VERTEX-ALGEBRAIC STRUCTURE OF THE PRINCIPAL SUBSPACES OF CERTAIN A (1)

VERTEX-ALGEBRAIC STRUCTURE OF THE PRINCIPAL SUBSPACES OF CERTAIN A (1) 1 -MODULES I: LEVEL ONE CASE arxiv:07041759v1 [mathqa] 13 Apr 2007 C CALINESCU J LEPOWSKY AND A MILAS Abstract This is the first in a series of papers in which we study vertex-algebraic structure of Feigin-Stoyanovsky s principal subspaces associated to stard modules for both untwisted twisted affine Lie algebras A key idea is to prove suitable presentations of principal subspaces without using bases or even small spanning sets of these spaces In this paper -modules These convenient presentations were previously used in work of Capparelli-Lepowsky-Milas for the purpose of obtaining the classical Rogers-Ramanujan recursion for the graded dimensions of the principal subspaces we prove presentations of the principal subspaces of the basic A (1) 1 1 Introduction One of the central problems in combinatorial representation theory is to find (combinatorial) bases of stard modules for affine Kac-Moody Lie algebras Finding a basis of a stard module is closely related to finding a basis of the vacuum subspace for an appropriate Heisenberg algebra by means of vertex operators Z-algebras (see in particular [LW1] [LW4] [LP1] [LP2] [MP1] [MP2]) In [FS1] [FS2] Feigin Stoyanovsky associated to every stard A (1) n -module L(Λ) another distinguished subspace the principal subspace W(Λ) = U( n) v Λ where n is the nilradical of a Borel subalgebra of sl(n+1) n = n C[tt 1 ] Λ v Λ are the highest weight a highest weight vector of L(Λ) Of course this definition extends to an arbitrary highest weight module for an affine Lie algebra Compared to the vacuum subspaces of L(Λ) for Heisenberg Lie algebras the principal subspace W(Λ) is a somewhat simpler object yet interestingly enough it still carries a wealth of information about the stard module (see [FS1] [FS3]) From the definition W(Λ) can be identified with the quotient U( n)/i Λ where I Λ is the kernel of the natural map f Λ : U( n) W(Λ) a a v Λ 2000 Mathematics Subject Classification: Primary 17B69; Secondary 17B65 05A17 CC gratefully acknowledges partial support from the Center for Discrete Mathematics Theoretical Computer Science (DIMACS) Rutgers University JL gratefully acknowledges partial support from NSF grant DMS 0401302 1

2 C CALINESCU J LEPOWSKY AND A MILAS It is an important general problem to find a presentation of W(Λ) (ie to precisely describe I Λ ) to compute the corresponding graded dimension of W(Λ) especially from our point of view to reveal new vertex-algebraic structure underlying such issues A central aim of this paperis to supplyanew proofof thenatural presentation of the principal subspaces W(Λ i ) of the basic A (1) 1 -modules L(Λ i) i = 01 established in [FS1] [FS2] There are a number of proofs all of which have essentially used detailed structure such as bases of the stard modules One wants to know how to prove this result using less such structure For a recent proof in this direction see [Cal] The presentation result was used in [CLM1] as a step in proving the classical Rogers-Ramanujan recursion for the graded dimensions of the spaces W(Λ i ) The proof of the presentation that we provide in this paper does not use any bases of the stard modules or of the principal subspaces or even small spanning sets that turn out to be bases (that is spanning sets whose monomials satisfy the difference-two condition on their indices) so gives new insight into the program of obtaining Rogers-Ramanujan-type recursions initiated in [CLM1] (for level one modules) [CLM2] (for higher level modules) In those papers exact sequences among principal subspaces were built yielding q-difference equations in turn formulas for the graded dimensions of the principal subspaces Vertex (operator) algebras intertwining operators among stard modules were used together with certain translations in the affine Weyl group of sl(2) It turns out that we have been able to use these same ingredients in a new way combined with the viewpoint of generalized Verma modules to prove the presentation result by induction on conformal weight Our main results are described in Theorems 21 22 in Theorem 41 we offer a reformulation of part of Theorem 22 using the notion of ideal of a vertex algebra The ideas arguments presented here certainly generalize as we will show in a series of forthcoming papers [CalLM1] [CalLM3]; the paper [CalLM1] covers the case of the principal subspaces of the higher level stard sl(2)-modules for which the Rogers-Selberg recursions were obtained in [CLM2] 2 Formulations of the main result We use the setting of [CLM1] (cf [FLM] [LL]) We shall work with the complex Lie algebra with the brackets g = sl(2) = Cx α Ch Cx α [hx α ] = 2x α [hx α ] = 2x α [x α x α ] = h Take the Cartan subalgebra h = Ch The stard symmetric invariant bilinear form ab = tr (ab) for ab g allows us to identify h with its dual h Take α to be the (positive) simple root corresponding to the root vector x α Then under our identifications h = α αα = 2 The corresponding untwisted affine Lie algebra is (21) ĝ = g C[tt 1 ] Ck

with the bracket relations VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 3 (22) [a t m b t n ] = [ab] t m+n +m ab δ m+n0 k for ab g mn Z together with the condition that k is a nonzero central element of ĝ Let where d acts as follows: Setting consider the following subalgebras of ĝ: g = ĝ Cd [da t n ] = na t n for a g n Z [dk] = 0 n = Cx α g = g t 0 n = n C[tt 1 ] n = n t 1 C[t 1 ] n 2 = n t 2 C[t 1 ] The Lie algebra ĝ has the triangular decomposition (23) ĝ = (Cx α g t 1 C[t 1 ]) (h Ck) (Cx α g tc[t]) It also has the triangular decomposition (24) ĝ = ĝ <0 ĝ 0 where ĝ <0 = g t 1 C[t 1 ] ĝ 0 = g C[t] Ck We consider the level 1 stard ĝ-modules L(Λ 0 ) L(Λ 1 ) (cf [K]) where Λ 0 Λ 1 (h Ck) are the fundamental weights of ĝ ( Λ i k = 1 Λ i h = δ i1 for i = 01) Denote by v Λi the highest weight vectors of L(Λ i ) used in [CLM1] namely (25) v Λ0 = 1 v Λ1 = e α/2 v Λ0 (See Section 2 of [CLM1]) The principal subspaces W(Λ i ) of L(Λ i ) are defined by (26) W(Λ i ) = U( n) v Λi for i = 01 in [FS1] By the highest weight vector property we have (27) W(Λ i ) = U( n ) v Λi Now we set (28) W(Λ 1 ) = U( n 2 ) v Λ1 Then (29) W(Λ 1 ) = W(Λ 1 )

4 C CALINESCU J LEPOWSKY AND A MILAS because (x α t 1 ) v Λ1 = 0 (recall (22)) For i = 01 consider the surjective maps (210) F Λi : U(ĝ) L(Λ i ) a a v Λi Let us restrict F Λi to U( n ) F Λ1 to U( n 2 ) denote these restrictions by f Λi f Λ 1 : (211) (212) f Λi : U( n ) W(Λ i ) a a v Λi f Λ 1 : U( n 2 ) W(Λ 1 ) a a v Λ1 Our main goal is to describe the kernels Ker f Λ0 Ker f Λ1 Ker f Λ 1 We proceed to do this Throughout the rest of the paper we will write a(m) for the action of a t m on any ĝ-module where a g m Z (recall [CLM1]) In particular we have the operator x α (m) the image of x α t m We consider the following formal infinite sums indexed by t Z: (213) R t = x α (m 1 )x α (m 2 ) m 1 +m 2 = t For each t R t acts naturally on any highest weight ĝ-module in particular on L(Λ i ) i = 12 In order to describe Ker f Λi Ker f Λ 1 we shall truncate each R t as follows: (214) Rt 0 = x α (m 1 )x α (m 2 ) t 2 m 1 m 2 1 m 1 +m 2 = t We shall often be viewing Rt 0 as an element of U( n) in fact of U( n ) rather than as an endomorphism of a module such as L(Λ i ) In general it will be clear from the context when expressions such as (214) are understood as elements of a universal enveloping algebra or as operators It will also be convenient to take m 1 m 2 2 in (213) to obtain other elements of U( n) which we denote by Rt: 1 (215) Rt 1 = x α (m 1 )x α (m 2 ) t 4 m 1 m 2 2 m 1 +m 2 = t One can view U( n ) U( n 2 ) as the polynomial algebras (216) U( n ) = C[x α ( 1)x α ( 2)] (217) U( n 2 ) = C[x α ( 2)x α ( 3)] Then (218) U( n ) = U( n 2 ) U( n )x α ( 1) we have the corresponding projection (219) ρ : U( n ) U( n 2 )

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 5 Notice that (220) Rt 1 = ρ(r0 t ) Set (221) I Λ0 = U( n )Rt 0 U( n ) t 2 (222) I Λ1 = U( n )Rt 0 +U( n )x α ( 1) U( n ) t 2 Observe that (222) can be written as (223) I Λ1 = U( n )Rt 1 +U( n )x α ( 1) t 4 Remark 21 Note that (224) I Λ0 I Λ1 in fact (225) I Λ1 = I Λ0 +U( n )x α ( 1) Also set (226) I Λ 1 = t 4 U( n 2 )R 1 t U( n 2) Observe that (227) ρ(i Λ1 ) = I Λ 1 in fact (228) I Λ1 = I Λ 1 U( n )x α ( 1) It is well known ([B] [FLM]) that there is a natural vertex operator algebra structure on L(Λ 0 ) with a vertex operator map (229) Y( x) : L(Λ 0 ) End L(Λ 0 ) [[xx 1 ]] v Y(vx) = m Zv m x m 1 which satisfies certain conditions with v Λ0 as vacuum vector In particular we have (230) Y(e α x) = m Zx α (m)x m 1 where e α is viewed as an element of L(Λ 0 ) It is also well known that L(Λ 1 ) has a natural L(Λ 0 )-module structure The vector spaces L(Λ i ) are graded with respect to a stard action of the Virasoro algebra operator L(0) usually referred to as grading by conformal weight For m an integer (231) wt x α (m) = m

6 C CALINESCU J LEPOWSKY AND A MILAS where x α (m) is viewed as either an operator or as an element of U( n) For any element λ of the weight lattice of g we have (232) wt e λ = 1 2 λλ (cf [CLM1]) In particular (233) wt v Λ0 = 0 wt v Λ1 = 1 4 (recall (25)) The vector spaces L(Λ i ) have also a grading given by the eigenvalues of the operator 1 2 α(0) = 1 2h(0) called the grading by charge This is compatible with the grading by conformal weight For any m Z x α (m) viewed as either an operator or as an element of U( n) has charge 1 Also e α/2 viewed as either an operator or as an element of L(Λ 1 ) has charge 1 2 We shall consider these gradings restricted to the principal subspaces W(Λ i ) For any m 1 m k Z (234) x α (m 1 ) x α (m k ) v Λ0 W(Λ 0 ) (235) x α (m 1 ) x α (m k ) v Λ1 W(Λ 1 ) have weights m 1 m k m 1 m k + 1 4 respectively Their charges are k k + 1 2 respectively See Section 2 of [CLM1] for further details background notation Remark 22 It is clear that for i = 01 Also R 0 t R 1 t have conformal weight t: L(0) Ker f Λi Ker f Λi L(0) Ker f Λ 1 Ker f Λ 1 L(0)R 0 t = tr0 t for all t 2 L(0)R 1 t = tr1 t for all t 4 In particular the subspaces I Λ0 I Λ1 I Λ 1 are L(0)-stable We also have that R 0 t R1 t have charge 2 the spaces Ker f Λi Ker f Λ 1 I Λi I Λ 1 for i = 01 are graded by charge Hence these spaces are graded by both weight charge the two gradings are compatible We will prove the following result describing the kernels of f Λi f Λ 1 (recall (218) (228)): Theorem 21 We have (236) Ker f Λ0 = I Λ0 also Ker f Λ 1 = I Λ 1 or equivalently (237) Ker f Λ1 = I Λ1

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 7 We will actually prove a restatement of this assertion (see Theorem 22 below) For this reason we need to introduce the generalized Verma modules in the sense of [L1] [GL] [L2] for ĝ as well as what we shall call the principal subspaces of these generalized Verma modules The generalized Verma module N(Λ 0 ) is defined as the induced ĝ-module (238) N(Λ 0 ) = U(ĝ) U(bg 0 ) Cv N Λ 0 where g C[t] acts trivially k acts as the scalar 1 on Cv N Λ 0 ; v N Λ 0 is a highest weight vector From the Poincaré-Birkhoff-Witt theorem we have (239) N(Λ 0 ) = U(ĝ <0 ) C U(ĝ 0 ) U(bg 0 ) Cv N Λ 0 = U(ĝ <0 ) C Cv N Λ 0 = U(ĝ <0 ) with the natural identifications Similarly define the generalized Verma module N(Λ 1 ) = U(ĝ) U(bg 0 ) U where U is a two-dimensional irreducible g-module with a highest-weight vector v N Λ 1 where g tc[t] acts trivially k by 1 By the Poincaré-Birkhoff-Witt theorem we have the identification N(Λ 1 ) = U(ĝ <0 ) C U For i = 01 we have the natural surjective ĝ-module maps (240) (cf (210)) F N Λ i : U(ĝ) N(Λ i ) a a v N Λ i Remark 23 The restriction of (240) to U(ĝ <0 ) is a U(ĝ <0 )-module isomorphism for i = 0 a U(ĝ <0 )-module injection for i = 1 Set (241) W N (Λ i ) = U( n) v N Λ i an n-submodule of N(Λ i ) (cf (26)) We shall call W N (Λ i ) the principal subspace of the generalized Verma module N(Λ i ) By the highest weight vector property we have (242) W N (Λ i ) = U( n ) v N Λ i Also consider the subspace (243) W N (Λ 1 ) = U( n 2 ) v N Λ 1 of W N (Λ 1 ) Remark 24 In view of Remark 23 the maps (244) are n -module isomorphisms the map (245) U( n ) W N (Λ i ) a a v N Λ i U( n 2 ) W N (Λ 1 ) is an n 2 -module isomorphism a a v N Λ 1

8 C CALINESCU J LEPOWSKY AND A MILAS In particular we have the natural identifications (246) W N (Λ 1 ) W N (Λ 1 )/U( n )x α ( 1) v N Λ 1 U( n 2 ) in view of (218) We have the natural surjective ĝ-module maps (247) for a ĝ i = 01 Set Π Λi : N(Λ i ) L(Λ i ) a v N Λ i a v Λi (248) N 1 (Λ i ) = Ker Π Λi The restrictions of Π Λi to W N (Λ i ) (respectively W N (Λ 1 ) ) are n-module (respectively n 2 - module) surjections: (249) π Λi : W N (Λ i ) W(Λ i ) for i = 01 (250) π Λ 1 : W N (Λ 1 ) W(Λ 1 ) (recall (29)) As in the case of L(Λ i ) there are natural actions of the Virasoro algebra operator L(0) on N(Λ i ) for i = 01 giving gradings by conformal weight these spaces are also compatibly graded by charge by means of the operator 1 2 α(0) = 1 2h(0) We shall restrict these gradings to the principal subspaces W N (Λ i ) The elements of W N (Λ i ) given by (234) (235) with v Λi replaced by vλ N i have the same weights charges as in those cases Remark 25 Since the maps π Λi i = 01 π Λ 1 commute with the actions of L(0) the kernels Ker π Λi Ker π Λ 1 are L(0)-stable These maps preserve charge as well so that Ker π Λi Ker π Λ 1 are also graded by charge In view of Remark 23 the following assertion is equivalent to that of Theorem 21: Theorem 22 We have (251) Ker π Λ0 = I Λ0 v N Λ 0 ( N 1 (Λ 0 )) Moreover (252) Ker π Λ 1 = I Λ 1 v N Λ 1 ( N 1 (Λ 1 )) or equivalently (253) Ker π Λ1 = I Λ1 v N Λ 1 ( N 1 (Λ 1 )) 3 Proof of the main result Continuing to use the setting of [CLM1] we have Consider the linear isomorphism V P = L(Λ 0 ) L(Λ 1 ) (31) e α/2 : V P V P

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 9 Its restriction to the principal subspace W(Λ 0 ) of L(Λ 0 ) is (32) e α/2 : W(Λ 0 ) W(Λ 1 ) We have (33) e α/2 x α (m) = x α (m 1) e α/2 on V P for m Z (34) e α/2 v Λ0 = v Λ1 so that (35) e α/2 (x α (m 1 ) x α (m k ) v Λ0 ) = x α (m 1 1) x α (m k 1) v Λ1 for any m 1 m k Z Then in particular (32) is a linear isomorphism We shall follow [CLM1] to construct a lifting (36) e α/2 : W N (Λ 0 ) W N (Λ 1 ) of e α/2 : W(Λ 0 ) W(Λ 1 ) in the sense that the following diagram will commute: W N (Λ 0 ) π Λ0 d e α/2 W N (Λ 1 ) e W(Λ 0 ) α/2 W(Λ 1 ) However here we shall use our current notation which involves generalized Verma modules principal subspaces of these modules For any integers m 1 m k < 0 we set (37) π Λ 1 e α/2 (x α (m 1 ) x α (m k ) v N Λ 0 ) = x α (m 1 1) x α (m k 1) v N Λ 1 which is well defined since U( n ) viewed as the polynomial algebra C[x α ( 1)x α ( 2)] maps isomorphically onto W N (Λ 0 ) under the map (244) Since W N (Λ 1 ) = U( n 2 ) v N Λ 1 = êα/2 (U( n ) v N Λ 0 ) the linear map êα/2 is surjective By Remark 24 it is also injective Denote by (38) (êα/2 ) 1 = ê α/2 : W N (Λ 1 ) W N (Λ 0 ) its inverse Then êα/2 is indeed a lifting of e α/2 as desired ê α/2 is correspondingly a lifting of the inverse (39) e α/2 : W(Λ 1 ) W(Λ 0 ) Lemma 31 We have (310) e α/2 (I Λ0 v N Λ 0 ) = I Λ 1 v N Λ 1

10 C CALINESCU J LEPOWSKY AND A MILAS Proof: For any t 2 e α/2 (Rt 0 vn Λ 0 ) = êα/2 = m 1 m 2 1 m 1 +m 2 = t m 1 m 2 1 m 1 +m 2 = t x α (m 1 )x α (m 2 ) vλ N 0 x α (m 1 1)x α (m 2 1) v N Λ 1 = R 1 t+2 v N Λ 1 In view of the definition of the map êα/2 of the descriptions (221) (226) of the ideals I Λ0 I Λ 1 we have (310) Now we restrict (31) to the principal subspace W(Λ 1 ) obtain (311) e α/2 : W(Λ 1 ) W(Λ 0 ) which is only an injection Since (312) e α/2 v Λ1 = e α v Λ0 = x α ( 1) v Λ0 (recall (34) [CLM1]) we have using (33) (313) e α/2 (x α (m 1 ) x α (m k ) v Λ1 ) = x α (m 1 1) x α (m k 1)x α ( 1) v Λ0 for any m 1 m k Z As above we construct a natural lifting (314) of the map (311) making the diagram commute by taking e α/2 : W N (Λ 1 ) W N (Λ 0 ) W N (Λ 1 ) π Λ 1 W(Λ 1 ) e dα/2 W N (Λ 0 ) π Λ0 e α/2 W(Λ 0 ) (315) e α/2 (x α (m 1 ) x α (m k ) vλ N 1 ) = x α (m 1 1) x α (m k 1)x α ( 1) vλ N 0 for m 1 m k 2 Then like (311) the map (314) is an injection not a surjection Lemma 32 We have (316) e α/2 (I Λ 1 v N Λ 1 ) I Λ0 v N Λ 0 Proof: As in the proof of the previous lemma we calculate that for any integer t 4 e α/2 (R 1 t v N Λ 1 ) = R 0 t+2x α ( 1) v N Λ 0 x α ( t)r 0 3 v N Λ 0 2x α ( t 1)R 0 2 v N Λ 0 which proves our lemma Our next goal is to prove the main result Theorem 21 or equivalently Theorem 22 It is sufficient to prove (251) (252) We notice first that (317) I Λ0 v N Λ 0 Ker π Λ0 I Λ 1 v N Λ 1 Ker π Λ 1

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 11 Indeed as is well known the square of the vertex operator Y(e α x) is well defined (the components x α (m) m Z of this vertex operator commute) equals zero on L(Λ i ) in particular on W(Λ i ) for i = 12 The expansion coefficients of Y(e α x) 2 are the operators R t t Z: (318) Y(e α x) 2 = ( ) x α (m 1 )x α (m 2 ) x t 2 t Z m 1 +m 2 =t (recall (213) (230)) this proves (317) Now we define a shift or translation automorphism (319) τ : U( n) U( n) by τ(x α (m 1 ) x α (m k )) = x α (m 1 1) x α (m k 1) for any integers m 1 m k ; since U( n) C[x α (m) m Z] the map (319) is well defined For any integer s the s th power (320) τ s : U( n) U( n) is given by τ s (x α (m 1 ) x α (m k )) = x α (m 1 s) x α (m k s) Remark 31 Assume that a U( n) is a nonzero element homogeneous with respect to both the weight charge gradings We also assume that a has positive charge so that a is a linear combination of monomials of a fixed positive charge (degree) Then τ s (a) has the same properties (321) wt τ s (a) > wt a for s > 0 (322) wt τ s (a) < wt a for s < 0 by (231) If a is a constant that is a has charge zero then of course (323) τ s (a) = a τ s (a) a have the same weight charge We also have: Remark 32 Using the map τ we can re-express (32) (36) (311) (314) as follows: (324) e α/2 (a v Λ0 ) = τ(a) v Λ1 a U( n) (325) e α/2 (a v N Λ 0 ) = τ(a) v N Λ 1 a U( n ) (326) e α/2 (a v Λ1 ) = τ(a)x α ( 1) v Λ0 a U( n) (327) e α/2 (a v N Λ 1 ) = τ(a)x α ( 1) v N Λ 0 a U( n 2 ) (recall (35) (37) (313) (315)) The corresponding inverse maps of course involve τ 1

12 C CALINESCU J LEPOWSKY AND A MILAS Lemma 33 We have Proof: Let t 2 Then τ(i Λ0 ) I Λ0 +U( n )x α ( 1) = I Λ1 τ(r 0 t) = R 0 t+2 2x α ( t 1)x α ( 1) Recall from [CLM1] the role of intertwining vertex operators among triples of the L(Λ 0 )- modules L(Λ 0 ) L(Λ 1 ) the role of certain terms factors of these intertwining operators We will use the constant term of the intertwining operator Y(e α/2 x) of type ( L(Λ 1 ) L(Λ 1 ) L(Λ 0 ) denote it by Y c (e α/2 x) This is the operator denoted by o(e α/2 ) in Theorem 41 of [CLM1] Then (328) Y c (e α/2 x) : W(Λ 0 ) W(Λ 1 ) it sends v Λ0 to v Λ1 it commutes with the action of n (see the beginning of the proof of Theorem 41 in [CLM1]) Proof of Theorem 22: In view of (317) it is sufficient to prove (329) Ker π Λ0 I Λ0 v N Λ 0 Ker π Λ 1 I Λ 1 v N Λ 1 First we show that (252) follows from (251) whose truth we now assume Let a v N Λ 1 Ker π Λ 1 where a U( n 2 ) so that a v Λ1 = 0 in W(Λ 1 ) By (324) we have τ 1 (a) v Λ0 = 0 in W(Λ 0 ) so τ 1 (a) v N Λ 0 Ker π Λ0 = I Λ0 v N Λ 0 by assumption Thus e α/2 (τ 1 (a) v N Λ 0 ) êα/2 (I Λ0 v N Λ 0 ) by (325) Lemma 31 we obtain a v N Λ 1 I Λ 1 v N Λ 1 Thus we have the inclusion Ker π Λ 1 I Λ 1 v N Λ 1 so we have (252) It remains to prove (330) Ker π Λ0 I Λ0 v N Λ 0 We will prove by contradiction that any element of U( n ) v N Λ 0 that lies in Ker π Λ0 lies in I Λ0 v N Λ 0 as well Suppose then that there exists a U( n ) such that (331) a v N Λ 0 Ker π Λ0 but a v N Λ 0 / I Λ0 v N Λ 0 By Remarks 22 25 we may doassume that a is homogeneous with respect to the weight chargegradings Notethataisnonzeroinfactnonconstant soithaspositiveweight charge We choose such an element a of smallest possible weight that satisfies (331) We claim that there is in fact an element lying in U( n )x α ( 1) in addition having all of the properties of a By (218) we have a unique decomposition a = bx α ( 1)+c with b U( n ) c U( n 2 ) The elements b c are homogeneous with respect to the weight charge gradings In fact (332) wt b = wt a 1 wt c = wt a )

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 13 similarly the charge of b is one less than that of a the charges of c a are equal We have (333) τ 1 (a) = τ 1 (b)x α (0)+τ 1 (c) with (334) τ 1 (c) U( n ) Since a U( n ) a v Λ0 = 0 by applying the linear map Y c (e α/2 x) using its properties (recall (328)) we have (335) a v Λ1 = 0; what we have just observed is that (336) Ker f Λ0 Ker f Λ1 (recall the notation (211)) By (335) (324) we obtain so τ 1 (a) v Λ0 = 0 in W(Λ 0 ) (337) τ 1 (a) v N Λ 0 Ker π Λ0 ; this shows that if u lies in U( n) (not necessarily in U( n )) then (338) u v N Λ 1 Ker π Λ1 τ 1 (u) v N Λ 0 Ker π Λ0 But (333) (337) imply (339) τ 1 (c) v N Λ 0 Ker π Λ0 We now show that (340) τ 1 (c) v N Λ 0 I Λ0 v N Λ 0 If (340) did not hold then since τ 1 (c) U( n ) is doubly homogeneous wt τ 1 (c) < wt c = wt a (recall (322) (332)) we would be contradicting our assumption that a U( n ) is doubly homogeneous of smallest possible weight satisfying (331) Hence (340) holds from (334) Remark 24 we obtain τ 1 (c) I Λ0 Now Lemma 33 applied to τ 1 (c) implies (341) c I Λ0 +U( n )x α ( 1) InviewofRemark22wehavec = d+e whered I Λ0 e U( n )x α ( 1) arehomogeneous with respect to both gradings in fact wt d = wt e = wt c = wt a similarly for charge Now a = (bx α ( 1)+e)+d since d v N Λ 0 I Λ0 v N Λ 0 Ker π Λ0 a v N Λ 0 satisfies (331) we have (342) (bx α ( 1)+e) v N Λ 0 Ker π Λ0 but (bx α ( 1)+e) v N Λ 0 / I Λ0 v N Λ 0 Thus we have proved our claim by means of the doubly homogeneous element bx α ( 1) +e U( n )x α ( 1) which has the same weight as a

14 C CALINESCU J LEPOWSKY AND A MILAS In view of the claim all we have left to do is to show that there cannot exist an element a U( n )x α ( 1) satisfying (331) homogeneous with respect to both the weight charge gradings of smallest possible weight among all the elements of U( n ) satisfying (331) Take such an element a We may in fact assume that a = bx α ( 1) with b U( n 3 ) where n 3 is the Lie subalgebra of n defined in the obvious way Indeed we have the decomposition U( n ) = U( n 3 ) (U( n )x α ( 2)+U( n )x α ( 1)) into doubly-graded subspaces both x α ( 1) 2 v N Λ 0 x α ( 2)x α ( 1) v N Λ 0 are in I Λ0 v N Λ 0 Note that since b U( n 3 ) (343) τ 1 (b) U( n 2 ) τ 2 (b) U( n ) We also note that (344) wt b = wt a 1 that the charge of b is one less than that of a Since a v N Λ 0 = bx α ( 1) v N Λ 0 Ker π Λ0 we have bx α ( 1) v Λ0 = 0 in W(Λ 0 ) Then by the second equality in (312) we have By using (33) twice we obtain so that be α v Λ0 = 0 in W(Λ 0 ) τ 2 (b) v Λ0 = 0 in W(Λ 0 ) (345) τ 2 (b) v N Λ 0 Ker π Λ0 in W N (Λ 0 ) On the other h (346) τ 2 (b) v N Λ 0 / I Λ0 v N Λ 0 Indeed if τ 2 (b) v N Λ 0 I Λ0 v N Λ 0 then e α/2 (êα/2 (τ 2 (b) v N Λ 0 )) êα/2 (êα/2 (I Λ0 v N Λ 0 )) I Λ0 v N Λ 0 by Lemmas 31 32 However by (325) (327) (343) we have e α/2 (êα/2 (τ 2 (b) v N Λ 0 )) = êα/2 (τ 1 (b) v N Λ 1 ) = bx α ( 1) v N Λ 0 = a v N Λ 0 Thus we obtain a v N Λ 0 I Λ0 v N Λ 0 contradicting (331) This proves (346) But (347) wt τ 2 (b) wt b = wt a 1 < wt a (recall (322) in case b has positive charge (323) when b is a constant also (344)) Thus we have constructed a doubly homogeneous element namely τ 2 (b) U( n ) of weight less than that of a satisfying (331) This contradiction proves our theorem hence Theorem 21 as well

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 15 Remark 33 As an immediate consequence of Theorem 21 we observe that any nonzero doubly homogeneous element a U( n ) such that a Ker f Λ0 = I Λ0 has charge at least 2; that is there are no nonzero linear elements x α (m) in Ker f Λ0 m 1 Similarly there are no nonzero linear elements x α (m) in Ker f Λ 1 for m 2 We observe similarly that any homogeneous element of charge 2 that lies in Ker f Λ0 (respectively Ker f Λ 1 ) is a multiple of Rt 0 for some t 2 (respectively Rt 1 for some t 4) 4 A further reformulation In this section we shall present a reformulation of formula (251) of Theorem 22 in terms of ideals of vertex algebras We shall refer to [LL] for definitions results regarding ideals of vertex (operator) algebras regarding vertex operator algebra module structure on generalized Verma modules for ĝ It is known that N(Λ 0 ) carries a natural structure of vertex operator algebra with a vertex operator map Y( x) : N(Λ 0 ) End N(Λ 0 )[[xx 1 ]] v Y(vx) = m Z v m x m 1 satisfying certain conditions with v N Λ 0 as vacuum vector with a suitable conformal vector (see Theorem 6218 in [LL]) The conformal vector gives rise to the Virasoro algebra operators L(m) m Z in the stard way Recall L(0) from Section 2 Here we use the Virasoro algebra operator L( 1) which acts as a linear operator on N(Λ 0 ) such that (41) L( 1)v N Λ 0 = 0 (42) [L( 1)Y(vx)] = d dx Y(vx) (cf [LL]) Formula (42) gives (43) [L( 1)v m ] = mv m 1 for v N(Λ 0 ) m Z Thevector spacen(λ 1 )hasanaturalmodulestructureforthevertex operatoralgebran(λ 0 ) as described in Theorem 6221 of [LL] Formulas (42) (43) hold on N(Λ 1 ) as well Stard arguments show that W N (Λ 0 ) is a vertex subalgebra of N(Λ 0 ) (cf the last part of the proof of Proposition 42 below); W N (Λ 0 ) does not contain the conformal vector of N(Λ 0 ) Moreover W N (Λ 1 ) is a W N (Λ 0 )-submodule of N(Λ 1 ) Also L(0) preserves W N (Λ i ) for i = 01 L( 1) preserves W N (Λ 0 ) Recall from (247) (248) the natural surjective ĝ-module maps Π Λi i = 01 their kernels N 1 (Λ i ) ThenN 1 (Λ i ) istheuniquemaximal proper(l(0)-graded) ĝ-submoduleofn(λ i ) (44) N 1 (Λ 0 ) = U(ĝ)x α ( 1) 2 v N Λ 0 = U(Cx α g t 1 C[t 1 ])x α ( 1) 2 v N Λ 0 (45) N 1 (Λ 1 ) = U(ĝ)x α ( 1) v N Λ 1 = U(Cx α g t 1 C[t 1 ])x α ( 1) v N Λ 1 (cf [K] [LL])

16 C CALINESCU J LEPOWSKY AND A MILAS For the reader s convenience we recall the definition of the notion of ideal of a vertex algebra (Definition 397 in [LL]): Definition 41 An ideal of the vertex algebra V is a subspace I such that for all v V w I (46) Y(vx)w I((x)) (47) Y(wx)v I((x)) that is v n w I w n v I for all v V w I n Z Remark 41 In view of the skew-symmetry property Y(ux)v = e xl( 1) Y(v x)u for uv V under the condition that L( 1)I I the left-ideal right-ideal conditions (46) (47) are equivalent In particular (46) (47) are equivalent for a vertex operator algebra (cf Remark 398 in [LL]) Proposition 41 The space N 1 (Λ 0 ) is the ideal of the vertex operator algebra N(Λ 0 ) generated by x α ( 1) 2 v N Λ 0 Proof: By Remark 6224 Proposition 6617 in [LL] we have that N 1 (Λ 0 ) is an ideal of N(Λ 0 ) Since x α ( 1) 2 v N Λ 0 belongs to both N 1 (Λ 0 ) the ideal generated by x α ( 1) 2 v N Λ 0 for any a g m Z a(m) is a component of the vertex operator Y(a( 1) v N Λ 0 x) the statement follows As in ring theory we call an ideal of a vertex (operator) algebra generated by one element a principal ideal Thus Proposition 41 says that N 1 (Λ 0 ) is the principal ideal of N(Λ 0 ) generated by the null vector x α ( 1) 2 v N Λ 0 (cf (44) for the Lie-algebraic statement) The restrictions of the maps (247) to the principal subspaces W N (Λ i ) are the linear maps π Λi (249) introduced in Section 2 Thus (48) Ker π Λi = N 1 (Λ i ) W N (Λ i ) for i = 01 Remark 42 We observe that N 1 (Λ 0 ) W N (Λ 0 ) is an ideal of W N (Λ 0 ) Indeed since N 1 (Λ 0 ) is an ideal W N (Λ 0 ) is a vertex subalgebra of N(Λ 0 ) Y(vx)w (N 1 (Λ 0 ) W N (Λ 0 ))((x)) Y(wx)v (N 1 (Λ 0 ) W N (Λ 0 ))((x)) for v W N (Λ 0 ) w N 1 (Λ 0 ) W N (Λ 0 ) Also N 1 (Λ 1 ) W N (Λ 1 ) is a W N (Λ 0 )-submodule of W N (Λ 1 ) The intersection N 1 (Λ 0 ) W N (Λ 0 ) which equals I Λ0 v N Λ 0 by (48) Theorem 22 is also a principal ideal generated by the same null vector: Proposition 42 We have that I Λ0 v N Λ 0 is the ideal of the vertex algebra W N (Λ 0 ) generated by x α ( 1) 2 v N Λ 0

VERTEX-ALGEBRAIC STRUCTURE OF PRINCIPAL SUBSPACES 17 Proof: We denote by I the ideal of W N (Λ 0 ) generated by x α ( 1) 2 v N Λ 0 By (221) we have (49) I Λ0 v N Λ 0 = t 2 U( n )R 0 t v N Λ 0 this space contains x α ( 1) 2 v N Λ 0 We first show the inclusion (410) I Λ0 v N Λ 0 I Since x α ( 1) 2 v N Λ 0 I I is an ideal of W N (Λ 0 ) we have But since (411) Y(x α ( 1) 2 v N Λ 0 x) = t Z we see that Y(x α ( 1) 2 v N Λ 0 x)v N Λ 0 I((x)) ( m 1 +m 2 = t (412) R 0 t vn Λ 0 I x α (m 1 )x α (m 2 ) ) x t 2 for each t 2 Finally since each x α (m) m Z is a component of the vertex operator Y(x α ( 1) v N Λ 0 x) the inclusion (410) holds Itremains only toshow that I Λ0 v N Λ 0 is an ideal of W N (Λ 0 ) We firstobserve that theoperator L( 1) preserves I Λ0 v N Λ 0 Indeed by using (41) (42) (411) we obtain (413) L( 1)(R 0 t vn Λ 0 ) = (t 1)R 0 t+1 vn Λ 0 I Λ0 v N Λ 0 for any t 2 More generally from (43) for any m 1 m r 1 we have this shows that L( 1)(x α ( m 1 ) x α ( m r )Rt 0 vn Λ 0 ) r = m j x α ( m 1 ) x α ( m j 1) x α ( m r )Rt 0 vλ N 0 j=1 +(t 1)x α ( m 1 ) x α ( m r )R 0 t+1 v N Λ 0 I Λ0 v N Λ 0 (414) L( 1)(I Λ0 v N Λ 0 ) I Λ0 v N Λ 0 (recall (49)) In view of Remark 41 (414) in order to prove that I Λ0 v N Λ 0 is an ideal of W N (Λ 0 ) it is enough to show that v n w I Λ0 v N Λ 0 for any v W N (Λ 0 ) = U( n ) v N Λ 0 w I Λ0 v N Λ 0 n Z This will follow once we show that v n w U( n)w To do this note that v is a linear combination of monomials in x α ( m) with each m 1 applied to v N Λ 0 We first assume that v = v N Λ 0 Then (415) Y(vx)w = Y(v N Λ 0 x)w = w

18 C CALINESCU J LEPOWSKY AND A MILAS (since Y(v N Λ 0 x) is the identity operator) so that v n w U( n)w Now we assume that v is a linear combination of elements x α ( m 1 ) x α ( m r ) v N Λ 0 with m 1 m r 1 which we write as x α ( m 1 )u for u = x α ( m 2 ) x α ( m r ) v N Λ 0 Since x α ( m 1 )u = (x α ( 1) v N Λ 0 ) m1 u the iterate formula (cf formula (3111) more specifically formula (3812) in [LL]) gives (416) Y(vx)w = Y ( (x α ( 1) vλ N 0 ) m1 ux ) w ( ( ) ) 1 d m1 1 = Y(x α ( 1) vλ N (m 1 1)! dx 0 x) Y(ux)w ( ) 1 = d m1 1 x α (j)x j 1 Y(ux)w (m 1 1)! dx Now (416) induction on r 0 imply that v n w U( n)w completing the proof Using the notation (v) V for the ideal generated by an element v of a vertex (operator) algebra V we have succeeded in reformulating (251) in Theorem 22 as follows: Theorem 41 We have (417) Ker π Λ0 = (x α ( 1) 2 v N Λ 0 ) N(Λ0 ) W N (Λ 0 ) = (x α ( 1) 2 v N Λ 0 ) W N (Λ 0 ) In particular the intersection with the vertex subalgebra W N (Λ 0 ) of the principal ideal of N(Λ 0 ) generated by the null vector x α ( 1) 2 v N Λ 0 coincides with the principal ideal of the vertex subalgebra W N (Λ 0 ) generated by the same null vector j Z References [A] G Andrews The Theory of Partitions Encyclopedia of Mathematics Its Applications Vol 2 Addison-Wesley 1976 [B] R E Borcherds Vertex algebras Kac-Moody algebras the Monster Proc Natl Acad Sci USA 83 (1986) 3068-3071 [Cal] C Calinescu Interwtining vertex operators certain representations of ŝl(n) Comm in Contemp Math to appear [CalLM1] C Calinescu J Lepowsky A Milas Vertex-algebraic structure of the principal subspaces of certain -modules II: higher level case to appear [CalLM2] C Calinescu J Lepowsky A Milas Vertex-algebraic structure of certain subspaces of level one A (2) 2 -modules in preparation [CalLM3] C Calinescu J Lepowsky A Milas Vertex-algebraic structure of principal subspaces of certain modules for affine Lie algebras in preparation [CLM1] S Capparelli J Lepowsky A Milas The Rogers-Ramanujan recursion intertwining operators Comm in Contemp Math 5 (2003) 947-966; arxiv:mathqa/0211265 [CLM2] S Capparelli J Lepowsky A Milas The Rogers-Selberg recursions the Gordon-Andrews identities intertwining operators The Ramanujan Journal 12 (2006) 379-397; arxiv:mathqa/0310080 [DL] [FS1] A (1) 1 C Dong J Lepowsky Generalized Vertex Algebras Relative Vertex Operators Progress in Mathematics Vol 112 Birkhäuser Boston 1993 B Feigin A Stoyanovsky Quasi-particles models for the representations of Lie algebras geometry of flag manifold; arxiv:hep-th/9308079

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20 C CALINESCU J LEPOWSKY AND A MILAS E mail address: calinescu@mathohio-stateedu On leave from the Institute of Mathematics of the Romanian Academy Department of Mathematics Rutgers University Piscataway NJ 08854 E mail address: lepowsky@mathrutgersedu Department of Mathematics Statistics University at Albany (SUNY) Albany NY 12222 E mail address: amilas@mathalbanyedu