Available at: http://www.ictp.trieste.it/pub_off IC/98/122

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

HIGHEST WEIGHT IRREDUCIBLE REPRESENTATIONSOF THE LIE SUPERALGEBRA gl(l\oo)

T.D. Palev a~> and N.I. Stoilova 6)

Institute for Nuclear Research and Nuclear Energy,

1784 Sofia, Bulgaria c)and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Two classes of irreducible highest weight modules of the general linear Lie superalgebra g/(l|oo) are

constructed. Within each module a basis is introduced and the transformation relations of the basis under

the action of the algebra generators are written down.

MIRAMARE - TRIESTE

August 1998

a> E-mail: tpalev@inrne.bas.bgb> E-mail: stoilova@inrne.bas.bgc> Permanent address.

I. INTRODUCTION

We construct two classes of irreducible representations of the infinite-dimensional general linear Lie

superalgebra (7/(1 |oo). Both of them are classes of highest weight representations, corresponding to two

different orderings of the basis in the Cartan subalgebra. Related to this it is convenient to define (7/(1 |oo)

in two different, but certainly equivalent ways. We denote them as glo(l\co) and g/(oo|l|oo) (see the end of

the Introduction for the notation that follow).

Definition 1. The Lie superalgebra glo(l\oo) is a complex linear space with a basis {eij}ijeN- The Z2-

gradmg on gl^(\\(X)) is defined from the requirement that ey , eji, j = 2, 3, . . . are odd generators, whereas all

other generators are even. The multiplication (=the supercommutator) [ , ] on g/(l|oo) is a linear extension

of the relations:

{eihekl\ = 8jken - {-l)de^'i)^a^) Siiekj, i,j,k,le N. (1)

As a basis in the Cartan subalgebra Ho we choose {e8-j}jgN with a natural order between the generators:ea <- ejj> if' <- J- Then £Q = {eij}i<jeN (resp. £Q = {eij}i>j£N) a r e the positive (resp. the negative) root

vectors and {ej]8-+i}jgN a r e the simple root vectors.

Definition 2. The Lie superalgebra 17/(00 |l|oo) is a complex linear space with a basis {Eij}ij£Z- The Z2-

grading on 17/(00 |l|oo) is defined from the requirement that Eoj, Ejo, 0 7 j G Z are odd generators, whereas

all other generators are even. The supercommutator on 17/(00|l|oo) is a linear extension of the relations:

lEijtEkl} = 6jkE« - (-l)des(^)deg(Ekl)SilEkjt itjtk,le Z. (2)

As a basis in the Cartan subalgebra 7i we choose {Ea}i^z with a natural order between the generators:

En < Ejj, Hi < j . £+ = {Eij}i<j£z (resp. £~ = {-Eljj}j>jgz) a r e the positive (resp. the negative) root

vectors in 17/(00|l|oo) and {Eii+i}iez are the simple root vectors.

Both algebras are isomorphic. In order to see this let g : Z —> N be a bijective map, defined as

g(z) = 2\z\ + e(z)£N, V z e Z . (3)

Then it is easy to verify that the map <p, which is a linear extension of the relations

= eff(,-),ff(j), i j ' G Z , (4)

is an isomorphism of ^f/(oo|l|oo) on glo(l\co). Therefore both ^f/o(l|oo) and (7/(00|l|oo) are two different

realizations of one and the same algebra, namely gl(l\co). Note that <p is a map of 7i onto "Ho; it is not

however a map off"1" into £^. For instance take E-10 G £+ • Then ip(E-io) = e^i G £ Q • Hence a highest

weight representation of (7/(00|l|oo) may be not a highest weight representation of (7/0(1 |oo).

The reasons for studying representations of this particular superalgebra, namely (7/(1 |oo), stem from

physical considerations. Our motivation originates from an attempt to introduce new quantum statistics

both in quantum mechanics1'2 (in this case the superalgebras are finite-dimensional) and in quantum field

theory (QFT).3 '4 In order to see where the connection to the statistics comes from, we recall shortly the

origin of the Lie superstatistics.

2

The starting point is based on the observation that any n pairs bf, . . ., bf of Bose creation and annihi-

lation operators (CAOs), namely (below and throughout [x, y] = xy — yx, {x, y] = xy + yx)

&•,#] = % Pr.&j] = [&+,&+] = o, (5)

considered as odd elements, generate a representation, the Bose representation p, of the Lie superalgebra

osp(l\2n) = B(0|n).5 Denote by B1 , . . ., Bf those generators of B(0\n), which in the Bose representation

coincide with the Bose operators, p(Bf) = bf. Similarly as the Chevalley generators do, the operators

Bf, . . ., Bf and the relations they satisfy, namely

[ { B ^ B ] } , B%] = (e - ()6ikB] + ( e - r , ) 6 j k B l C,V, e = ± or ± 1, (6 )

define uniquely the LS B(0|n).5 The operators Bi are odd root vectors of B(0\n), whereas {Bf, 57} belong

to the Cartan subalgebra. The operators (6) are known in quantum field theory: these are the para-Bose

operators, generalizing the statistics of the tensor fields.6 The important conclusion is that the representation

theory of n pairs of para-Bose (pB) operators is equivalent to the representation theory of the Lie superalgebra

B(0\n). Certainly in QFT the algebra is 5(0|oo), it is infinite-dimensional.

The identification of the para-Bose statistics with a well known algebraic structure provides a natural

background for further generalizations. In QFT the commutation relations between the CAOs are deter-

mined from the translation invariance of the field under consideration.7 In momentum space the translation

invariance of a scalar (or tensor) field ty(x) is expressed as a commutator between the energy-momentum

Pm, m = 0,1, 2, 3 and the CAOs of of *(«):

[Pm,af] = ±kTaf, (7)

where the index i replaces all (continuous and discrete) indices of the field and

To quantize the field means, loosely speaking, to find solutions of Eqs. (7) and (8), where the unknowns are

the CAOs ai . The Bose operators (5) and their generalization, the pB operators (6), certainly satisfy (7).

By no means however they do not exhaust the set of the possible solutions.

The first possibility for finding new solutions and hence for further generalization of the statistics stems

from the observation that the commutation relations between the Cartan elements and the root vectors, in

particular Eq. (7), remain unaltered upon q-deformations. The deformations of the parastatistics along this

line was studied in Refs. 8-11 and more generally in Ref. 12.

Another opportunity, closely related to the present paper, is based on the observation that 5(0|n)

belongs to the class B superalgebras in the classification of Kac.13 Therefore it is natural to try to satisfy

the quantization equations (7) and (8) with CAOs, generating superalgebras from the classes A, C and D or

generating other superalgebras from the class B. In Refs. 3, 4 it was shown that this is possible indeed. For

charged tensor fields the main quantization condition (7) can be satisfied with CAOs, which generate the

LS g/(oo|l|oo), namely a LS from the class A. Up to now however this new statistics, the A—superstatistics,

did not achieve any further development. The reason is that so far the Fock spaces corresponding to the

A—superstatistics were not constructed. Here we come to the relation between the A—superstatistics and

the present investigation. The Fock spaces are representation spaces of gl(l\oo). In order to study the

physical consequences of the A—superstatistics in QFT one has to develop first the representation theory of

17/(001l|oo) (for charged scalar fields) and of (7/0(1 |oo) (for neutral fields). This is what we do in the present

paper. The reason to study only highest weight representations reflects the fact that there should exist a

state with a lowest energy, a vacuum, which turns to be the highest weight vector in the corresponding

gl(l\oo)— module.

So far the A—superstatistics was tested only in finite-dimensional cases, namely in the frame of a

(noncanonical) quantum mechanics. We have in mind the Wigner quantum systems, introduced in Refs. 1

and 2, which attracted recently some attention from different points of view.14'15'16 These systems possess

quite unconventional physical features, properties which cannot be achieved in the frame of the canonical

quantum mechanics. The (n + 1)—particle WQS, based on s/(l/3n),1 7 exhibits a quark like structure:

the composite system occupies a small volume around the centre of mass and within it the geometry is

noncommutative. The underlying statistics is a Haldane exclusion statistics,18 a subject of considerable

interest in condensed matter physics. The osp(3/2) WQS, studied in Ref. 14, leads to a picture where two

spinless point particles, curling around each other, produce an orbital (internal angular) momentum 1/2.

One can expect that also in QFT the Lie superstatistics could lead to new features.

In the literature one does not find many papers dealing with representations of infinite-dimensional

simple Lie superalgebras.20'21 Implicitly however such algebras and their representations were used in theo-

retical physics since the QFT was created. On the first place we have in mind the ordinary Fock space W\ of

infinitely many pairs of Bose CAOs {&*}igz- As mentioned above, the Bose operators are (representatives

of) the odd generators of 5(0 |oo) and their Fock space W\ is one particular irreducible 5(0|oo)— module.

The Fock spaces Wp of para-Bose operators {Btf}igz, corresponding to order of the parastatistics p £ N,6

are also irreducible and inequivalent to each other B(0|oo)— modules. The Clifford construction in Ref. 21 is

a generalization to the case when both bosons {bi }igz, considered as odd variables, and fermions {fi }igz,

which are even generators, are involved. The assumption is that the bosons anticommute with the fermions.

Then any n pairs of Bose CAOs and m pairs of Fermi CAOs generate (a representation of) the Lie superal-

gebra B(m\n).22 Therefore the Fock representation of {bf, ff}i^z is an irreducible 5(oo|oo)— module. Its

restriction to (7/(00100) leads to a set of irreducible representations of this superalgebra.

In the paper we use essentially results from the representation theory of gl(l\n). The finite-dimensional

irreducible modules (fidirmods) of the latter are, one can say, well understood. A character formula for all

typical13 and atypical23 modules has been constructed. The dimensions of all fidirmods are known.24'25 A

basis, similar to the GZ basis of gl(n), was defined and its transformation under the action of the Chevalley

generators was written down.26'27 The results were even generalized to the quantum algebra Uq[gl(l\n)].28

This is in contrast to the more general case of gl(m\n) and Uq[gl(m/n)], where most of the above problems

are still waiting to be solved although partial results do exist.29 '30 '31 '32 '33

The irreducible highest weight representations of (7/0(1 |oo), which we consider, are a generalization to

the infinite-dimensional case of the finite-dimensional essentially typical representations of gl(l\n) in the

Gel'fand-Zetlin basis (GZ basis). In order to see where the possibility for a generalization comes from we

recall (Sect. II.A) the way the Gel'fand-Zetlin basis was introduced.31 This basis is, however, inappropriate

for a generalization to the case of highest weight g/(oo|l|oo) modules. Therefore in Sect. II.B we modify it,

introducing a new basis, which we call a C—basis. It is an analogue of the C—basis for gl^ .34>35 Section

III is devoted to the irreducible g/(l|oo) modules. In Sect. III.A we extend the Gel'fand-Zetlin basis to the

infinite-dimensional case and apply it to glo(l\oo). The highest weight irreducible g/(oo|l|oo) representations

are defined in Sect. III.B. They appear as a generalization of the essentially typical representations of gl(l\n)

in the C—basis. The transformations of the basis under the action of the algebra generators are explicitly

written down.

Throughout the paper we use the notation:

LS, LS's - Lie superalgebra, Lie superalgebras;

CAOs - creation and annihilation operators;

fidirmod(s) - finite-dimensional irreducible module(s);

GZ basis - Gel'fand-Zetlin basis;

N - all positive integers;

Z - all integers;

Z_|_ - all non-negative integers;

Z2 = {0, 1} - the ring of all integers modulo 2;

C - all complex numbers;

[p; q] = {p,p + l,p+ 2, . . ., q — 1, q], if q — p £ Z + and [p; q] = 0 otherwise; (9)

[m]k = [mlk,m2k, • • -mkk], where mik £ C; (10)

[M]2k+e = [M_ki2k+e,M_k+h2k+e,...,Mk_1+e,2k+e], 0 e {0,1}, i e N ; (11)

hj=mij + l, lij = -rriij + i - 1, i £ [ 2 ; j ] ; (12)

L0,2k+e = M0i2k+e, 0 e { 0 , 1 } ,

Li2k+e = -Mit2k+e + i + l, # £ { 0 , 1 } , i e [ - A r ; - l ] , (13)

[m] = [ m 1 , m 2 , . . . , m k , . . ] = { m i \ m i E C,i £ N } ; (14)

[ . . . , M _ p , . . . , M _ i , M o , M i , . . . , M , , . . . ] = {M,-|M,- e C , i e Z } ; (15)

II. FINITE-DIMENSIONAL ESSENTIALLY TYPICAL REPRESENTATIONS OF gl{l\2n)

As in the case of ^f/(l|oo) it is convenient to use two different notation for the finite-dimensional super-

algebras from this class. In the first notation glo(l\N) is the same as in Definition 1, but the indices i,j run

from 1 to N + 1.

Then e n , e22, • • •, ejv+i,jv+i is a basis in the Car tan subalgebra %Q. Denote by e 1 , . . ., eN+l the dual

basis, el(ejj) = 8lj. The correspondence root vector -B- root reads: e8-j f> f' - fJ, i 7 j = 1, . . . , # + 1;

A 0 = {e% - e3}i7Lj€[1;N+1] is the root system; A°+ = {e% - e:1}i<je[1]N+1] and

are the standard systems of positive roots and simple roots, respectively. The special linear superalgebra

sZo(l|iV) is a subalgebra of glo(l\N) spanned by all glo(l\N) root vectors and the Car tan elements e n + en

for all i 7 1.

Similarly, gl(M\\\N) is the same as in Definition 2, but i,j = -M, -M + 1, . . .,N and M, N £ Z+. In

particular {-E1M}J£[_M;JV] is a basis in the Car tan subalgebra % with {£8}8-e[_M;JV] its dual. The simple root

vectors are {£ 1 i i i + i } i e [_ M ; j V _i ] . Hence

7T = { 5 M - £

is the system of simple roots.

We have written explicitly the systems (19) and (20) in order to underline that they contain different

number of odd roots: TT° has only one, e1 — e2, whereas the odd roots in TT are £~l — £°, £° — £l. Therefore the

systems of the simple roots of s/o(l|2n) and s/(n|l|n) are different, despite of the fact that these algebras are

isomorphic. This property demonstrates one of the essential differences between the Lie algebras and the Lie

superalgebras. For each simple Lie algebra there exists (up to a transformation from the Weyl group) only

one system of simple roots. This is not the case for the basic Lie superalgebras, where several inequivalent

simple root systems can be in general defined (for more details see Ref. 36, 37, 38). As a result one and the

same irreducible gl(l\2n) module can be described with different signatures. We shall have to take this into

account in the definition of the C—basis.

A. GZ basis31

Let y([m]jv"+i) be a highest weight finite-dimensional irreducible glo(l\N) module (fidirmod) with a

highest weightJV+l

[m]N+1 = [mliN+1,m2lN+i, • • • ,mN+liN+1] = ^ miiN+1e\ (21)

i=i

where

mj<N+1 £ C, j = 1,...,N+ 1, miiN+i -mi+iiN+i e Z+, i = 2 , 3 , . . . , # . (22)

If «JV+I is the highest weight vector in ^([mjjv+i), then enXN+i = mj-jv+i^JV+i-

Consider the chain of subalgebras

g l o ( l \ N ) D g l o ( l \ N - 1 ) D g l o ( l \ N - 2) D ... D g l o ( l \ 2 ) D g l o ( l \ l ) D g l o ( l \ 0 ) = g l o ( l ) . (23)

Then ^([mjjv+i) is said to be essentially typical, if it is completely reducible with respect to any one of the

subalgebras in the chain (23). Each essentially typical module V^fmjjv+i) carries a typical representation13

of the special linear superalgebra s/o(l|n), but the inverse is in general not true.

6

Set

/I ,JV+I = mltN+1 + 1; litN+i = -mitN+1 + i - 1, i = 2, 3, . . .,N + 1.

Proposition I.31 The glo(l\N) module V([m]N+i) is essentially typical if and only if

h,N+i $ [1'2,N+I;IN+I,N+I]-

Let y([m]jv"+i) be an essentially typical glo(l\N) module and let

V{[m]N+1) D V{[m]N) D ^ ( H i v - i ) D . . . D V{[m]k+1) D . . . V{[m]2) D

(24)

(25)

(26)

be aflagof fl/0(l|A:) fidirmods ^ ( ), & = 0, 1, 2, . . . , # , where

k+i

(27)

is the signature of y ( ). In the ordered basis

C l l , e 2 2 , • • • , Ck + l,k + l ( 2 8 )

of the Cartan subalgebra of glo(l\k), niik+i is the eigenvalue of en on the highest weight vector xk+i £

enxk+i = mitk+ixk+i, i = l,...,k + 1. (29)

Since we consider only essentially typical modules and the fidirmods of £f/o(l) are one dimensional, the flag

(26) defines a vector )- It turns out this vector is uniquely defined (up to, certainly, a

multiplicative constant) by the signatures [m]jv+i,

[m]jv+i

[m]N

m =

[m}2

m]jy, • • •, \_rn\2, mn. Therefore one can set

TI2 A^+l • • • Tnjy A^+l ^^A^+l N-\-l

m,2 N • • • mN N

m22

(30)

The vectors (30), corresponding to all possible flags (26), constitute a basis r([m]jv+i) in the glo(l\N)

fidirmod ^([mjjv+i)- This is the GZ basis introduced in Ref. 31 (for the more general case of gl(M/N)).

Proposition 2.31 The GZ basis F([m]jv"+i) in the essentially typical module ^([mjjv+i) is given by all

tables (30) for which

1. the numbers niiN+i, i = 1,2, . . .N + 1 are fixed for all tables and satisfy (22), (24), (25).

2. m i , - -mi , , - _ i = (?,-_i £ { 0 , 1 } , i = 2 , 3 , . . . , # + l, (31)

3. rriij+i — niij E Z + ; mtj — mi+i j+i e Z + , 2 < i < j < N. (32)

The transformations of the basis r([m]jv+i) under glo(l\N) are completely defined from the action of the

Chevalley generators

eu\m) = (> mki - ) mki-i)\m), i = 1,2, . . ., N + 1,

e12\m) = e2i|m) = (1 -

(33)

(34)

r i — 1

Eri + l

8 - 1 ri + 1

1/2

(35)

ra)- (i,0

m)_iji), i = Z , . . . , . (36)

where l\j = my + 1; /8-j = —rriij + i — 1, i 7 1 anc? the table \m)±^j-j is obtained from the table \m) by

the replacement rriij —> rriij ± 1.

If a vector from the r.h.s. of (35) or (36) does not belong to the module under consideration, then

the corresponding term is zero even if the coefficient in front is undefined; if an equal number of factors in

numerator and denominator are simultaneously equal to zero, they should be canceled out.

The glo(l\N) highest weight vector ;EJV+I in ^([ra]jv+i) is a vector from the GZ basis

= \rn), for which ma = * = 1,2, . . ., N,

i.e.,

fn) =

rnN+i,N+i

In this case

(37)

(38)

= mitN+1\rh), i = 1,2, ...,N+ 1, e f e ) f e + i |m) = O, k = 1, 2, . . ., N. (39)

B. C-basis

Let Eij, i, j = —n, —n + l , . . . , n b e the genera tors of gl(n\l\n). Define a sequence of suba lgebras

g l ( k \ l \ k - 1 + 6) = l i n . e n v . { E i j \ i , j e [ - k ; k - 1 + 6]} V 0 G { 0 , 1 } , k e [ l - 6 ; n ] . (40)

As an ordered basis in the C a r t a n suba lgebra of gl(k\l\k — 1 + 6) t ake

E-k,-k, E-k+i,-k+i, • • •, Ek-i+e,k-i+e- (41)

Proposition 3. The map <p, which is a linear extension of the relations

=eg(i),g(j), i,j = -n,-n+l,...,n, (42)

is an isomorphism of gl(n\l\n) on glo(l\2n). Its restriction on gl(k\l\k —1 + 6) is an isomorphism of

gl{k\l\k- 1 + 6) on glo(l\2k- 1 + 6) for each 6 £ { 0 , 1 } and k £ [l-6;n\. The chain of subalgebras

gl{n\l\n) D gl(n\l\n - 1) D gl(n - l | l | n - l ) D gl(n - l | l | n - 2 ) D . . . D gl(l\l\l) D gl(l\l) D gl(l), (43)

( # / ( l | l | 0 ) = fif/(l|l), fif/(0|l|0) = gl(l)) is transformed by ip into the chain (23)

glo{l\2n) D glo{l\2n - 1) D glo{l\2n - 2) D . . . D gh{l\2) D gh{l\l) D glo{l). (44)

The proof is straightforward.

The isomorphism <p allows one to turn any glo(l\2k —1 + 6) irreducible module Vr([m]2fe+e) into a

gl{k\l\k- 1 + 6) module:

, Mx e V([m\2k+e)- (45)

The relevant for us point is that each V {[m]2k+e) can be labeled also with its highest weight with respect

to gl(k\l\k — 1 + 6). By definition it consists of the eigenvalues of the representatives of the Cartan generators

(41), namely

f{E-k,-k), f{E-k+i,-k+i), • • •, f{E-2,-2), fiE-1,-1), (p(Eoo), ip(Eii),..., (p(Ek-i+e,k-i+e) (46)

onthe gl(k\l\k — l + 6) highest weight vector i)2k+9 £ V([m]2k+0)- The latter is defined from the requirements

= 0, i<j = -k,-k + l,...,k-l + 6, (47)

f(Eii)y2k+e = Mit2k+ey2k+e, i = -k,-k + 1 , . . .,k - 1 + 6. (48)

Set

= [M_k,2k + 8, M _ k + l,2k + 8 , • • • , M k - 1 + 8,2k + e]- (49)

The new signature [Af]2fc+# defines, as mentioned above, uniquely V([m]2k+e)- Hence

V{[m]2k+e) = V([M}2k+e). (50)

9

Consider now a GZ basis vector \m) corresponding to the flag

V([m]2n+1) D V([m]2n) D V([m] 2 n _i ) D . . . D V([m]2fc+(s,) 3 . . .V( [m] 2 ) D T

namely the vector (30) with N = In. In view of (50) the same flag can be written as

V ( [ M ] 2 n + 1 ) D V([M]2n) D V ( [ M ] 2 n _ i ) D . . . D V ( [ M ] 2 k + s ) D . . . D V([M]2) V{MX1

and therefore the vector \m) is completely defined by the signatures [M]2n+i,

we can write any GZ basis vector (30) also in the form

M— n 2 n M— n + l,2n • • • ™ —l,2n ™0,2n ™ l , 2 n

), (51)

(52)

Therefore

\M) =

Mn- i)2n

-\ 2n-l

-<03 M-13

M_ i , 2

(53)

Obviously (30) (with TV = 2n) and (53) are two different labelings for one and the same vector \m) = \M).

We call the basis, written in the notation (53), a C—basis in Vr([M]2n+i) = Vr([m]2n+i) and denote it as

r( [M] 2 n + 1 ) .

In order to use effectively the basis F([M]2n+i) w e need to determine all signatures [M]2fc+#, namely

to find the values of the entries in (53). To this end we have to determine as a first step the highest weight

vector y2k+e within each gl(k\l\k — 1 + 9)— module V([m]2k+e) in the chain (51) and subsequently, using

(48), to compute its gl(k\l\k —1 + 9) signature [M]2k+e-

Proposition 4- The gl(k\l\k —1 + 9) highest weight vector y2k+e in V([m]2k+e) (from the chain (51)) is

the GZ vector \m)2k+e> for which

mh2r+T + k - r = mh2k+e, V r £ {0, 1}, r £ [1 - r ; k - r ] ; (54)

m r _ i i 2 f e -2 j+r = mrt2k+e, V r e [3 - 9; k + 1], r e {0 ,1} , j e [1 - 9; r - 2]; (55a)

m r _ i i 2 f e -2 j+r = mrt2k+e, VrG[Ar + 2;2Ar], r e {0 ,1} , j e [1 - 9; 2k - r + r ] . (556)

Proof: It is easy to verify that the conditions (54) are equivalent to

0 2 i - i = 1, ie[l;k], (56a)

6»2i = 0 , i £ [ l ; k - 1 + 9], (566)

whereas the conditions (55) can be replaced by

h,2i+i-h,2i = 0, i £ [ l ; f c - l + 6»], s£ [2 ;2 i ] , (57a)

I H - I , 2 . - - J . , 2 , - - I - 1 = 0, i e [2; A], s £ [2; 2* — 1]. (576)

10

We need to show that (47) holds for y2k+8 = \rn)2k+6- It certainly suffices to verify it only for the

gl(k\l\k —1 + 9) simple root vectors, namely to prove that

ip(E_i_i+l)\m)2k+0 = 0, ie [l;k],

<p(Eiii+1)\m)2k+9 = 0, i e [0; k - 2 + 9].

(58)

(59)

The validity of the latter follows from the observation that (p(E-io) = e2i, cp(Eoi) = [ei2,e23],

ip(E-it-i+1) = [e2it2i-i,e2i-i,2i-2\, i G [2; A;], ip(Ei-lti) = [e2i-it2i,e2i,2i+i\, i G [ 2 ; A r - 1 + 0] a n d E q s .

(34)-(36). This completes the proof.

We are now ready to determine the g/(A:|l|A: — 1 + 0) signature of V{[m]2k+e) for any 0 £ {0, 1} and

k G [l',n]. Taking into account (54), (55) and (45) and using the transformation relation (33), one obtains

Lp{Eii)\m)2k+e = e2\i\t2\i\\m)2k+e = {mi+k+2,2k+e + l)|m)2A;+e, i £ [-A;; - 1 ] , (60a)

ip(EOo)\m)2k+e = eii\m)2k+e = {mi,2k+e - k)\m)2k+e, (606)

Lp{Eii)\m)2k+e = e2i+i,2i+i\m)2k+e = rrii+k+i,2k+e\m)2k+e, i G [1;A;- 1 + 0], (60c)

Comparing (60) with the definition (48) we obtain the g/(A:|l|A: —1 + 0) signature [M]2fe+e of V([m]2k+e)'-

Mit2k+e = nii+k+2,2k+e + 1, i £ [-k; - 1 ] ,

M0,2k+e = rnit2k+e - k,

Mit2k+e = mi+k+li2k+e, i e[l,k-l + 9],

MOi =

(61a)

(616)

(61c)

(61a7)

We have added the evident relation (61d) for completeness, since it is not contained in (61a-c). The above

relations hold for any 0 G {0, 1} and k G [1; n]. In particular,

Mit2n+i = mi+n+2t2n+i + 1, « G [-n; - 1 ] ,

Mot2n + 1 = rril,2n + l — n,

(62a)

(626)

(62c)

The gl(n\l\n) highest weight vector j/2n+i = \M) is the one from (53), for which M8J- = Mj^n+i f° r anY

admissible i and j :

\M) =

11

(63)

From (31) and (32) one derives the "in-betweenness conditions", which define completely the new basis

(53). The transformations of the C—basis are most easily written in terms of the following variables:

(64)

We formulate the result as a proposition.

Proposition 5. The 2n + 1 —tuple \M~\2n+i = [^_n,2n+i, ^_n+i,2n+i, • • •, Mn2n+i\ is a signature of an

essentially typical gl(n\\\n) module V{\M~\2n+i) if and only if

Mit2n+i G C, ie [-n;n], (65a)

M , > t i - M i + W i E Z + , i e [ - n ; - 2 ] U [ l ; n - l ] , (656)

M _ i i 2 n + i - M i i 2 n + i £ N , (65c)

Mo,2n + 1 = Lot2n + 1 *$. [L-n,2n + l', - n,2n + l] •

The C-basis F([Af]2 n+i) in V([M]2n+i) consists of all tables (53) for which the labels

Mit2k+e, # e { 0 , l } , ke[l-9;n], i e[-k;k - 1 + 9], (66)

take all possible values consistent with the "m-betweenness conditions"

Mit2k+i-Mit2keZ+, Are[l;n], i £ [-k; -1] U [1; k - 1], (67a)

M , - i 2 f c - i - M , - i 2 f c e Z + , A r e [ 2 ; n ] , i £ [-k + 1; - 1 ] U [1; k - 1], (676)

M , - _ i , 2 f c - M , - i 2 f c - i e Z + , A r e [ 2 ; n ] , i £ [ -* + 1; - 1 ] U [2; k - 1], (67c)

M i - i i 2 f c - M , i 2 H i e Z + , £ e [ l ; n ] , i £ [-k + 1; - 1 ] U [2; A:], (67a7)

M _ i , 2 f c - M i i 2 f c - i e N , A r e [ 2 ; n ] , (67e)

M _ l i 2 f e - M i i 2 f c + i G N , i £ [ l ; n ] , ( 6 7 / )

M0i2fe+i - M0,2fe = V>2fe e { 0 , 1 } , A r e [ l ; n ] , (67flf)

^ 0 , 2 ^ - ^ 0 , 2 ^ - 1 = fok-i e { 0 , - 1 } , A r e [ l ; n ] , (67A)

The transformations of the C-basis under the action of the inverse images ip~1(en), ip~1(e{{+i) and

i)c~1(ej-_|_i,j-) of the ^f/o(l|2n) Chevalley generators follow from (33)-(36) and (61),(62). The result reads (we

write Eij instead of <p(Eij)):

( \i\ + 6(i)-l \i\-l

J 2 MJ,2\i\+e(i)- J 2 M j , 2 \ i \ + 6 ( i ) - i \ \ M ) , i e [ - n ; n ] , (68)j=-\i\ j=-\i\+i-e(i) )

, (69)

12

(70)

E

l)(-Lo,2s-2 —

2s-l — ij,2j-l)(-^fe,2i-l — Lj2i-1 + 1)

J i - i ) , » e [ 2 ; n ] , (71)

8 - 1

{Lo 2i — Lj 2i){Lo 2i — Lj 2i + 1) , . , , . M , , _ „ .x 7 \7~^ r l ™ / ( i , 2 j ) ; * £ [1 ; MJ, (7/)

fe£0 i + l(^02i + l - Lk2i-l) Yl'ki0 i(L0,2i + l ~ Lk,2i + l)

} ^ lM)-(o>2O- ife,2i -

8 — ij,2i — l)(ife,2i — Lj2i)

i £ [ l ; n ] , (73)

1

i£[2;n], (74)

We have written the transformation relations of the C—basis under the action of generators, which are differ-

ent from the gl(n\l\n) Chevalley elements. These generators however define completely all other generators.

In this sense Eqs. (68)-(74) are complete. We shall use them in order to derive the transformations of the

17/(001l|oo) irreducible modules under the action of the Chevalley generators.

Remark. We are thankful to the referee for pointing out that Proposition 4 can be proved also

without using the transformation relations (34)-(36). To this end note (see Eqs. (58)-(59)) that the

gl(k\l\k —1 + 9) highest weight vector y2k+e = \m)2k+e £ V{[m]2k+e) is determined from the require-

ment to be annihilated by the generators {<)C(£1_J)_J_|_I)| i 6 [1; k]} U {(p(Eii+i)\ i 6 [0; k — 2 + 6]}, i.e., by

{e2k,2k-2, e2k-2,2k-4, • • •, &A2, &21, ei3, e35, • • •, &2k+20-^,2k+20-1}• The roots, corresponding to the above root

vectors, namely

^k+e = {e2k - e2k~2, e2k~2 - e2k~\ . . ., e4 - e2, e2 - e\ e1 - e3, e3 - e5, . . ., e2k+20-3 - ^ + 2 * - i } ; ( 7 5 )

c a n b e t a k e n a s a n e w s y s t e m of s i m p l e r o o t s of gl(l\2k + 6—1) w i t h a s y s t e m of p o s i t i v e r o o t s

13

Let k'zk+e = [m]'2k+e = J2"i=i mi,2k+e^' be the standard signature (= the highest weight) of V([m\2k+e),

namely the signature corresponding to the choice of simple roots

n2k+e = - e2, e2 - e3, . . ., - e2k+e}. (76)

Denote by A2f+e the corresponding to it system of positive roots. The problem is to determine the signature

(= the highest weight) A2k+g of Vr([m]2fe+e) with respect to A + . This problem can be solved on the ground

of results from Refs.39'40 Given a subset of positive roots A _ of gl(l\2k + 6 — 1) and a simple root a 6 A^_,

one constructs a new system of positive roots A'_[ by a simple a reflection (a):39 '40

ra(A'+), if a is even;

(A'+U{-a})\{a}, if a is odd,(77)

where ra is an element from the Weyl group of gl(l\2k + 9—1), corresponding to a.

If V2k+e is an essentially typical gl(l\2k + 9—1) module with a highest weight A', corresponding to A'+ ,

then the highest weight with respect to A^. is

X" = ra(X') if a is an even root and X" = X'— a if a is an odd root.

L e t Yli=i(ai) = (« i ) («2) • • .{««>• Then

(78)

k 2 s - 1i 2k + 9 _A + —

From (77)-(79) one derives that

k+lA2k+S = (mh2k+s -

3=2

2k+6

j=k+2

i.e.,

{mit2k+e + l)\m)2k+e,

- k)\m)2k+0,

e2i-2k-i,2i-2k-i\m)2k+e = nii,2k+e\m)2k+e, i & 2;2k + 0].

(80)

(81a)

(816)

(81c)

Eqs. (81) are the same as (60) (written in somewhat different notation). Hence one obtains the gl(k\l\k+9—l)

signature as given in (61) and the corresponding to it highest weight \m)2k+e (Proposition 4)-

III. IRREDUCIBLE REPRESENTATIONS OF gl(l\co)

Here we construct representations of glo(l\co) and g/(oo|l|oo), which appear as a generalization to the

case n —> oo of the results obtained in the previous section. In both cases the representations (or the

corresponding modules) are labeled with infinite sequences of (in general different) complex numbers. Due

to the isomorphism <p (see (4)) each ^f/o(l|oo) module is also a g/(oo|l|oo) module and vice versa. Therefore

14

we can also say that we describe bellow two classes of representations of the "abstract" Lie superalgebra

gl(l\oo). For definiteness we refer to the class of representations of glo(l\oo) as to Gel'fand-Zetlin (GZ)

representations (Sect. III.A), whereas the representations of 17/(00|l|oo) are said to be C-representations.

A. Gel'fand-Zetlin representations

The extension of the results of Sect. II to the case n —> 00 is rather evident. We collect the results in a

proposition.

Proposition 6. To each sequence of complex numbers

[m] = [m1,m2, • • .,mk, • •] = C,i G N}, (82)

such that

where

mi - mi+i £ Z + , i = 2 , 3 , . . . ,

l 1 = m i + 1 ; U = - m i + i - l , i = 2 , 3 , . . . ,

(83)

(84)

there corresponds an irreducible highest weight glo(l\oo) module ^([m]) with a signature (82). The basis

F([m]) in V(\_rn\), which we call a GZ basis, consists of all tables

m) =

mi m2

j m2j m

[m\2

characterized by an infinite number of coordinates

m,j, V j e N , i=l,2,...,j,

which are consistent with the conditions:

1. for each table \m) there exists a positive (depending on \m)) integer N[\m)] G N such that

rriij = rrii, Vj > N[\m)], i = 1, . . . ,j;

2. mu - mi,,-_i = 6i-i £ {0,1}, i = 2, 3 , . . . ;

3. mij+i — niij E Z + ; m^ - mi+i j+i e Z + , 2 < i < j e N.The transformation of the basis (85) is determined from the action of the Chevalley generators

(85)

(86)

(87)

(88)

(89)

= (^2 mki ~8 - 1

), i G N ,k = l k = l

(90)

15

e12|m) = 0i|m)u, e2i|m) = (1 - (91)

— hi){h,i-l — Iji + 1)

r»+i n

(92)

ei+iti m) = c/j_i(l - m )_i— iki

1/2

(93)hi)

T/«e highest weight vector \m) is the one from (85) for which

rriij = mi, Vj £ N, i £ 1, 2, . . ., j .

Proof: Let

(94)

m =

m

HJV+I

H 2

er(H). (95)

Then

(i)

(ii)

, njjv"+i,JV+i], # = 1, 2, . . ., is said to be the (N + I)"1—signature of \m

m

and \m)low{N+ls> = (96)

are said to be the (N-\-\)th — upper and the (N-\-\)th — lower part of \m), respectively. Consider the subalgebra

g l o ( l \ N ) = {eij\i,j = 1, . . . , N + 1 } C g l 0 ( l \ o o ) . (97)

Observation 1: Let e be a glo(l\N) generator or any polynomial of glo(l\N) generators. Then, for any

m) £ r([m]), e\m) is a linear combination of vectors from F([m]) with one and same (N + l)th— upper part

16

Denote by

T([m]i\i>N+ 1)C T([m])

the set of all vectors (85), that have one and the same [m]8 signatures, for all i > N + 1. Let

(98)

(99)

be the linear span of F([m]8|i > N + 1). From (90)-(93) it follows that T^([m]8|i > N + 1) is invariant with

respect to glo(l\N). To each vector \m) G F([m]8|i > N + 1) put in correspondence its (N -\-1)"1—lower part:

\low(N+l) V er([ i > A T

Let

T([m]N+1) = {/(|m)) | \m) £ F([m]*l i > N + 1)}.

(100)

(101)

T h e n / m a p s bijectively F([m]8)|i > N+l) on F([m]jv+i)- Obviously F([m]jv+i) consists of all GZ tables of an

essentially typical glo(l\N) module with a signature [m]jv+i- Define an action of glo(l\N) on \m) £ r([m]jv+i)

with the relations (33)-(36). Then the linear envelope ^([mjjv+i) of F([m]jv"+i) is an essentially typical

glo(l\N) module with a signature [m]jv+i- After comparing the relations (90)-(93) with (33)-(36) and having

in mind Observation 1 we have:

Observation 2. The subspace y([m]8|i > N + 1) C ^([m]) is an essentially typical finite-dimensional

glo(l\N) module with a signature [m]jv+i and a GZ basis r([m]8|i > N + 1).

Let eij, tki be any two generators from ^f/o(l|oo) and \m) be an arbitrary vector from F([m]). Considereij,eki a s elements from glo(l\N) C <//o(l|00), where N + 1 > max(i,j,h,l). Then \m) is a vector from the

1^) fidirmod y([m]8|i > N + 1) C V([m]) and therefore (Observation 2)

kl)ekieij)\m) = (Sjkeu - (-1 (102)

Therefore the linear space ^([m]) is a ^f/o(l|oo) module.

Consider any two vectors x, y G ^([™]),

x = y = ), K)er(H),

a8 G C, i = 1, . . ., q.

Let

N =

(103)

(104)

According to (87) all vectors \ml), i = 1, . . ., q, have one and the same k — 1 signatures, for every k — 1 >

N. Therefore |m8') E V([m]k_i\k - 1 > N) C ^([m])- H e n c e «,2/ e ^ ( H f e - i l ^ ~ x > ^ ) - T n e s P a c e

^([m]fc-i|^ ~ 1 ^ N) is a ^f/o(l|7V) fidirmod (Observation 2) and, therefore, there exist a polynomial P of

the glo(l\N) generators such that y = Px. Hence ^([m]) is an irreducible glo(l\co) module.

Consider the vector \m) £ F([m]) [see (91)]. From Eqs. (90)-(93) we have

e88|m) = m8|m), Vi £ N,

17

(105)

and

efcifc+i|m) = O, VfceN. (106)

Therefore the irreducible (7/0(1 |oo) module ^([m]) is a highest weight module with a signature

[m\ = [mi,m2,...,mk,...] (107)

and a highest weight vector \m). This completes the proof.

B. C-representations

Most of the preliminary work for constructing the representations of 17/(00|l|oo) was done in Sect. II.B.

It remains to give a precise definition of the C—basis in the infinite-dimensional case and to write down the

transformation of the basis under the action of the Chevalley generators.

Let

[M] = [..., M_ p , . . . , M_i, Mo, Mi, M2 , . . .] = {M,-},-eZ (108)

be a sequence of complex numbers such that

Mi - Mi+1 £ Z+, ie [-00; -2] U [1; 00], (109a)

M_i - Mi £ N, (1096)

Mo + Mi g Z. (109c)

Here and throughout

[—00; a] = {a, a — 1, a — 2, . . ., a — i, . . .} = {a — i}iez+, (HO)

[6; 00] = {6, 6 + 1, 6 + 2 , . . . , 6 + i , . . .} = {6 + i}ieZ+, (111)

A table |M), consisting of infinitely many complex numbers

M,-i2fc-HSi-i, V ^ N J e {0,1}, i = [-* - 9 + 1; A - 1], (112)

will be called a C—table, provided the following conditions hold:

(1) There exists a positive, depending on |M), integer 7V[|M)] such that

Mit2k+e_1 = Mi, \/k> N[\M)], # £ { 0 , 1 } , iE [ 1 - 0 - M - l ] ; (113)

(2) The coordinates Mi^k+e-i, 9 6 {0, 1}, take all possible values

eZ+, Are [ l + 0;oo], i e [ - A r + 0 ; - l ] U [ l ; A r - l ] , (114a)

2(SieZ+ , Are [ l + 0;oo], i e [ - A r + l ; - l ] U [ 2 ; A r - 0 ] , (1146)

8 e N , ke[l + 9;oo], ( 1 1 4 c )

M0 ,2fc+i-(Si - M0,2fc-(Si = i>2k-e G { 0 , 1 - 2 ( 9 } , A e [ l ; o o ] . (114a 7 )

18

Order the complex numbers Mi^k+e-i, k £ N, 9 £ {0, 1}, as in the table below

, Mi-g-k, •••, M-i, Mo, Mi, . . . ,

-1, Mot2k + 0-l, Mi2k + 9-l, •••, Mk-lt2k + 9-l

M13M-i

M-i, 3 ,

,2)

Mo3,

M02

Moi

\M) =

(115)

We are ready now to state our main and final result.

Proposition 7. To each sequence (108) (see also (109)) there corresponds an irreducible highest weight

£f/(oo|l|oo) module V([M]) with a signature [M\. The basis T([M]) in V([M]) consists of all C-tables (115).

The transformations of the basis under the action of the g/(oo|l|oo) Chevalley generators read:

/\k\+e(k)-i \k\-i

Ekk\M)=l Y^ Mit2\k\+9(k)- Yl\ i = -\k\ i = -\k\ + l-6

M), (116)

(117)

E-ho\M) = -MLo,2 - i_i,2)|M)_ (118)

Eoi\M) = -Ml + Wi)\M

+ (1 + ^i) (-(i-1,3 - L-it2)(Li3 - L-it2))

Eio\M) = - ( - 1 ) ^ ( 1 -

1/2 (L02 - L-it2)

(£03 — £-i,2)(£oi — £-i,2)(£oi — £-1,2 + 1)

03

(-(L_i,3 " £-1,2 " I)(£l3 " £-1,2 "

(01)

(120)

- fa>k)(l

1/2

-\M)

k+1

~ Lj,2kj,2k + 2

19

,2fc + 2 , - , x (

l

i . ,x(j,2feG.2&

Are[ l ,oo ] , (121)

E.k+lt.k\M) = -(1

fe-2 T \i,2fe-l ~ Lj,2k-2>

I

(Lo,2k-2 -

l)(L0,2k-3 ~ ij,2fc-2

k-2- Ljt2k-2)(Lit2k-2- Ljt2k-2

, . ,Wj,2fe-2

' J ( 0 . 2 f e " 1

k i

1p2k-2W2k-3

V

fe ~ Lj,2k-1

-l ~ Lj,2k-l)(Lit2k-l ~ Lj>2k-l +1 )

2)I . ,x(0,2fe-

V ^ V ^ p( • n

I/OZ-HI j#=-Hi V

/i-rfe-2 / r r

[U L L

T2fe-i

(Lo,2k-1 — Li2k-l)(Lo,2k-l — Li2k-l + l)(Loy2k-2 ~ ^j,2k-'l) (Lo,2k-2 ~ Lj,2k-2

(L0,2k ~

Ek+ltk\M) = -(-1

0,2fc-l " Ljt2k-2){L0t2k-3 ~

Are[2,oo], (122)

0,2k + 2 - L{,2k ~ 4>2k + l -

. _(0,2A;\M)-(0)2k

20

~ Lj,2k ~ Lj,2k+3)M)-(0,2k

k

E1/2

£ - 1

-(j,2

1/2

,2fe + 2 — Li2k + 2

1/2

(123)

i l-2

,2fe - Li,2k-2) ,2k — Li 2k)

Yli7i0 = -k + l(L0,2k ~ U,2k-

k-2 ( T-rfe-2

} y 1p2k-2W2k-l IjjL0 = -k + l \

ot2k ~ l>i,2k-\

1/2

T&-2^O fc - Lj,2k-2

r f e -1

,2fe-2 — Lj2k-2 — l)(£j,2fc-2 •

i1/2

^0,2A:-l ~ Lj,2k-l) Ylj7LQ = -k + 2 (L0,2k- 1 ~

l \L0,2k-l -

k-1 k-2

PCfc-2 ~ U,2k-l) [ ~ U,2k-l)

1/2

T & - 1,2k-1 — Li2k-l — l)(ij,2fe-l — Li2k-l)

1/2^k-l ~ Ljjk-2 ~

Tk-2k-2 ~ Ljt2k-2 ~

M]-(j,2k-2)M)-(l,2k-l)>

k e [2,00]. (124)

21

The above transformation relations (116)-(124) were derived first for g/(n|l|n) from (68)-(74) and the

supercommutation relations. Therefore they give a representation of g/(n|l|n) for any n. An essential

requirement, when passing to n —> oo, is given with the condition (113). It is straightforward to check that

^([M]) is invariant under the action of the generators. The rest of the proof, which we skip, is rather similar

to that of Proposition 6, although technically it is more involved.

IV. CONCLUDING REMARKS

We have constructed two classes of highest weight irreps of the infinite-dimensional Lie superalgebra

gl(l\oo). It should be noted that the GZ representations are inequivalent to the C-representations. More than

that: the C-representations, being highest weight irreps of 17/(00|l|oo), are not highest weight representations

of <7/o(l|oo) and vice versa. Indeed, assume that the (7/0(1 |oo) module ^([m]) is also a highest weight

(7/(001l|oo) module with a highest weight vector y. Then y has to be a highest weight vector of any of the

subalgebras gl(k\l\k -1 + 9). Hence Eqs. (54) and (55) have to hold for any 9 = 0, 1, k G [1 - 0;oo].

Therefore y£V([m\) (see (87)).

Our primary interest in the present investigation is related to its eventual applications in a generalization

of the statistics in quantum field theory. From this point of view our results are however very preliminary.

The first observation in this respect is that the algebra (for definiteness) 17/(00|l|oo) is not large enough. It

does not contain important physical observables (like the energy-momentum of the field Pm, see (8)), which

are infinite linear combinations of the generators of (7/(00|l|oo). In order to incorporate them one has to go

to the completed central extension a(oo|l|oo) of (7/(00|l|oo) in a way similar as for the Lie algebra (7/0041 or

the Lie superalgebra gloo\oo-'2° This is only the first step. The next one will be to determine those 17/(00|l|oo)

modules ^([Af]), which can be extended to a(oo|l|oo) modules.

The most important and perhaps the most difficult step will be to express the transformations of the

17/(001l|oo) modules in terms of natural for the QFT variables, namely via the creation and the annihilation

operators af of (7/(00|l|oo), which are just its odd generators.4 This is however not simple and, may be, even

not necessary in the general context of the representation theory. The physical state spaces, the Fock spaces,

have to satisfy several additional physical requirements.42 In particular any such space has to be generated

from the vacuum (the highest weight vector) by polynomials of the creation operators, which are only a part

of the negative root vectors. This imposes considerable restriction on the physically admissible modules.

Hence in the applications one has to select first the Fock spaces from all (7/(00|l|oo) modules and then study

their transformation properties under the action of the physically relevant operators, in particular of the

CAOs.

An additional problem is related to the circumstance that in QFT the indices of the CAOs are not

elements form a countable set. Therefore as a test model one can try to consider first the (7/(00|l|oo)

statistics in the frame of a lattice quantum filed theory or locking the field in a finite volume.

22

ACKNOWLEDGMENTS

N.I.S. is grateful to Prof. M.D. Gould for the invitation to work in his group at the Department of

Mathematics in University of Queensland. T.D.P. is thankful to Prof. S. Okubo for the kind invitation to

conduct a research under the Fulbright Program in the Department of Physics and Astronomy, University

of Rochester. We wish to thank Prof. Randjbar-Daemi for the kind hospitality at the High Energy Section

of ICTP.

This work was supported by the Australian Research Council, by the Fulbright Program of U.S.A.,

Grant No 21857, and by the Contract $ — 416 of the Bulgarian Foundation for Scientific Research.

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