REFERENCE: IC/8V55 - streaming.ictp.itstreaming.ictp.it/preprints/P/84/055.pdfreference: ic/8v55 international centre for theoretical physics new expressions for the eigenvalues of

REFERENCE: IC/8V55

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

NEW EXPRESSIONS FOR THE EIGENVALUES OF THE INVARIANT OPERATORS

OF THE GENERAL LINEAR AND THE ORTHOSYMPLECTIC LIE SUPERALGEBRAS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

CO. Nwachuku

and

M.A. Rashid

1984 MIRAMARE-TRIESTE

IC/8V55 I. INTRODUCTION

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organisation

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

HEW EXPRESSIONS FOR THE EIGENVALUES OF THE INVARIANT OPERATORS

OF THE GENERAL LINEAR AIID THE ORTHOSYMPLECTIC LIE SUPERALGEBRAS *

C O , ttvachuku **

International Centre for Theoretical Physios, Trieste, Italy

and

M.A. Rashid

Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria.

ABSTRACT

We obtain expansions for the eigenvalues, C , of the invariant

operators (Casimir Operators) of the general linear, and the orthosymplectic

Lie superalgebras in terms of products of suitably defined graded power sums P .

The resulting expressions are closed and provide unified formulae for computing

the C 's for those superalgebras as well as their corresponding Lie algebras.

The formulae are remarkably simple to suggest that the power sums used in this

text could play a more basic role in the understanding of the pattern of the

expansion coefficients. Explicit illustrations are given for the various

series for p £. 8.

MIRAMAHE - TRIESTE

June 19814

* To be submitted for publication.

•* On sabbatical leave from the Department of Mathematics, University of Benin,

Benin City, Nigeria.

The problem of constructing the generalized invariant operators of the

general linear, the special linear and the orthosymplectic Lie superolge"braSl)-h)

have been successfully accomplished . The related problem of finding

the generalized eigenvalues of these invariants in simple and convenient forms

continues to receive attention

Recently Scheunert , and Bincer , working in tensor basis(were

able to obtain expressions for the eigenvalues of these superalgebras in a

particularly useful closed form, analogous to those of

Pereloraov and Popov for the unitary, and the present authors for the

orthogonal and the symplectic Lie algebras (see Eq.2.8 of this paper).

As recognized by earlier authors * , this form of expression though

useful for studying the analytic structures of the C , is not particularly

suited for practical calculations even for low values of p, since the summation

as well as the product runs through to the rank of the algebra. The

equation also contains fractional terms, whereas the C is a polynomial in the

\'s, and there must be cancellations. Some successful attempts have been made

in the past for the various series of the classical Lie algebras to rewrite

the C using generating functions, as a polynomial expansion in the X's.

Perhaps the most convenient form so far as a result of this endeavour is the

power sums product expansion formula for the C , obtained for the u(n) by ._>

Popov , and for the o(n) and the sp(2n) by one of the present authors

In what follows we attempt to express directly the eigenvalues of the

general linear, u{n,m), the special linear, su(n,m), and the orthosymplectic,

osp(N,2m), Lie superalgebras, as sums of products of suitably defined power

sums, obtaining expressions for these Lie superalgebras analogous to those of

Eefs.(lO) and (15) for the Lie algebras, thus tying together the treatment of

this aspect of the problem.

In so doing we have chosen to expand in terms of the naturally occurring

graded power sum P k defined in Eq.(2.15). With this power sum, half of

the terms in the coefficients of the expansion vanish resulting in considerable

time save. The final result appears remarkably simple to suggest that the

P^'s and therefore their non-graded limits P could play a useful role in

the study of the general patterns for these coefficients. Our result alsog 5)

affords a direct proof of the ansatz obtained by Balantekin and Bars ' for

the su(n), and conjectured by them for the o(tl). Finally, for the case of

the sp(2n) we obtain the corresponding ansatz.

-2-

In order to fix the notations and keep the presentation as self-

contained as practicable, ve outline briefly in Sec.II some of the earlier

results which we ; -ed to derive our main equations In Sec.III. Illustrative

examples are outlined in Sec.IV which also contains pertinent discussions.

There is an appendix which outlines the properties of the coefficients

Q (v) and 0 (v) for completeness.

II. NOTATIONS AHD DEFINITIONS

The infinitesimal generators X , in the canonical two index formJ

(Racah basis) of the general linear, and the orthosymplectic Lie superalgebras

satisfy the commutation relations, which can he written in a single form as

(2.1)

vhere the supercomrautator is defined by

(2.2)

with the index grading

- n i l<

and

(2.3)

-n £ i f

-3-

Eq.(2.l) as it star.ds in full with the indices riii'.ninp, from -n-m

to n+m describes the orthosymp]eotic Lie superalgehra. osp(N,Pn). Hero

K stands for 2n or 2n+l, and zero is included only when N is odd. If

the n-'s are set equal to zero (one) we recover the lie algebra o(N), {

If the indices are restricted to vary from I to n+m only, the general

linear Lie superalgebra u(n,m), is obtained. In this case the Lie algebra

u(n) is recovered "by setting the n.'s equal to zero.

The p-th order invariant (Casimir) operator C for each of the super-

algebras is defined by

(£.5)

n, (2.6)

and the summation runs over the appropriate ranges of the indices in each case.

Clearly, these operators commute accordingly vith all the generators of the

algebra, and "by the generalized Schur Lemma, they are. constant multiples

of the identity matrix in any irreducible representation, except possibly vhen

the dimensions of the bosonic and fermionic subspaces of the superalgebra are

sexual. This case is therefore excluded from consideration in this paper

For representations with unique highest veight, by acting recursively

p times on the highest weight vith the operator T^.tp) defined by

(X'• (2.7)

I,

and contracting the i and j Indices afterwards, one obtains the expression

for the eigenvalues C of the Casimir operators of the superalgebras as sums of

the elements of the corresponding Perelomov-Popov matrix A. raised to the

J-J g) Y)power p. On diagonalizing the A , the C can be written in the form *

(2.8)

-U-

where the summation and the product run over the respective ranges as indicated

earlier for each superalgebra.

In Eq.(2.8}

, for the u(n,m)

1 +

10-' A . -

osp(2n,2m) (2.9)

, " " osp(2n+l ,2m) .

Here the A.'s are the diagonal elements of A given byi ij

+ r.̂ + ̂ (n - m - a±) , f o r t h e u(n,m)

o . f . + r . + ^-(Sn - 2m - 1 - a.) , " " osp(2n,2ta)

cr.f. + r . + — (2n - 2m - a. + <5. „) , " " osp(2n+l ,2m)i i i 2 1 l O

(2.10)

where f. i s the number of boxes in thei-throv of the Young supertableaujc

corresponding to the highest veight of the irreducible representation of interest.

The quantities

For the u(n,m)

The quantities r take different values for the different superalgebras

(2.11)

<i J

while for the osp(2n,2m), and the osp(2n+l,2m), respectively,

(2.12)

-5-

and

(2.13)

with the subsidiary conditions r . = -r. in the two latter cases.

It remains to define the graded pover sums. For this purpose it is

convenient to introduce the symbol

pi = ' ViJ ± - ^ - u H i . • (2.114)

Then for each of the superalgebras, the graded power sums P are defined by

(2.15)•- ' —j y

with P = 0 for the su(n,m) and the osp(B,2m). Here the summation again

runs over the ranges of i.

Ill THE PRODUCT POWER SUM EXPANSION FORMULAE

3.1 The integral representation and generating functions

In this subsection it is convenient to separate the cases,

(i) The u(n,m) and the su(n,m)

For the u(n,m) and the su(n,m), Eq.(2.8) together with Eq.(2.9) can

be recast in the integral form

cf (3.1)Twhere the integration is over a large circle with the origin as centre on the

complex X-plane, containing all the poles . By setting \ = 1/z, we have

2iri p+2 ( 3 .2 )

-6-

which implies the identity

1 - f(z)

where

C zP

p+1 (3.3)

p " 0

(3.i

and the integration in Eq.<3.2) is along a closed path containing the origin

tut excluding all the poles of f(z).

Furthermore^ fQ(z) be the corresponding function f(z) for the

identity representation for which f± = 0 for all 1. Then

= t(.- (3.5)

(3.6)

(ii) The osp(£n,2m)

For this case, Eqs.(2.8) and (2.9) become

(3.7)

(3.8)

Here, t h e extra term in Eq.(3.8) takes care of the pole in the integrand at

\ = n-m-g- which i s not included in the summation E q . ( 3 . 7 ) ; and we have in

use of t h e i d e n t i t y

-7-

TT (> - fe) =(3.9)

In terms of the variable z = I/A, Eq.'3.8)

(3.10)

which implies the identity

b+l(3.11)

p = lwhere

it* 1-1 i -(3.12)

Furthermore, as in the previous case

( i i i ) The osp(2n+l,2m)

In a similar way, the C for the osp(2n+l,2m) can be written in

the form

r -

(3.15)

where

» I- (n -m- t i )?(3.16)

s o t h a t f r o m E q s . ( ^ . U ) a n d ( 3 . 5 ) , ( 3 . 1 2 ) a n d ( 3 . 1 3 ) , ( 3 . l 6 ) a n d ( 3 . 1 9 ) ,

and the corresponding identity

n + rrt

I I I 1 " 7^A L I , (3.17)

has been used.

Eq.(3.l6) implies the Taylor sum

(3.18)

The corresponding fo(z) is given by

3.2 The expansion formula

If for each superalgebra we take the logarithm of f(z) and use the

corresponding relations, Eqs.(3.6), (3.13} and (3-19), respectively, an

expansion involving the exponential of functions of power sums can be obtained

in a form given by Scheunert which is analogous to the results in Refs.(9) and

(lit) for the u(n), and the o(N) and sp(2n) respectively.These expressions

toe in themselves tedious to handle. Instead, we seek an expansion in terms

of the elementary products of the power sums P k in which the combining

coefficients are completely determined.

With f(z) and f0(

2) defined in the preceding subsection, it is nov

convenient to treat the three cases together. For each superalgebra let F(z) he

defined by the relation

f(z) = fQ(Z) (3.20)

HHi Dz ( -

(3.21)

where

(3.22)

and T is the set of all odd positive integers.

If we extend the definition of £ , to include JL=O, by requiring that

? «1, Eq.(3.2l) can be written in a more convenient form

In Ff*) - -k>\

where i, (z) has the formal form

(3.23)

From Eq.(3.23),

•o

(3.25)

-9--10-

Now let us introduce the abbreviations

[•,]] = I T vv!

(3-26)

(3.27)

Finally, on substituting Eqs.(3.29) and (3.20) into l!qs.{3.3), (3.U)

and (3.18) for the various cases, we obtain the desired expansion for the Cj;

cF

and let (v) denote the set of al l non-negative integers satisfying the

constraint equation

K + 1 (k + l)v (3.26)"k t

integer

for some/K ( ^ = 0 for the su(n,m) and the osp(lf,2m)). The Eq..(3.25)

can be written in a form in which a l l the coefficients of a typical product

lterm P . . . P, are collected together:

In this equation, for the u(n,m),

(3-35)

where

d = n-m, I = p-K > 0 and p > K > 1 , (p > K> 2 for the su(n ,m)) ,

and for the osp(N,2m),

(3.36)

where

(V)

i s t n e rank independent function satisfying

{3.29)

with

t-"

d = N - 2 m , £ = p - K > 0 and p > K > 2 (3 .38)

(3.30)

The Q (v)'s play a universal role in this problem in the sense that for a

well-defined set of power sums and for given £ and (v), they have the same

value for all the Lie superalgebras considered, and also for their corresponding

Lie algebras. It is this remarkable property that enables considerable

unification in the treatment of this aspect of the problem.

Eq. (3.30) implies the following special values for the (^(vVsj

V0) =

for all (v)

"• -

r > 0

(3.31)

(3.32)

(3.33)

IV. RESULTS AHD DISCUSSIOHS

Further properties of the coefficients Q-Av) and B (v) and illustrations

are given in the Appendix for more specific cases. Using these we compute

the C for p. <8. The cases of the su(n,ro) and the osp(N,2m) are

further simplified by the fact that P = 0 • v.. Table I contains the

C 's for the su(n,m), and Table II the 'correction effects' due to the u{n,m).

In Table III we show the corresponding values for the osp(N,2m).

Our results agree with those of Seheunert when expressed in terms

of the power sums

ti+M

Q L = V (r.ii.-r; i >

- 1 2 -

(H.I)

-11-

mnnmw "E

Using the relation

ft U.2)

L=*for the u(n,m) and the osp(H,2m), the factor of 2 being applicable only

in the case of the osp(N,2m). In Eq.. (lt.l)

In the non-graded l i m i t in which <J. = 1 , and d i s replaced by n(u)

for the u(n)(o(M)),

•,»*)"-ir;-it]where X.° and p. are the corresponding values of X^ and Pi in this

limit. We recover the results of Popov for the u(n) and those of Ref.(15)

for the o(N), by using the relations

?: =Ui

{It.5)

[° — f>* J Sx = 0, for su(n), o(N) . (It.6)

This is a direct proof of the ansatz of Ref.(5) for the u(n) and the o(w).

If we replace o. by -1, A. by -A and d by -2m, we obtain for

the case of the sp(2m),

*.7)

according to which

a - < >N

(it.8)

-13-

where C is the eigenvalue of the Casimir operator/order p for theP

sp(2m) in Table II of Ref.(l5). This provides the equivalent ansatz for

obtaining the osp(2n,2m) results from those of the sp(2m) and vice versa:

multiply the C of sp(2m) by (-i)P+1 and replace 2m by d*=2n-2m.

Generally, when compared with earlier results, our results show thatan

the C 's appear in their simplest form revealing/interesting pattern whenP * 0

expressed in terms of the graded power sums P , or equivalently p, for

the corresponding Lie algebras, a fact hardly apparent if one confined ones

investigations to the Lie algebras and blindly applied existing ansatz.

Also in terms of the P. or P, half the coefficients Qr(v) appearing in

the main expansion formula vanish resulting in considerable time save. These

power suras could play a more significant role in the detailed study of the

general pattern of these coefficients.

ACKNOWLEDGMENTS

One of the authors (C.O.H.) would like to thank Professor Abdus Salem,

the International Atomic Energy Agency and UNESCO for hospitality at the

International Centre for Theoretical Physics, Trieste.

- l i t -

APPENDIX (i) For t - 1, set ^ = k,

1. The coefficients Qf(v)

More speci

using the relation

More specific forms for the coefficients Q.(v) can be established T \( i i ) For t = 2 ,

(A.3)

where com. means all possible combinations of k ,. . .k in all the terms

appearing in the square bracket [ ]. This relation can readily be established

by induction on t using the defining Eq. (3.210,

If in Eq.{A.l4) k = kg = k, we have

( i i i ) For t = 3,

(A.5)

Nov let QjtP^ "" Pk ' denote Q£(u) for the particular set (v)

which satisfies the constraint Sq.(3.£8) for the values v « v » .,. = v, = 1,1 2 t

all other v's being equal to zero. Then using Eqs.(3.30) to (3.33) in

(A.l), we obtain, for I = 2r \T positive integer),

t / k,+zr+i-t(A. 2)

Eq,(A.2) determines the values of a l l the Q,(v) needed for the ca lcu la t ion

of 6 (v ) . For example.

Q fPDp-tr

\ ir + i j + \ zrtj (A.6)

with quan t i t i e s Q (P P 2 ) and CU (P 3 ) obtained by se t t i ng k - k = k,

and k = k = k = k, r espec t ive ly in t h i s equation.

2. The coef f ic ien t s $ (v)

These are determined d i r e c t l y from Kqs.(3. 35) and (3.3T)- I t i s ,

however, sometimes conven Lent t o use^ i jnpler forms derived from these equations

as i l l u s t r a t e d below for the case of"7"u(n,m).

-15- -16-

rresponds to tile partition (v) for which,

t

Thus, 4 = p-'- _ 0, and from Eg.. (3.25)

(i) The B (P ) : The sum P co

v = 1 \> = 0 i / k , which satisfies the constraint Eq.(3.?:B) for K = k.k i

p-k, even (A.?a)

p-k, odd , (A.7b)

(ii) The S (P, P, ): The product sum P P, occurs when v = v = 1,

v. = 0, i # k , k . In t h i s case K = k + k£ + 1, and J, = P - ( ^ + kg + l ) > 0,

so tha t

(A.8a)

From EqB.(A.B) 0 (Pk ) is obtained 'by setting ky = k£ = Is and dividing

through by 2!; and so on for higher products.

l)

2)

3)

h)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16}

P.D. J a r n s and U.S. Green, J . Math. Phys. 2Q_, 2115 (1979).

B. Balantekin and I. Ears, J. Math. Phys. 22, llU9 (198l).

M. Bednar ajid V. Sachl, J. Math. Phys. 1£, 11*87 (1978).

For the construction of generalized Casirair operators for the classical

Lie Algebras see for example:

G. Racah, "Group theory and spectroscopy", Princeton Institute for

Advanced Study Lectures (1951), reprinted in Springer Tracts in Modern

Phys. 31, 28 (1965);

H.B.G. Casimir, Proc. Roy. Acad. Amsterdam _3j*_, Sthl (1931);

L.C. Biedenharn and J .D. Louck, Comm. Math. Phys. 8̂ , 89 (1968).

B.Balantekin, J . Math. Phys. 2J4_, 1(86 (1982);

see a l s o E e f s . d ) , (2) and (3 ) .

H. Scheunert, J. Math. Phys. 2k_, 2681 (1983).

A.M. Bincer, J. Math. Phys. 2k_, 25^6 (1983).

A.M. Perelomov and V.E. Popov, J. Mucl. Phys. US3E 3., 92U (1966); 1, 1127

(1966).

A.M. Perelomov ana V.E. Popov, J. Hucl. Phys. USSR 3_> 693 (1967).

V.S. Popov, Theor. Mat. Fiz. 2£, 357 (1976), (Theor. Math. Phys. 29_,

(1976) translated Aug. 1977).

J.D. Louck and L.C. Biedenharn, J . Math. Phys. 11, 2368 (1970).

S. Okubo, J . Math, Phys. 16, 528 (1975).

CO. Nwachuku and M.A. Rashid, J . Math. Phys. 17, l 6 l l (1976).

CO. Nvachuku and M.A. Rashid, J . Math. Phys. 18, 1387 (1977).

CO. Nvachuku, J . Math. Phys, SC_, 1260 (1979).

V. Hittenberg, "A guide to Lie superalgebras", Invited talk at the

VI International Colloquium on Group Theoretical Methods in Physics,

Tubingen, July 18-22,1977.

-17-

TABLE T

TAELF II

C (p < 8) for the su(n,m), d = n-n 'Correction effects' for the C Cp S.8) for the u(n,m), d - n-m

P P3 2

P P5 2

-d

-d I-d

3T5

-d

716

32

-d

-1

Td

7d

32

73

Jl12

-d

-1

-1

1

K

1

3

It

5

6

7

8

pk

Pl

Pl 2

P2P1

P3P1

P l 3

PUP1

p P 2

2 1

P P5 1

PP 2F3 1

P2£P1

P ^

p6pl

P P 2pltPl

P3P2P1

P 2 P 1 3

Cl

-

C2

-d

C3

1

5"

12

d

d2

-1

%

1

16

38

d

-1

16

C6

d

3d8

9"8

d

d

-1

12

C7

1

532

9dB~

1912

12

d

a2

-1

12

12

1~2l*

C8

d

5d32

~8

19d12

d2

17"5"

1

d

d2

d2

d21*

-1

12

1

1

-19-

*} The C for the u(n,m) is obtained by adding the terms here for a givenP

p to the corresponding terms in Table I for the su(n,m).

-20-

TABLE I I I

K

2

3

It

5

6

7

8

P 2

P 3

*U

P 5

P6

P3P2

P7

V 2

P 3 2

P8

P5P2

v 3P 2

3

C2

1

C 3

| - d

1 £-*

C5

|(3-6d+d2)

^ + *>

1

12

^ ( i S ^ d - S d 2 ^ 3 )

1,13 26 J , 2 ,

^ + d)

2 " d

1

- 1

1

C7

' ^(e-i^d2-^^)

| ( 6 - Td + d2)

^(6 + d)

f **a>

1

- 1

l2

C8

»i ^ —2+lnd.—Xod. +12d. "- 5^ +d /

^.Ilta.^dV)

j4{8 + 8d - 3d2 + d 3 )

1 2

tI + ~T + d 2 )

- ( i - d ) i

•¥i -d)

i

- i

- i

i