A New Family Of Algebras Underlying The Rogers--ramanujan Identities And Generalizations

A New Family of Algebras Underlying the Rogers--Ramanujan Identities and Generalizations James Lepowsky, and Robert Lee Wilson PNAS 1981;78;7254-7258 doi:10.1073/pnas.78.12.7254 This information is current as of March 2007. This article has been cited by other articles: www.pnas.org#otherarticles E-mail Alerts

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Proc. NatL Acad. Sci. USA

Vol. 78, No. 12, pp. 7254-7258, December 1981 Mathematics

A new family of algebras underlying the Rogers-Ramanujan identities and generalizations (Euclidean Kac-Moody Lie algebras/standard modules/principal Heisenberg subalgebras/vacuum spaces)

JAMES LEPOWSKYt AND ROBERT LEE WILSONt School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Communicated by G. D. Mostow, September 8, 1981

ABSTRACT The classical Rogers-Ramanujan identities have been interpreted by Lepowsky-Milne and the present authors in terms of the representation theory of the Euclidean Kac-Moody Lie algebra A('). Also, the present authors have introduced certain "vertex" differential operators providing a construction of A(') on its basic module, and Kac, Kazhdan, and we have generalized this construction to a general class of Euclidean Lie algebras. Starting from this viewpoint, we now introduce certain new algebras 2v which centralize the action ofthe principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra g on a highest weight gmodule V. We state a general (tautological) Rogers-Ramanujantype identity, which by our earlier theorem includes the classical identities, and we show that 2v can be used to reformulate the general identity. For g = A('), we develop the representation theory of 2v in considerable detail, allowing us to prove our earlier conjecture that our general Rogers-Ramanujan-type identity includes certain identities of Gordon, Andrews, and Bressoud. In the process, we construct explicit bases of all of the standard and Verma modules of nonzero level for A('), with an explicit realization of A() as operators in each case. The differential operator constructions mentioned above correspond to the trivial case 2v = (1) of the present theory.

1v commutes with the action of B and, hence, preserves [1 and

acts richly enough on it essentially to "untwist" the action of g to the tensor product of two commuting actions. A striking feature of 1v is that, in its action on V, it satisfies identities that are themselves the "generating functions" of infinite systems of identities. In this paper, these identities are presented for g = A(') We obtain a new proof of the main result (Theorem 1) of ref. 6, interpreting the classical Rogers-Ramanujan identities as the

formula

Z X41[n/A[n-11) for the level 3 standard A(') modules, where f[nj designates the I-filtration of fQ (6). The content of the Conjecture of ref. 6, =01)

n-0

whose proof is presented here, is that this same formula, for general standard A(')-modules, coincides with the generalized Rogers-Ramanujan identities of Gordon, Andrews, and Bressoud. However, we do not have an independent proof of these identities. The first section of this study is devoted to the definition and general properties of the algebras 2v, in the setting of Euclidean Lie algebras as discussed in ref. 2. We also note a slight simplification of the proof of the main result of ref. 2, from the present viewpoint. The second section contains the deeper analysis of the case g = A('). The details will appear elsewhere.

In this paper, we launch a program to give explicit constructions of general standard modules of general Euclidean Lie algebras and, hence, to produce a wide variety of new realizations of these Lie algebras as algebras of operators. The first construction (1) of a Euclidean Lie algebra, namely A('), by differential operators on a "Fock space" and its sequel (2) for a general class of Euclidean Lie algebras turn out to be the "trivial" cases of the present theory, in a sense to be made precise below. The discovery of the ideas presented here was motivated by a desire to understand more deeply the Lie theoretic significance of the Rogers-Ramanujan identities, continuing a program begun in refs. 3-7. In this paper, by introducing new algebras 2v associated with arbitrary Euclidean Lie algebras g and certain g-modules V, we reduce the problems of interpreting Rogers-Ramanujan-type identities and of explicitly constructing the modules V to the representation theory of Zv. For g = A(), we develop the representation theory of 2v deeply enough to prove the Conjecture in ref. 6 relating the ?-filtrations of the standard A(')-modules to the generalized Rogers-Ramanujan identities of Gordon, Andrews, and Bressoud. In the process, we obtain explicit bases of all the standard and Verma modules of nonzero level and an explicit recursive description of the action of Iv and, hence, of A('), in each case. The modules V under consideration may each be viewed as the tensor product of a "Fock space" with a "vacuum space" Ql, for the principal Heisenberg subalgebra ? of g. Our algebra The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

THE ALGEBRAS £v We shall introduce algebras 2v in the generality of Euclidean Lie algebras. We shall generally use the notation ofref. 2, whose results will be summarized below. Let A = (ay).=0 be a Euclidean generalized Cartan matrix (8, 9)-i.e., one listed in tables 1, 2, or 3 of ref. 2. Let the corresponding Euclidean Kac-Moody Lie algebra g = g(A) over C (which we say is oftype 1, 2, or 3, respectively) have canonical generators ei, fi, hi,, i = 0, ---, N. Define a Z-gradation of g, called the principal gradation, by the conditions

deg ei = -degfi = 1, deg hi = 0, i = 0,', N. Let z (normalized as in ref. 2) span the center of , and let lr:g g/Cz denote the canonical map. Set N

e

ei, and

=

=

T-' (X(9y7(e))

i=O

address: Department of Mathematics, Rutgers University, New Brunswick, NJ 08903.

t Present

7254

Mathematics:

where the superscript denotes centralizer. As in ref. 2, denote by bl, b2, *-- the sequence

m(e) + dh('), j = 1, *--, r; d = 0, 1,2, *arranged in nondecreasing order; here e is the type, and the positive integers hWe) and m -, m(t) are the (generalized) Coxeter numbers and exponents listed in table Eo of ref. 2. (For e = 1, these are the classical Coxeter number and exponents of the underlying finite-dimensional simple Lie algebra; for e > 1, see ref. 10.) PROPOSMION 1 (1, 2). The algebra B is a (principally) graded Heisenberg subalgebra of g with basis {z, pi, qi i = 1,2, }, where

[pipj]

=

0=

Proc. NatL Acad. Sci. USA 78 (1981)

Lepowsky and Wilson

[qiqj], [pi,qj] = 6ijz for a ij = 1,2,

and deg z = 0, deg pi = bi = -deg qj for all i = 1,2, We call B the principal Heisenberg subalgebra of g. The principal gradation of g induces in the natural way the gradation U(g) = 3 U, of the universal enveloping algebra of g. For k E C, we say that a g-module has level k if z acts as the scalar k on it. Denote by wk the category of level k g-modules V with a nonpositive integer grading V = V. such that each Vj is finite-dimensional and Uj * Vj 5 Vi+j for alij E Z (taking Vj = (0) forj > 0). A highest weight vector in a g-module is a nonzero vector annihilated by eo, ., eN and whose span is preserved by ho, -, hN. Let V be a graded highest weight g-module, i.e., a gmodule generated by a highest weight vector, say vo, such that the sum Ej
x(W) = >

jpo

(dim W-j)qJ

in the indeterminate q. Let Wbe a graded subspace ofa module V in %k-i.e., W = ESO W_, where Wj = wnvJ. If Y is a graded subspace of W, then W/Y has an obvious gradation. In

particular, y(W) and A(W/Y) are defined. Characters associated with the principal gradation ofa graded highest weight module are called principally specialized characters (cf. ref. 4). Denote by B+ the subalgebra @D>j, Bj of B and by p the subalgebra Cz (3 B+. For k 8E C*, let Ck, denote the one-dimensional p module on which z acts as k and ?+ acts as 0. The induced ? module Mk = U(0) ®u(P) Ck is irreducible. For V E Ck, denote by fQv (or by fQ, if there is no confusion) its vacuum

7255

(u E U(B), w E flv) is a well-defined b-module map. Note that the A-module U(?) ®U(t) flv is semisimple and, in fact, is a direct sum of copies of Mk. PROPOSION 2 (cf. refs. 6 and 7). Suppose that k E C*. Then the map f is an b-module isomorphism. In particular,

x(V) = F * x(flv), where F = H| (1

-

qbJ)

j21

The principally specialized characters y(V) of the standard modules V have known product expansions given by the "numerator formula" (4, 10, 12). Thus, the characters xf() for these modules have known product expansions. For the case g = A('), see refs. 3 and 4 (cf. ref. 6, Proposition 3). For Verma modules, we easily obtain: PROPOSMON 3. If V is a Verma module (i.e., a universal highest weight module), then

AXfi) = f| (1I qj21

Let V be a graded highest weight g-module with highest weight vector vo. The &-filtration of V, 0 = V_1] 5 V[o]C V[1l c- C V, is defined as follows (6): For all n 2 0, V[n( is the span of all the expressions x, *- x; v, j 2 0, where each xe E g and at most n of the xe lie outside ?. Each V[n] is clearly graded. For n 2 -1, set fl[n = Q f V[n1. Then each f[n] is graded, and [1]] C ROl []CCfl is a filtration of fQ such that fQ = Un £l[n]. Hence (6): PROPOSMON 4. We have =

A0) =

> n-0

Ai[nl/fi[n-1l)-

[1]

Eq. 1, which is a tautology as it stands, includes the classical Rogers-Ramanujan identities, as the cases of the level 3 standard A(')-modules, by the detailed study of the sum side for these cases in refs. 6 and 7. The primary goal of the present paper is to introduce new algebras that implement the sum side of Eq. 1 in great generality. The completion C(g) of g is defined to be the vector space HJEEZ gj. Note that g acts on its completion. PROPOSmON 5 (1, 2). The completion C(g) contains elements

X(M) = jez

In -_) 1, *) N

(subscript denoting homogeneous component) such that g has basis {z, pi, qi, Xjm)

space

> 0; m = 1,

,

N;j E Z}

(see Proposition 1) and such thatfor certain scalars Qv =

{v 8 V

V =

which is graded. The map

fi U(M) ®U(P)) QV u

0 w k-+ u * w

0,

Ami, vmi (m

=

1, -, N; i > 0),

[pi,X(m)] =-AmiX(m) and [qi, X(m)] = vmAX(m) V

in C(g). Let k E C and V E %k, and define the completion C(V) of V to be the vector space 1j,5O Vj. Then g acts on the completion

7256

Mathematics: Lepowsky and Wilson

of V, and every element of C(g) may be viewed as a linear operator from V to C(V). Assume that k E C*. For m = 1, , N, define the following linear operators (cf. ref. 6):

Ez E(m) = exp I umiqi/km

End C(V)

i>O

E(+m) = exp E Vmip/k)

E End V,

i>O

where exp denotes the formal exponential series, and Z(m) = E(m)X(m)E+) 8 Hom(V,C(V)). For each j E Z, let

z7m) E End V be the homogeneous component of degree j (in the obvious sense) of Z(m), so that Z(m) = jzEZZ(m) Definition: Let k E C*, and let V E wk. Denote by 2v or 2 the subalgebra of End V generated by {7J) j E Z, m = 1, *E, N} Note that the associations V Z~m), V -+ v, etc., have obvious functorial properties. Propositions 1 and 5 and elementary properties of exponentials readily imply: PROPOSITION 6. The algebra O centralizes the action ofBon V. In particular, 2 preserves fQ. PROPOSITION 7. For all m = 1, , N, X(m) = (E-))rlZ(m) (E+ ))-' 8 Hom(V, C(V)). By looking at homogeneous components and applying Propositions 2, 6, and 7, we now readily obtain: PROPOSITION 8. The correspondences W ~-+ U(B) *W and Y + Y n n define mutually inverse bijections between the set of all 2-submodules W of 51 and the set of all g-submodules Y of V. In particular, V is g-irreducible if and only if 51 is !-irreducible. Suppose now that V is a graded highest weight module with highest weight vector vo. We define the ff-filtration of fl = f1v

Proc. Nad. Acad. Sci. USA 78 (1981) X(M)

51

=

v O, and the identity 1 for V can be equivalently for-

mulated using the 2-filtration. Now assume that V is irreducible under B. This can occur only if either g has symmetric Cartan matrix or g is of type 2 or 3 and, in these cases, occurs if and only if V is a basic g-module (2), i.e., a standard g-module of level 1. (The equivalence of V as an b-module with the irreducible induced &-module Ml defined above follows from the numerator formula, together with Theorems I and 2 of ref. 13 for g $ A2 and from ref. 10 for g - A2). See ref. 2.) In this case, 51 is clearly one-dimensional, so that for each m and each J # 0, 7m) =- on Q and, hence, on V. Also, Z(m) = cm on 51 and, hence, on V for some cm GC. Thus, Z(m) = cm, and we obtain the main result of ref. 2 by a slightly shorter argument: PROPOSMION 10 (1, 2). If V is a basic g-module, then there are (nonzero) scalars c1, *-@, CN such that

cm(Em)) (Em))

for each m.

As in refs. 1 and 2, we now may view B and the X(m), and hence g, as differential operators.

THE STANDARD MODULES FOR A(') Now we specialize to the case g = A('). We obtain explicit bases of all the standard modules and Verma modules of nonzero level, and an explicit description of the action ofA(1) in each case. As noted above, the original construction of Al') on its basic module through differential operators (1) amounts to the trivial one-dimensional case dim fv = 1, dim Lv = 1, of the present

theory.

Because N = 1, we write Z for Z(m). THEOREM 1. Let k E C* and V E8 Ck For an indeterminate, A, set

Z(C)= >

,

jez

aformal Laurent series in C with coefficients in End V. Let {1, C2 be two commuting indeterminates. We have

(1 C1/)2 k(1 + C/f2)lkZ(lIZ(.2) -

-(1 -

2/1)2/k(1 + 2/1)-/kZ(C2)Z(C1)

=

k>

j(Y-/;2)j

[2]

jEz

The coefficients of Z(C1)Z(;2) and Z('2)Z(C1) on the left-hand side of Eq. 2 are to be understood as formal power series in C/ C2 and C2/hC, respectively. The right-hand side is a formal Laurent series in C1/C2. Eq. 2 is to be interpreted as the "generating function" for the infinite system of identities obtained by equating the coefficients of all the monomials a[ a (rs E Z) on the two sides. Each such identity involves formal infinite sums of endomorphisms of V, and each such sum acts as a well-defined endomorphism of V in view of the fact that the grading of V is truncated from above. Theorem 1 may thus be reformulated as follows: THEOREM 2. Define the numbers 1 = ao,aj,a2, *X by the expansion

° = a[-,] C d°] C fl1l' c ... C ai

by the condition that for all n . 0, f[n] is the span of all the elements xl xi vo, 0 ' i ' n, where each xe is one of the Z*m) (j E Z, m = 1, ', N). We have: PROPOSITION 9. The 2-filtration of 51 coincides with the b-filtration of51, i.e., fl[nl = l[n] for all n 2 0. In particular,

=

(1

-

{)2/k(j

+

{)-2/k

=

Ea j-O

(aan indeterminate). Then for all e, m 8 Z with e # 0,

2 aj(Zm-jZe-m+j Ze-m-jzm+j) = 0

[3]

E a, (Zm-jZ-m+j -Z-m-jzm+j) = (-l)mkm.

[4]

j'O

and j0o

Remark: It is easy to see that for j 2 1, a1 = -(4/k)2F1(l + (2/k), 1 - j;2;2), where 2F1 is the hypergeometric function. Proposition 3 asserts that if V is a Verma module, then

X(5v) = > jpo

pjqi

where p is the classical partition function. Using this and Theorem I or 2, we can prove:

Mathematics:

Proc. Natl. Acad. Sci. USA 78 (1981)

Lepowsky and Wilson

THEOREM 3. Let V be a Verma module of level k E C* and highest weight vector vo. Then f1v has basis fzilZ., Zin vol n 2 0,. ii i2 * in < °}, with the identities 2, or equivalently 3 and 4, giving effective recursive definitions of the action of the generators Zj of t on Qv with respect to this basis. Remark: Combining Theorem 3 with Propositions 2 and 7, we observe that we have obtained a new explicit construction of A(') for each Verma module of nonzero level. Remark: For Vas in Theorem 3, Eq. 1 coincides with the wellknown identity fj (I qj)-1 ja1

q n/(l

-

q) (I

-

n-0

t-I

t-1

Q(v) lot+ E qjVp

Hl (v- k+ 4e).

=

=

C=o

j=o

Then

I

Zt +

qjot (D{jt}Zl)

j
j-tmod2

E qjat(C{t

+ e

-

l,t}ot_(D{j,t 1}z7 ))= 0. [5] -

j
Suppose that k is even, and let t-1

q2) (1 qn) ...

rjij = H1 (M-k + 4e),

E

R(v)= v` +

e=o

j
jet mod 2

of Euler. The standard A(')-modules require a much deeper analysis. In what follows, the symbols and formulas are to be interpreted as in the. discussion after Theorem 1. be a sequence of commuting indeterminates. Let , 23), 2, For ij >O, i # j, set [ij] = (1 + WVjx)(' - J)-', ---

t-2

sP = vJ7 (V- k+ 2+ 4f).

S(v)= vt+

S

e=0

j
j-t mod 2

Then zt + (1/2) +

C(j) = [ji] + [ji], D(ij) = [ij]-' + [ji]'.

7257

E j
(rj

+

sj)o-t(D{jt}z)

[6]

(1/2)E E (rj sj)o-t(C{t 2,t}t-2(D{j,t -2}z2-)) = 0 -

-

j
and

For i ' j, define H [i] HI [im], My= 1ca
(zt

eZt-L1)

-

+

E

j
Nj = fo [em], ce<msj

C{ij} = C(i,i + 1)C(i + 1,i + 2) ... CU - 1j, D{ij} = D(i,i + 1)D(i + 1,i + 2) ... DU - 1,j).

Also define

C{Gj} = C{1j}, D{Gj} = D{lj}. Given an expression fl1, , i) involving the first i indeterminates, write

fl 1), Ofi = Orif(g1, 0,) = (1/i!) N

,

{
sfo-i_(Dj 1,t

-

-

1} (zeal

-

eZRAi))

=

0.

[7]

There is a generalized Rogers-Ramanujan identity, due to Gordon (14), Andrews (15-17), or Bressoud (18, .19), whose product side coincides with XIfv) for V the most general standard A(')-module of.positive level (3, 4; cf. ref. 6). (Bressoud's unified proof (18) of these identities does indeed include the case ko = k1 excluded in ref. 6.) Combining these identities with the infinite systems of new identities for which Eqs. 5, 6, and 7 are the generating functions, we can prove the Conjecture of ref. 6: THEOREM 5. In the notation of Theorem 4, let ko= min(A(ho),A(hl)), and let J = 0 if k is even, J = 1 if k is odd. Then for all n 2 0,

IT

where the and set

sum ranges over

the symmetric group of {1,

,

i},

x(

[n]Q[n-1])

=

ECneq ego

where Cnt denotes the number of partitions d, + + dn of e such that 0 < di di+,; di + 2 id,+,,; if di+t_2 di + 1, then di + *i + di+, 2-ko (mod 2-J); and at most ko of the di = 1. In particular, Eq. 1 for V coincides with the generalized Rogers-Ramanujan identity of Gordon, Andrews, or Bressoud. In particular, we have a new proof of the main result (Theorem 1) of ref. 6, our interpretation of the classical RogersRamanujan identities by the ?-filtration of fl for the level 3 standard Ar1-modules. At the same time, we obtain a new explicit construction of A(,') for each standard module V of positive level, via an explicit basis for fQv: THEOREM 6. In the notation of Theorems 4 and 5, let vo be a highest weight vector of V. Then Qv has basis -

0,

THEOREM 4. Let V be a standard A(')-module of level k > with highest weight A. Let E = (-1)A ho) and t = [k/2] + 1. Set

zo

1, and for j

>

= Z) Z, *, vj)

0

=

define

2icoj (MjNj

(where the noncommutative product is Z(;1) =

2

jNj2/k

H

Hz(;i)) ...

Z(;j)), so that also

Z(-v))

Suppose that k is odd, and define the polynomial

{Z-dnZ-dn-

.

Z-di Vo}

7258

Mathematics: Lepowsky and Wilson

where n 2 0 and the di vary as in Theorem 5. The identities 5, 6, 7 give effective recursive definitions of the action of the generators Z ofL on fQv with respect to this basis, and Propositions 2 and 7 give a corresponding explicit construction of A(') Remark: The case k = 2 of the present theory is much simpler than the cases of larger k because for k = 2, M2N2-2/k in Theorem 4 is the constant 2, and our algebra 2 is just an infinitedimensional Clifford algebra. The vacuum space Q is the Fock space for2, and the module Vis realized in this case very simply as the tensor product of a symmetric Fock space, on which e acts through creation and annihilation operators, with an antisymmetric Fock space, on which the generators ofZ act through creation and annihilation operators. Remark: The similarity ofthe polynomials in Theorem 4 with "conical polynomials" (20, 21) is tantalizing. We are indebted to H. Garland for informing us of his and I. Bars' new proof of our original construction (1) of A('). Their proof involved inserting formal variables in a manner which inspired our directions of investigation. We are grateful to D. Bressoud for informing us that the ko = k, identities that we conjectured on page 700 of ref. 6 had already been proved by him (ref. 18). We also thank I. Frenkel and S. Milne for stimulating conversations. Both authors gratefully acknowledge the hospitality of the Institute for Advanced Study during part of the preparation of this work and partial support from the Rutgers University Faculty Academic Study Program and National Science Foundation Grant MCS 80-03000.

Proc. Natl. Acad. Sci. USA 78 (1981)

Lepowsky, J. & Wilson, R. L. (1978) Comm. Math. Phys. 62, Kazhdan, D. A., Lepowsky, J. & Wilson, R. L. (1981) Adv. Math. 42; 83-112. 3. Lepowsky, J. & Milne, S. (1978) Proc. NatL Acad. Sci. USA 75, 1.

43-53. 2. Kac, V. G.,

578-579.

4. Lepowsky, J. & Milne S. (1978) Adv. Math. 29, 15-59. 5. Feingold, A. & Lepowsky, J. (1978) Adv. Math. 29, 271-309. 6. Lepowsky, J. & Wilson, R. L. (1981) Proc. NatL Acad. Sci. USA 78, 699-701. 7. Lepowsky, J. & Wilson, R. L. (1982) Adv. Math., in press. 8. Kac, V. G. (1968) Izv. Akad. Nauk SSSR Ser. Mat. 32, 1323-1367. (English translation: Math. USSR-Izvestija 2, 1271-1311. 9. Moody, R. V. (1968)J. Algebra 10, 211-230. 10. Kac, V. G. (1978) Adv. Math. 30, 85-136. 11. Kac, V. G. (1974) Funkt. AnaL Ego Prilozheniya 8, 77-78 (English translation: Funct. AnaL AppL 8, 68-70. 12. Lepowsky, J. (1980) Adv. Math. 35, 179-194. 13. Lepowsky, J. (1978) Adv. Math. 27, 230-234. 14. Gordon, B. (1961) Am. J. Math. 83, 393-399. 15. Andrews, G. E. (1976) The Theory ofPartitions, Encyclopedia of Mathematics and its Applications, ed. Rota, G.-C. (AddisonWesley, Reading, MA), Vol. 2. 16. Andrews, G. E. (1967), J. Combin. Theory 2, 422-430. 17. Andrews, G. E. (1967) J. Combin. Theory 2, 431-436. 18. Bressoud, D. (1979) J. Combin. Theory 27, 64-68. 19. Bressoud, D. (1980) Mem. Am. Math. Soc. 24, No. 227. 20. Lepowsky, J. (1975) Trans. Am. Math. Soc. 208, 219-272. 21. Lepowsky, J. (1975) Ann. Math. 102, 17-40.