two (nonequivalent) indecomposable pairs, one for K = + 1 and the other for K = - 1. The a-symmetric and a-skew-symmetric matrices are particular cases of a more general class of a-normal matrices. A matrix X is a-normal if it commutes with its a-image (see Subsec. 11 of Sec. 2). For instance, elements of the classical Lie groups are a normal with a(X) = X - I. Let us use Theorems 3 and 4 to establish normal forms for the pairs (Y,K ) formed by an a-normal matrix Yand a matrix K establishing an involution

the corresponding normal form of Y is obtained as either

with PIEtR[t], [Pd < N /2, or

(b) Y = aIN + bXI

+ X 2P 2 (X2) + X3 with P 2ER[t], [P2 ] < - 1 + N /2. K=

of D N x N relative to a t operation. We generalize Theorems 3 and 4 without detailed proof. Theorem 5: The F-algebra

A

generated by an a-indecomposable a-normal element Yof D N x N and its image a( Y) under the involution a with matrix K are equivalent to one of the following normal forms: 1. If A can be generated by a single element X over FIN then we have two cases: (a) and (b): X,K are chosen as in Theorem 3A with either (a) mo(t) = t or (b) mo(t) = t 2 + 1;

where the polynomial Pofdegree [P] in t over Fis one of the following: If Fa = F,p = 1 then P = t. If Fa = F,p = - 1, then P = t + PI(t Z) where PIEtF(t), [PI] 0),

where in all cases P I ,P2 are polynomials of F[t] for which [PI] I, and is given as

(aER,bER>o),

where the polynomiaIPER(t) is one of the following. Ifp = 1 then P = t. Ifp = - 1 then P = t + P I [t2], wherePIEtR[t], [PI] O) (4.9)

(a)Y=aIN+bXI+X2+PI(X3)

~)®IN/2 = /2

-a(XtJ,

(0) = a(X2 ),

n/2 (0)

= - a(X3);

J. Math. Phys .. Vol. 24. No.6, June 1983

(4.8)

(4.15)

Proof Theorem 6 just is the outcome of a full a decomposition ofa given pair (Y,K). We remark that the normal form is unique up to permutation of blocks. For the proof of Theorem 5 one makes use of Dickson's theorem by which A = B + J (A ) is the direct sum of a uniquely determined semisimple algebra B over F and the maximal nilpotent ideal J (A ) of A. Because of the a-invariance of A also Band J (A ) are a invariant. Because of the aindecomposability of the commutative algebra B it follows Djokovic et al.

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that B is a field, either B = FIN or F = R, B~ C. Hence, Y = YI + X 2 with YI in B, X 2 in J (A ), B

=

(IN,YI),J(A)

=

(FX2 ,FX3 ) withX3 =a(X2 ). IfJ(A)

cannot be generated by a single element over F then J(A) = (FX2 ) ~ (FX3 ).

If J (A ) is one-generated over F then we have J (A ) = (FX2 > andeitherJ(A) = (FX t >orJ(A) = (FX 2-)' Theremainder of the theorem follows by application of Theorems 3 and 4 to the more general situation of Theorem 5. Finally let us point out several classification problems that are related to the one solved here. II With the notations of this paper let G = GL(N,D) and fl(€,p) = !(X,K): Kt=€K, (XK)t=€pXKj, fl'(€,p) = !(Y,K): Kt=€K, yt=€pYj,

where X, Y,K are N X N matrices over D. Then G acts on these sets as follows: A.(X,K) = (AXA

-I,

AKA t),

A.(Y,K) = (AYA t,AKA t).

Themapw:fl (€,p)---+fl '(€,p)definedbyw(X,K) = (XK,K)isG equivariant. Let flo(€,p), resp.fl o(€,p), be the subset of fl (€,p), resp. fl '(€,p),consistingofallpairs(X,K), resp. (Y,K), withKinvertible. The restriction of w gives a bijection wo:flo(€,p)---+fl 0(€,p).

We have the following problems: (A)

(A') (A 0)

Classify G-orbits in fl (€,p), Classify G-orbits in fl '(€,p), Classify G-orbits in flo(€,p), . Classify G-orbits in fl o(€,p).

In our paper we solved the problem Ao. Using Wo it is clear that Ao is equivalent to A O. It is also clear that A ' is more general than A O.

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J. Math. Phys., Vol. 24, No.6, June 1983

The problemA', in the cases D = R or C and t = T, has been solved by Ermolaev in Ref. 11. One obtains a solution of Ao in two steps: 10 choose those pairs (YUKi) from Ermolaev's list for which Ki is invertible; 20 for each pair (Yi ,Ki) chosen in 10 form the pair (YiK i - I ,Ki) = (Xi ,K;). These pairs (Xi,Ki ) give the list of indecomposable pairs for our problem Ao. Although this is easy to do we have not done it systematically since these forms (Xi ,Ki ) are not as nice as our forms. 'A. I. Malcev, Foundations 0/ Linear Algebra (Freeman, San Francisco, 1963). IN. Burgoyne and R. Cushman, J. Algebra 44, 339 (1977). 3H. Zassenhaus, Canad. Math. Bull. I, 31, \01, 193 (1958); I. K. Cikunov, Ukr. Math. Zh.18, 79,122 (1966); J. Milnor, Invent. Math. 8, 83 (1969); T. A. Springer and R. Steinberg, in Lecture Notes in Mathematics, Vol. 131 (Springer, New York, 1970), p. 167; J. G. F. Belinfante and P. Wintemitz, J. Math. Phys. 12, 1041 (1971). 4J. Williamson, Amer. J. Math. 59,141 (1936); N. Burgoyne and R. Cushman, Celest. Mech. 8, 435(1974); A. J. Laub and K. Meyer, Celest. Mech. 9,213 (1974); M. Moshinsky and P. Wintemitz, J. Math. Phys. 21,1667 (1980); D. Z. Djokovic, Trans. Am. Math. Soc. 270, 217 (1982). 'J. Patera and C. Rousseau, J. Math. Phys. 23, 705 (1982); J. Patera, C. Rousseau, and D. Schlomiuk, J. Phys. A, \063 (1982); J. Patera, C. Rousseau, and D. Schlomiuk, J. Math. Phys. 23, 4109 (1982). oW. Miller Jr., Symmetry and Separation 0/ Variables (Addison-Wesley, Reading, Mass., 1977); W. Miller Jr., J. Patera, and P. Wintemitz, J. Math. Phys. 22, 251 (1981); P. Wintemitz,I. Lukac, Ya. A. Smorodinskii, Yad. Fiz. 7,192(1968) [SOy. J. Nucl. Phys. 7, 139(1968)]. 7R. L. Anderson, Lett. Math. Phys. 4, I (1980); R. L. Anderson, J. Hamad, and P. Wintemitz, Lett. Math. Phys. 5,143(1981); Physica D 4, 164 (1982);P. Wintemitz, Physica A 114, \05 (1982); J. Hamad, P. Wintemitz, and R. Anderson, "Superposition principles for matrix Riccati equations," J. Math. Phys. (to be published). "J. Patera, P. Wintemitz, and H. Zassenhaus, C. R. Math. Rep. Acad. Sci. Canada 2, 231, 237 (1980). oS. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978); R. Gilmore, Lie Groups, Lie Algebras and Some o/Their Applications (Wiley, New York, 1974). '"H. Zassenhaus, The Theory o/Groups, 2nd ed. (Chelsea, New York, 1958), p. 114, Theorem 7. "Yu. B. Ermolaev, Soviet Math. Doklady I, 523 (1960); R. Scharlau, Math. Z. 178, 359 (1981).

Djokovic et a/.

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