A note on principal subspaces of the affine Lie algebras in types B (1) l , C (1) l , F (1) 4 and G (1) 2
aa r X i v : . [ m a t h . QA ] J a n A NOTE ON PRINCIPAL SUBSPACES OF THE AFFINE LIEALGEBRAS IN TYPES B (1) l , C (1) l , F (1)4 AND G (1)2 MARIJANA BUTORAC
Abstract.
We construct quasi-particle bases of principal subspaces of standard mod-ules L (Λ), where Λ = k Λ + k j Λ j , and Λ j denotes the fundamental weight of affineLie algebras of type B (1) l , C (1) l , F (1)4 or G (1)2 of level one. From the given bases we findcharacters of principal subspaces. Introduction
This paper is a continuation of our study [Bu1, Bu2, Bu3, BK] of the principal sub-spaces associated to the standard module L ( k Λ ), for k ≥
1, of non-simply laced affineLie algebras of type B (1) l , C (1) l , F (1)4 and G (1)2 . In [G1], G. Georgiev constructed quasi-particle bases of principal subspaces of standard modules L (Λ) with the rectangularhighest weight, that is, Λ is of the form k Λ + k j Λ j , and Λ j denotes the fundamentalweight of level one, in the case of affine Lie algebra of type A (1) l . In [BK], we extendedGeorgiev’s approach to the principal subspaces which correspond to rectangular weightsof affine Lie algebras of type D (1) l , E (1)6 , E (1)7 and E (1)8 .The main result of this work is the construction of quasi-particle bases of principalsubspaces of standard modules L (Λ) with the rectangular highest weight for the remainingcases of untwisted affine Lie algebras (Theorem 2.1). From the constructed bases wefind characters of principal subspaces (Theorem 4.1). Obtained characters of principalsubspaces are connected with the characters of parafermionic field theories (see [AKS,FS, GG, Gep, KNS, G2]). This connection is further studied in the paper [BKP].Our construction follows closely the construction of quasi-particle bases of principalsubspaces of vacuum standard modules L ( k Λ ) in [Bu1, Bu2, Bu3, BK, G1]. The startingpoint in this construction is to find all relations among quasi-particles, which are thenused to find the spanning sets. In this paper we use these results to construct the spanningsets of principal subspaces of L (Λ). The main difference with the case of the principalsubspace of L ( k Λ ) is in the formulation of initial conditions (Lemma 2.0.1 and Lemma2.0.2) in terms of quasi-particles.The main idea of the proof of linear independence is, as in [Bu1, Bu2, Bu3, BK, G1],from a finite linear combination P a ∈ A c a b a v = 0 of quasi-particle monomial vectors b a v from the spanning set, obtain the following linear combination P a ∈ A c a b ′ a v = 0, where b ′ a v are still from the spanning set, such that b a < b ′ a , with the respect to the linear orderon quasi-particles. To do this we use coefficients of intertwining operators for vertexoperator algebra L (Λ ) associated with the affine Lie algebra from [Li1,Li2,Li3], together Department of Mathematics, University of Rijeka, Radmile Matejˇci´c 2, 51 000 Rijeka,Croatia
E-mail address : [email protected] .2000 Mathematics Subject Classification.
Primary 17B67; Secondary 05A19, 17B69.
Key words and phrases. principal subspaces, combinatorial bases, quasi-particles, vertex operatoralgebras, affine Lie algebras. ith simple current maps in the case of affine Lie algebras of type B (1) l and C (1) l , andWeyl group translation operators among standard modules of level 1.1. Preliminaries
Let g be a complex simple Lie algebra of type B l , C l , F or G with the triangulardecomposition g = n − ⊕ h ⊕ n + , where h denotes the Cartan subalgebra of g . Let h· , ·i bethe invariant symmetric nondegenerate bilinear form on g normalized so that long rootshave length √
2. Denote by R + ⊂ h ∗ the set of positive roots of g , by R the set of roots,by Q the root lattice and by { α , . . . , α l } the subset of simple roots. We fix the standardchoice of simple roots, which we now recall. Denote by { ǫ , . . . , ǫ l } the usual orthonormalbasis of the R l . Then in the case of B l , we have the following set of simple roots n α = ǫ − ǫ , . . . , α l − = ǫ l − − ǫ l , α l = ǫ l o , which correspond to the following labeling of Dynkin diagram α α . . . α l − α l ⇒ .In the case of C l , we will use the following notation for the basis of the root system n α = √ ǫ l , α = 1 √ ǫ l − − ǫ l ) , . . . , α l − = 1 √ ǫ − ǫ ) , α l = 1 √ ǫ − ǫ ) o , so that we have the following labeling of the Dynkin diagram α l α l − . . . α α ⇐ .In the case of F we have α α α α ⇒ ,where n α = ǫ − ǫ , α = ǫ − ǫ , α = ǫ , α = 12 ( ǫ − ǫ − ǫ − ǫ ) o , and in the case of G we have n α = 1 √ − ǫ + ǫ + ǫ ) , α = 1 √ ǫ − ǫ ) o , with the following Dynkin diagram α α ⇛ .For every α ∈ R ± , denote by x α the generator of n ± . Denote by { λ , . . . , λ l } the set offundamental weights of g , where λ i = ǫ + · · · + ǫ i for i = l, and λ l = 12 ( ǫ + · · · + ǫ l ) in the case of B l ,λ i = 1 √ ǫ + · · · + ǫ l − i +1 ) in the case of C l ,λ = ǫ + ǫ , λ = 2 ǫ + ǫ + ǫ , λ = 12 (3 ǫ + ǫ + ǫ + ǫ ) , λ = ǫ in the case of F ,λ = 1 √ − ǫ − ǫ + 2 ǫ ) , λ = 1 √ − ǫ + ǫ ) in the case of G . We identify h with h ∗ using form h· , ·i . Thus, the fundamental weights are viewed aselements of h (cf. [H]). he affine Kac-Moody Lie algebra e g associated with g is infinite-dimensional vectorspace e g = g ⊗ C [ t, t − ] ⊕ C c ⊕ C d, where c denotes the canonical central element and d denotes the degree operator, equippedwith the commutation relations[ x ( m ) , y ( n )] = [ x, y ] ( m + n ) + h x, y i mδ m + n c, [ d, x ( m )] = mx ( m ) and [ d, c ] = 0 , for all x, y ∈ g , m, n ∈ Z (cf. [K]). The generating functions for elements x ( m ) = x ⊗ t n of the affine algebra are defined by x ( z ) = X m ∈ Z x ( m ) z − m − . Denote by { α , α , . . . , α l } the set of simple roots, and by { Λ , Λ , . . . , Λ l } the set offundamental weights of e g .We consider rectangular weights, i.e. the highest weights of the formΛ = k Λ + k j Λ j , (1.1)where k , k j ∈ Z + and Λ j denotes the fundamental weight such that h Λ j , c i = 1. When e g is of type B (1) l we have j = 1 , l , in the case of C (1) l j = 1 , . . . , l , in the case of F (1)4 j is equal to 4 and in the case of G (1)2 j = 2 (cf. [K]). Denote by L (Λ) the standard (i.e.integrable highest weight) e g -module with a highest weight as in (1.1). With k = Λ( c )denote the level of e g -module L (Λ), k = k + k j .For every simple root α i , 1 ≤ i ≤ l denote by sl ( α i ) ⊂ g a subalgebra generated by x α i and x − α i , and let e sl ( α i ) = sl ( α i ) ⊗ C [ t, t − ] ⊕ C c α i ⊕ C d ⊂ e g be the correspondingaffine Lie algebra of type A (1)1 with the canonical central element c α i = 2 c h α i , α i i . The restriction of L (Λ) to e sl ( α ) is standard module of level k α i = 2 k h α i , α i i . For later use we introduce the following notation j t = (cid:26) ≤ t ≤ ν j k + ( ν j − k j , t > k α j j for ν j k + ( ν j − k j + 1 ≤ t ≤ k α j , (1.2)where ν j denotes h α j ,α j i .For each fundamental e g -module L (Λ j ) fix a highest weight vector v Λ j . By completereducibility of tensor products of standard modules, for level k > L (Λ) ⊂ L (Λ j ) ⊗ k j ⊗ L (Λ ) ⊗ k , with a highest weight vector v Λ = v ⊗ k j Λ j ⊗ v ⊗ k Λ . Consider e g -subalgebra e n + = n + ⊗ C [ t, t − ] . The principal subspace W L (Λ) of L (Λ) is defined as W L (Λ) = U ( e n + ) v Λ , (cf. [FS]). his space is generated by operators from U = U ( e n α l ) · · · U ( e n α ) , which act on the highest weight vector v Λ (see Lemma 3.1 in [G1] and also [Bu1, Bu2,Bu3, BK]), where e n α i = C x α i ⊗ C [ t, t − ] , ≤ i ≤ l. Quasi-particle bases
In this section, we first recall the notion and some basic facts about quasi-particlesfrom [Bu1, Bu2, Bu3, BK, G1]. Then we determine the spanning set of the principalsubspace W L (Λ) .Recall that the simple vertex operator algebra L ( k Λ ) associated with the integrablehighest weight module of e g with level k is generated by x ( − v k Λ for x ∈ g such that Y ( x ( − v k Λ , z ) = x ( z ) , where v k Λ is the vacuum vector, (cf. [FLM, LL]). Moreover, the level k standard e g -modules are modules for this vertex operator algebra.We will consider the vertex operators x rα i ( z ) = Y ( x α i ( − r v k Λ , z ) = X m ∈ Z x rα i ( m ) z − m − r = x α i ( z ) · · · x α i ( z ) | {z } r times (2.1)associated with the vector x α i ( − r v k Λ ∈ L ( k Λ ). Following [G1], for a fixed positiveinteger r and a fixed integer m define the quasi-particle of color i , charge r and energy − m as the coefficient x rα i ( m ) of (2.1).Note that charges of quasi-particles x rα i ( z ) in our quasi-particle basis monomial willbe less or equal to k α i , since x ( k αi +1) α i ( z ) = 0 (2.2)on L (Λ) (see [LL], [LP], [MP]). Also, from the definition (2.1) follows that x rα i ( z ) v k Λ ∈ W L (Λ) [[ z ]] . (2.3)In the case of L (Λ j ) we have the following relations Lemma 2.0.1.
In the case of affine Lie algebras e g of type B (1) l and C (1) l on L (Λ ) , wehave x α ( − v Λ = 0 , (2.4) x α ( − v Λ = 0 , (2.5) x α i ( − v Λ = 0 , for i = 1 , (2.6) x α i ( − v Λ = 0 , if h α i , α i i = 1 . (2.7) Proof.
Let us first assume that e g is of type B (1) l . Since the restriction of L (Λ ) to e sl ( α ) isa level one module and since we have h λ , α i = 1, it follows that e sl ( α ) v Λ is a standard A (1)1 -module L (Λ ). This gives us x α ( − v Λ = 0 x α ( − v Λ = 0 . The restriction of L (Λ ) to e sl ( α i ), where i = 1, is a level one module with trivial sl ( α i )-module on the top, and therefore it is a standard A (1)1 -module L (Λ ). From this follows(2.6). Relation (2.7) follows from the fact that the restriction of L (Λ ) to e sl ( α l ) is a evel two module with trivial sl ( α i )-module on the top, and therefore it is a standard A (1)1 -module L (2Λ ) (cf. [K]).In a similar way it can be verified that the claims of the lemma hold for the case of a C (1) l -module L (Λ ). (cid:3) Lemma 2.0.2.
In the case of affine Lie algebras e g of type B (1) l , C (1) l , F (1)4 , G (1)2 on L (Λ j ) ,where j = 1 , we have x α i ( − v Λ j = 0 , for ≤ i ≤ l, (2.8) x α i ( − v Λ j = 0 , for i = j and h α i , α i i = 1 , (2.9) x α j ( − v Λ j = 0 , for h α j , α j i = 1 , (2.10) x α j ( − v Λ j = 0 , for h α j , α j i = 1 , (2.11) x α j ( − v Λ j = 0 , for h α j , α j i = 23 , (2.12) x α j ( − v Λ j = 0 , for h α j , α j i = 23 , (2.13) x α j ( − v Λ j = 0 , for h α j , α j i = 23 . (2.14) Proof.
Let e g be of type B (1) l . The restriction of L (Λ l ) to e sl ( α i ), where i = l , is a standard A (1)1 -module L (Λ ). Therefore, x α i ( − v Λ l = 0 . On the other hand the restriction of L (Λ l ) to e sl ( α l ) is a level two module with twodimensional sl ( α l )-module on the top, so it must be a standard A (1)1 -module L (Λ + Λ ).From this follows x α l ( − v Λ l = 0 ,x α l ( − v Λ l = 0 ,x α l ( − v Λ l = 0 . In a similar way it can be verified that relations (2.8), (2.10) and (2.11) hold for thecase of a C (1) l -module L (Λ j ), where j = 2 , . . . , l and F (1)4 -module L (Λ ). In the caseof C (1) l -module L (Λ j ) and F (1)4 -module L (Λ ) we also have relation (2.9), which followsfrom the fact that the restriction of L (Λ j ) to e sl ( α i ), where i = j, i = 3 in the caseof F (1)4 ) is a level two module with trivial sl ( α i )-module on the top, and therefore itmust be a standard A (1)1 -module L (2Λ ). When e g is of type G (1)2 we have relations (2.8),(2.12), (2.13) and (2.14) on L (Λ ). These relations are a consequence of the fact that therestriction of L (Λ ) to e sl ( α ) is a level one standard L (Λ )-module and the restrictionof L (Λ ) to e sl ( α ) is A (1)1 -module L (2Λ + Λ ). (cid:3) From the last two lemmas we have x rα i ( z ) v ⊗ k j Λ j ⊗ v ⊗ k Λ ∈ z P rt =1 δ i,jt W L (Λ) [[ z ]] . (2.15)Our quasi-particle basis monomial will be of the form b = b α l · · · b α b α , (2.16)where b α i = x n r (1) i ,i α i ( m r (1) i ,i ) . . . x n ,i α i ( m ,i ) ,n r (1) i ,i ≤ · · · ≤ n ,i ≤ k α i and r (1) i ≥ r (2) i ≥ . . . ≥ r ( k αi ) i for i = 1 , . . . , l. ere n p,i and r ( t ) i represent parts of a conjugate pair of partitons C i = ( n r (1) i ,i , . . . , n ,i )and D i = ( r (1) i , r (2) i , . . . , r ( s ) i ) of some fixed n i . Following [G1] we call C i a charge-type ofa monomial b α i , D i a dual-charge-type and n i a color-type of a monomial b α i . We canvisualize charge-type and dual-charge type of monomial b α i using graphic presentation,as in the following example. Example 2.0.1.
For monomial x α i ( m ,i ) x α i ( m ,i ) x α i ( m ,i ) x α i ( m ,i ) of color-type n i = 11 , of charge-type C i = (1 , , , and dual-charge-type D i = (4 , , , we have the graphic presentation as given in Figure , where each quasi-particle of charge r is presented by a column of height r . Number of boxes in every row represents part ofa dual-charge-type D i . r (1) i r (2) i r (3) i r (4) i n ,i n ,i n ,i n ,i Figure 1.
Graphic presentationAnalogously to the situation with a single color, we define the charge-type C , the dual-charge-type D of b in (2.16) by C = ( C l ; . . . ; C ) , (2.17) D = ( D l ; . . . ; D ) , (2.18)and the color-type of b as the l-tuple ( n l , . . . , n ) where n i denotes the color-type of amonomial b α i . Moreover, by E = (cid:16) m r (1) l ,l , . . . , m ,l ; . . . ; m r (1)1 , , . . . , m , (cid:17) we denote the energy-type of b .Now, let b, b be any two quasi-particle monomials of the same color-type, expressed asin (2.16). Denote their charge-types and energy-types by C , C and E , E respectively. Wedefine the linear order among quasi-particle monomials of the same color-type by b < b if C < C or C = C and E < E , (2.19)where for (finite) sequences of integers we define:( x p , . . . , x ) < ( y r , . . . , y )if there exists s such that x = y , . . . , x s − = y s − and s = p + 1 r or x s < y s . (2.20)From (2.3) and (2.15) follows that energies in the expression obtained by applying(2.16) on the highest weight vector comply the following difference condition m p,i ≤ − n p,i − n p,i X t =1 δ i,j t , for 1 ≤ p ≤ r (1) i . (2.21)We strengthen this inequality by using relations among quasi-particles. he interactions among quasi-particles of different colors [Bu1, Lemma 2.3.2], [Bu2,Lemma 4.3, Lemma 5.3], [Bu3, Lemma 3.2], [BK, Lemma 5.3] we summarize as follows. Lemma 2.0.3.
For quasi-particles of fixed charges n i − and n i on W L (Λ) we have ( z − z ) M i x n i α i ( z ) x n i − α i − ( z ) = ( z − z ) M i x n i − α i − ( z ) x n i α i ( z ) , (2.22) where M i = min n ν αi ν αi − n i − , n i o . The interaction among quasi-particles of the same color is described by the followingassertion [F, Lemma 3.3], [JP, Lemma 4.4], [G1, (3.18)–(3.23)].
Lemma 2.0.4.
For fixed charges n , n such that n n and fixed integer M such that m + m = M the monomials x n α i ( m ) x n α i ( m ) , x n α i ( m − x n α i ( m +1) , . . . , x n α i ( m − n +1) x n α i ( m +2 n − of operators on W L (Λ) can be expressed as a linear combination of monomials x n α i ( j ) x n α i ( j ) such that j m − n , j > m + 2 n and j + j = M and monomials which contain a quasi-particle of color i and charge n + 1 . Moreover, for n = n the monomials x n α i ( m ) x n α i ( m ) with m − n < m m can be expressed as a linear combination of monomials x n α i ( j ) x n α i ( j ) such that j j − n and j + j = M and monomials which contain a quasi-particle of color i and charge n + 1 . Denote by B W the set of all quasi-particle monomials of the form as in (2.16) whichsatisfy the following difference conditions m p,i ≤ − n p,i + r (1) i − X q =1 min n ν αi ν αi − n i − , n i o − p − n p,i − P n p,i t =1 δ i,j t , for 1 ≤ p ≤ r (1) i , (2.23) m p +1 ,i ≤ m p,i − n p,i , for n p +1 ,i = n p,i , ≤ p ≤ r (1) i − , (2.24)where r (1)0 = 0 and j t is as in (1.2). We have Theorem 2.1.
The set B W = { bv Λ : b ∈ B W } forms a basis of the principal space W L (Λ) . The proof that B W is the spanning set goes as in [G1], by using induction on thecharge-type and the total energy of quasi-particle monomials. It remains to prove thelinear independence of the spanning set.3. Proof of linear independence
In the proof of linear independence of the set B W we will employ operators defined onlevel one standard modules L (Λ j ). The projection π D , which generalizes the projectionintroduced in [G1] (see also [Bu1, Bu2, Bu3, BK]), enables us to use these operators in thecase of higher levels. In Section 3.1 we will recall the main properties of π D . In Section3.2 we will introduce coefficients of intertwining operators among level one modules andin Section 3.3 we will recall important properties of Weyl group translation operators.Finally in Section 3.4 we prove linear independence of the set B W . .1. Projection π D . For a dual-charge-type D of monomial (2.16) denote by π D theprojection of W L (Λ) on the vector space W L (Λ jk )( µ ( k ) l ; ... ; µ ( k )1 ) ⊗ · · · ⊗ W L (Λ j )( µ (1) l ; ... ; µ (1)1 ) ⊂ W ⊗ k j L (Λ j ) ⊗ W ⊗ k L (Λ ) ⊂ L (Λ j ) ⊗ k j ⊗ L (Λ ) ⊗ k , where j t ∈ { , j } , 1 ≤ t ≤ k , W L (Λ jt )( µ ( t ) l ; ... ; µ ( t )1 ) denotes the h -weight subspace of the levelone principal subspace W L (Λ jt ) of weight µ ( t ) l α l + · · · + µ ( t )1 α ∈ Q with µ ( t ) i = ν i − X p =0 r ( ν i t − p ) i for 1 ≤ t ≤ k. (3.1)With the same symbol we denote the generalization of the projection π D to the space offormal series with coefficients in W ⊗ k j L (Λ j ) ⊗ W ⊗ k L (Λ ) . Let x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n ,l α l ( z ,l ) · · · x n r (1)1 , α ( z r (1)1 , ) · · · x n , α ( z , ) v Λ (3.2)be the generating function of the monomial (2.16), which acts on the highest weightvector v Λ . From relations (2.2) follows that the projection of (3.2) is: π D (cid:18) x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n , α ( z , ) v Λ (cid:19) (3.3)= C x n ( k ) r ( νl ( k − l ,l α l ( z r ( νl ( k − l ,l ) · · · x n ( k ) r ( νlk ) l ,l α l ( z r ( νlk ) l ,l ) · · · x n ( k )1 ,l α l ( z ,l ) · · ·· · · x n ( k ) r ( ν k − , α ( z r ( ν k − , ) · · · x n ( k ) r ( ν k )1 , α ( z r ( ν k )1 , ) · · · x n ( k )1 , α ( z , ) v Λ jk , ⊗ · · · ⊗⊗ x n (1) r (1) l ,l α l ( z r (11) l ,l ) · · · x n (1) r ( νlk ) l ,l α l ( z r ( νlk ) l ,l ) · · · x n (1)1 ,l α l ( z ,l ) · · ·· · · x n (1) r (1)1 , α ( z r (1)1 , ) · · · x n (1) r ( ν k )1 , α ( z r ( ν k )1 , ) · · · x n (1)1 , α ( z , ) v Λ j , where C ∈ C ∗ , and where0 ≤ n ( t ) p,i ≤ ν i , n p,i = k X t =1 n ( t ) p,i , for every 1 ≤ p ≤ r (1) i . For fixed color i the projection π D places at most ν i generating functions x α i ( z p,i ) oneach tensor factor v Λ jt , 1 ≤ t ≤ k . This property of π D is demonstrated in the followingexample for the case of affine Lie algebra e g of type G (1)2 . Example 3.1.1.
Consider the formal power series x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) v Λ (3.4) with coefficients in the principal subspace W L (Λ +Λ ) of level 2 standard module L (Λ +Λ ) of affine Lie algebra e g of type G (1)2 . The projection π D of (3.4) , where D = (3 , , ,
1; 2 , ,onto W L (Λ )(1;1) ⊗ W L (Λ )(7;2) is Cx α ( z , ) x α ( z , ) v Λ ⊗ x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) v Λ , (3.5) (C ∈ C ∗ ). Graphically, the image of (3.4) can be represented as in Figure , where boxesin columns represent n ( t ) p,i . Λ v Λ α α α α α Figure 2. π D ( x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) x α ( z , ) v Λ ) First note that we have n (1)1 , = n (2)1 , = 1 , since from the relation x α ( z , ) = 0 on L (Λ + Λ ) follows that with the projection π D every factor x α ( z , ) of the vertex operator x α ( z , ) is applied on the different tensor factor. With the projection the vertex operator x α ( z , ) is applied only on the rightmost tensor factor, so n (1)2 , = 1 and n (2)2 , = 0 . Therelation x α ( z , ) = 0 on L (Λ + Λ ) implies that with the projection π D three vertexoperators x α ( z , ) are applied on the rightmost tensor factor and one vertex operator x α ( z , ) is applied on the remaining tensor factor. From this follows that n (1)1 , = 3 and n (2)1 , = 1 . With the projection the vertex operator x α ( z , ) is applied only on the rightmosttensor factor, so n (1)2 , = 3 and n (2)2 , = 0 . Finally, the vertex operator x α ( z , ) is appliedon the rightmost tensor factor, therefore we have n (1)3 , = 1 and n (2)3 , = 0 . Coefficients of level intertwining operators and simple current maps. First let e g be of type B (1) l or of type C (1) l . Denote by I ( · , z ) the intertwining operator oftype (cid:0) L (Λ j ) L (Λ j ) L (Λ ) (cid:1) , defined by I ( w j , z ) w = exp( zL ( − Y ( w , − z ) w j , (3.6)where w j ∈ L (Λ j ) and w ∈ L (Λ ) (cf. [FHL]). Following [Bu1, Bu2] denote by A λ theconstant term of the intertwining operator I ( v Λ , z ). This coefficient commutes with theaction of quasi-particles (cf. [Bu1, Bu2]).Fix λ ∈ h as in Section 1. Following H. Li (cf. [Li1, Li2, Li3]), for any L ( k Λ )-module V we introduce the following notation( V ( λ ) , Y λ ( · , z )) = ( V, Y (∆( λ , z ) · , z )) , where ∆( λ , z ) = z λ exp X n ≥ λ ( n ) n ( − z ) − n ! . By Proposition 2.6 in [Li1] follows that V ( λ ) has a structure of a weak L ( k Λ )-module.In particular, L ( k Λ ) ( λ ) ∼ = L ( k Λ ) is a simple current L ( k Λ )-module.Following [Bu1] and [Bu2] denote by e λ simple current map, that is bijection e λ : L (Λ j ) → L (Λ j ) ( λ ) , such that x α ( m ) e λ = e λ x α ( m + α ( λ )) , (3.7)for all α ∈ R and m ∈ Z and e λ v Λ = v Λ , (3.8)(see [DLM], [Li3], or Remark 5.1 in [P]). rom (3.8) follows that the monomial vector π D bv Λ ∈ B W , where W L (Λ) is the principalsubspace of the standard module L ( k Λ + k Λ ) and b is of dual-charge type D , equals π D b ( e λ v Λ ) ⊗ k ⊗ v ⊗ k Λ . (3.9)From (3.7) and from the definition of the projection π D it follows that (3.9) is the coeffi-cient of the variables z − m r (1) l ,l − n r (1) l ,l r (1) l ,l · · · z − m , − n , , z − m r (1)1 , − n r (1)1 , r (1)1 , · · · (3.10) · · · z − m r ( k , − n r ( k , r ( k +1 , z − m r ( k , − n r ( k , +( n r ( k , − k ) r ( k , · · · z − m , − n , +( n , − k )1 , in π D (cid:18) x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n , α ( z , ) ( e λ v Λ ) ⊗ k ⊗ v ⊗ k Λ (cid:19) . (3.11)Denote this coefficient by e ⊗ k λ ⊗ ⊗ k (cid:0) π D b + v ⊗ k Λ (cid:1) , where b + = b α l · · · b α b + α (3.12)and b + α = x n r (1)1 , α ( m r (1)1 , ) · · · x n r ( k , α ( m r ( k +1 , ) (3.13) x n r ( k , α ( m r ( k , + n r ( k , − k ) · · · x n , α ( m , + n , − k ) . From (2.23) and (2.24) follows that energies of quasi-particle monomial vectors b + v k Λ satisfy difference conditions of energies of quasi-particle monomial vectors from the basisof principal subspace of the standard module L ( k Λ ), (see also [Bu2]).For the case when W L (Λ) is the principal subspace of the standard module L ( k Λ + k j Λ j ), j = 1, we will use the following lemma: Lemma 3.2.1.
In the case of affine Lie algebra e g of type B (1) l e λ v Λ l = x ǫ ( − v Λ l . (3.14) In the case of affine Lie algebra e g of type C (1) l and j = 1 e λ v Λ j = x √ ( ǫ l − j +1 + ǫ j ) ( − · · · x √ ( ǫ + ǫ l − ) ( − x √ ( ǫ + ǫ l ) ( − v Λ l − j +2 . (3.15) Proof.
Relations (3.14) and (3.15) are verified by arguing as in the proof in [P, Lemma5.3]. e − λ x ǫ ( − v Λ l is a weight vector with weight Λ l , since we have α ∨ (0) e − λ x ǫ ( − v Λ l = e − λ ( α ∨ (0) − c ) x ǫ ( − v Λ l = 0 ,α ∨ j (0) e − λ x ǫ ( − v Λ l = e − λ α ∨ j (0) x ǫ ( − v Λ l = 0 , for j = 1 , l,α ∨ l (0) e − λ x ǫ ( − v Λ l = e − λ α ∨ l (0) x ǫ ( − v Λ l = e − λ x ǫ ( − v Λ l ,x − θ (1) e − λ x ǫ ( − v Λ l = e − λ x − θ (2) x ǫ ( − v Λ l = 0 . From x α ( − x α ( − v Λ l = 0 follows that x ǫ (0) x α ( − x α ( − v Λ l = 2 x α ( − x ǫ ( − v Λ l = 0 , so we have x α (0) e − λ x ǫ ( − v Λ l = e − λ x α ( − x ǫ ( − v Λ l = 0 . We also have x α j (0) e − λ x ǫ ( − v Λ l = e − λ x α j (0) x ǫ ( − v Λ l = e − λ x ǫ ( − x α j (0) v Λ l = 0 , or j = 1 , l , and x α l (0) e − λ x ǫ ( − v Λ l = e − λ x α l (0) x ǫ ( − v Λ l = e − λ x ǫ + ǫ l ( − v Λ l = 0 , since the restriction of L (Λ l ) on e sl ( ǫ + ǫ l ) is a level one standard A (1)1 -module of highestweight Λ . Hence (3.14) holds and L (Λ l ) ( λ ) ∼ = L (Λ l ).In a similar way we prove that e − λ u := e − λ x √ ( ǫ l − j +1 + ǫ j ) ( − · · · x √ ( ǫ + ǫ l − ) ( − x √ ( ǫ + ǫ l ) ( − v Λ l − j +2 is a weight vector with weight Λ j , since we have α ∨ j (0) e − λ u = e − λ α ∨ j (0) u = e − λ u,α ∨ (0) e − λ u = e − λ ( α ∨ (0) − c ) u = 0 ,α ∨ i (0) e − λ u = e − λ α ∨ i (0) u = 0 , for i = 1 , j,x − θ (1) e − λ u = e − λ x − θ (2) u = 0 . From x α ( − x α ( − v Λ l − j +2 = 0 follows that x α ( − x √ ( ǫ + ǫ l ) ( − v Λ l − j +2 = 0 , so wehave x α (0) e − λ u = 0 . We also have x α j (0) e − λ u = e − λ x α j (0) u = 0 , and x α i (0) e − λ u = e − λ x α i (0) u = 0 , for i = j,
1. The last statement is true also in the case when x α i (0) doesn’t com-mute with monomials in u . In this case, by induction on j follows that x α i (0) u = u ′ x √ ( ǫ a + ǫ b ) ( − v Λ l − j +2 = 0, since the restriction of L (Λ l ) on e sl ( √ ( ǫ a + ǫ b )), for any a, b ∈ { , . . . , l } , is a level two standard A (1)1 -module of highest weight 2Λ . Hence (3.15)holds and L (Λ j ) ( λ ) ∼ = L (Λ l − j +2 ). (cid:3) By Proposition 2.4 in [Li1] there is an intertwining operator of type (cid:0) L (Λ j ) ( λ L (Λ j ) L (Λ ) ( λ (cid:1) ,which we will denote by I ( · , z ), such that I ( v Λ j , z ) e λ v Λ = e λ I (∆( λ , z ) v Λ j , z ) v Λ = e λ exp( zL ( − z h λ , Λ j i v Λ j = z h λ , Λ j i (cid:2) e λ v Λ j + e λ v Λ j zL ( −
1) + · · · (cid:3) . Denote by I ( · , z ) the intertwining operator of type (cid:0) L (Λ j ) ( λ L (Λ ) ( λ L (Λ j ) (cid:1) and by A λ the coef-ficient of ( − z ) h λ , Λ j i in I ( e λ v Λ , z ) v Λ j . From Lemma 3.2.1 follows A λ v Λ j = e λ v Λ j . (3.16)From the commutator formula (cf. formula (2.13) of [Li3]) we have[ x α i ( m ) , I ( e λ v Λ , z )] = X t ≥ (cid:18) mt (cid:19) I ( x α i ( t ) e λ v Λ , z ) = 0 . In the case when e g is of type F (1)4 or of type G (1)2 , recall from [BK] and [Bu3] theconstant term A λ of the operator x θ ( z ). We will use the same symbol to denote thecoefficient of z − in z − x θ ( z ), i.e. A λ = Res z z − x θ ( z ) = x θ ( − . (3.17) ow, consider the action of the operator( A λ ) s := 1 ⊗ · · · ⊗ | {z } k − s factors ⊗ A λ ⊗ ⊗ · · · ⊗ | {z } s − , for s = n , , where A λ is as above, on the vector π D bv Λ , where bv Λ ∈ B W and b is ofdual-charge type D . Since A λ commutes with the action of quasi-particles, it follows thatthe image ( A λ ) s ( π D bv Λ ) is the coefficient of the variables z − m r (1) l ,l − n r (1) l ,l r (1) l ,l · · · z − m , − n , , in( A λ ) s π D x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n , α ( z , ) v Λ . (3.18)Using (3.3) follows that in the s -th tensor factor (from the right) of (3.18), we have F s A λ v Λ js , where F s := x n ( s ) r ( νl ( k − l ,l α l ( z r ( νl ( k − l ,l ) · · · x n ( s ) r ( νlk ) l ,l α l ( z r ( νlk ) l ,l ) · · · x n ( s )1 ,l α l ( z ,l ) · · · x n ( s ) r ( s )1 , α ( z r ( s )1 , ) · · · x n ( s )1 , α ( z , ) . Using (3.16), in the case of affine Lie algebras of type B (1) l or of type C (1) l , we rewrite the s -th tensor factor (from the right) of (3.18) as F s A λ v Λ js = e λ F s v Λ js z r ( s )1 , · · · z , . (3.19)From (3.19), now follows ( A λ ) s ( π D bv Λ ) = ( e λ ) s ( π D b + v Λ ) , (3.20)where ( e λ ) s := 1 ⊗ · · · ⊗ | {z } k − s factors ⊗ e λ ⊗ ⊗ · · · ⊗ | {z } s − , and where b + = b α l · · · b α b + α (3.21)with b + α = x n r (1)1 , α ( m r (1)1 , ) · · · x n r ( s +1)1 +1 , α ( m r ( s +1)1 , ) (3.22) x n r ( s )1 , α ( m r ( s )1 , + 1) · · · x n , α ( m , + 1) . In the case of affine Lie algebras of type F (1)4 or of type G (1)2 , the role of e λ will play Weylgroup translation operator e θ , which we will introduce in the next section.3.3. Weyl group translation operators.
Denote by e α the Weyl group translationoperator e α = exp x − α (1) exp( − x α ( − x − α (1) exp x α (0) exp( − x − α (0)) exp x α (0)for every root α , (cf. [K]). We will use the following property of the Weyl group translationoperator e α : x β ( j ) e α = e α x β ( j + β ( α ∨ )) for all α, β ∈ R and j ∈ Z . (3.23)Let e g be of type F (1)4 or of type G (1)2 , and assume that α = θ . Since we have e θ v Λ j = x θ ( − v Λ j , (3.24)we rewrite the s -th tensor factor (from the right) of (3.18) as F s A λ v Λ js = e θ F s v Λ js z r ( s )1 , · · · z , . (3.25) ow we have ( A λ ) s ( π D bv Λ ) = ( e θ ) s ( π D b + v Λ ) , (3.26)where ( e θ ) s := 1 ⊗ · · · ⊗ | {z } k − s factors ⊗ e θ ⊗ ⊗ · · · ⊗ | {z } s − , and where b + is as in (3.21) and (3.22).If we continue to apply the procedure of action of operators ( A λ ) s and ( e λ ) s (or ( A λ ) s and ( e θ ) s in the case of affine Lie algebras of type F (1)4 or of type G (1)2 ), on the b + v Λ , afterfinitely many steps we will obtain the monomial vector˜ bv Λ = b α l · · · b α ˜ b α v Λ , where˜ b α = x n r (1)1 , α ( ˜ m r (1)1 , ) · · · x n r ( s )1 +1 , α ( ˜ m r ( s )1 +1 , ) x n r ( s )1 , α ( ˜ m r ( s )1 , ) · · · x n , α ( ˜ m , )= x n r (1)1 , α ( m r (1)1 , ) · · · x n r ( s )1 +1 , α ( m r ( s )1 +1 , ) x n r ( s )1 , α ( m r ( s )1 , − m , − s ) · · · x n , α ( − s ) . Note that the quasi-particle monomial ˜ b has the same charge-type and the dual charge-type as b + and belongs to B W .We use the fact that e α v Λ j = Cx α ( − v Λ j , (3.27)where C ∈ C \ { } . Hence, the vector π D b + v Λ equals the coefficient of the variables z − m r (1) l ,l − n r (1) l ,l r (1) l ,l · · · z − m r (1)1 , − n r (1)1 , r (1)1 , · · · z − m r ( s )1 +1 , − n r (1)1 +1 , r ( s )1 +1 , z − m r ( s )1 , − n r ( s )1 , + sr ( s )1 , · · · z − m , , (3.28)in C π D x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n , α ( z , ) (1 ⊗ ( k − s ) ⊗ e ⊗ sα ) v Λ . (3.29)By shifting the operator (1 ⊗ ( k − s ) ⊗ e ⊗ sα ) all the way to the left in (3.29), and by droppingit, from (3.23) we get π D ′ b ′ v Λ , where b ′ = b α l · · · b ′ α b ′ α (3.30)for b ′ α = x n r (1)1 , α ( ˜ m r (1)1 , + 2 n r (1)1 , ) · · · x n , α ( ˜ m , + 2 n , ) ,b ′ α = x n r (1)2 , α ( m r (1)2 , − n (1) r (1)2 , − · · · − n ( s ) r (1)2 , ) · · · x n , α ( m , − n (1)1 , − · · · − n ( s )1 , ) . Note, that the dual charge-type D ′ of b ′ equals D ′ = (cid:0) r (1) l , . . . , r ( k αl ) l ; · · · ; r (1)2 , . . . , r ( k α )2 ; r (1)1 − , . . . , r ( n , )1 − , , . . . , | {z } k − s (cid:1) . Finally, using the same arguments as in [Bu1,Bu2,Bu3,BK] one can check that b ′ belongsto B W . .4. Proof of linear independence.
Assume that we have a relation of linear depen-dence between elements of B W X a ∈ A c a b a v Λ = 0 , (3.31)where A is a finite non-empty set and c a = 0 for all a ∈ A . Furthermore, assume that all b a have the same color-type ( n l , . . . , n ). Let a ∈ A be such that b a < b a for all a ∈ A , a = a . Let b be of charge-type C as in (2.17) and dual-charge-type D as in (2.18).On (3.31) we act with the projection π D π D : W L (Λ) → W L (Λ jk )( µ ( k ) l ; ... ; µ ( k )1 ) ⊗ · · · ⊗ W L (Λ j )( µ (1) l ; ... ; µ (1)1 ) , where j t ∈ { , j } and µ ( t ) i , 1 ≤ t ≤ k is as in (3.1). From the definition of projection,it follows that by π D all monomial vectors b a v Λ with monomials b a which have highercharge-type than C with respect to (2.20) will be mapped to zero-vector. Therefore, weassume that in X a ∈ A c a π D b a v Λ = 0 , (3.32)all monomials b a are of charge-type C .In the case when e g be of type B (1) l or of type C (1) l and Λ = k Λ + k Λ we use the factthat v Λ = e λ v Λ . Now, from (3.32) we have0 = X a ∈ A c a π D b a ( e λ v Λ ) ⊗ k j ⊗ v ⊗ k Λ = ( e ⊗ k j λ ⊗ ⊗ k ) X a ∈ A c a π D b + a v ⊗ k Λ . If we drop the operator e ⊗ k j λ ⊗ ⊗ k , we will get X a ∈ A c a π D b + a v ⊗ k Λ = 0 , (3.33)where b + a is of the form as in (3.12) and (3.13). So, in the case of B (1) l or of type C (1) l andΛ = k Λ + k Λ , we have c a = 0, and the assertion of the Theorem 2.1 follows.Now, let e g be of type B (1) l , C (1) l , F (1)4 or of type G (1)2 and Λ = k Λ + k j Λ j , j = 1. On(3.32) apply the procedure described in Sections 3.2 and 3.3 until all quasi-particles ofcolor 1 are removed from the summand c a π D b a v Λ . This also removes all quasi-particlesof color 1 from other summands, so that (3.32) becomes X a ∈ A c ′ a π D b ′ a v Λ = 0 , (3.34)where b ′ a are of the form as in (3.30) and scalars c ′ a = 0. The summation in (3.34) goesover all a = a such that b a ( α ) = b a ( α ) since the summands such that b a ( α ) < b a ( α )will be annihilated in the process.In the case of B (1) l monomial vectors in (3.34) can be realized as elements of the principalsubspace W L ( k Λ + k l Λ l ) of the affine Lie algebra of type B (1) l − . In particular, when e g is oftype B (1)2 , with the described procedure we get monomial vectors which can be realized aselements of W L ((2 k + k )Λ +(2 k − k − k )Λ ) of the affine Lie algebra of type A (1)1 . In the case of C (1) l , monomial vectors in (3.34) can be realized as elements in W L ((2 k + k j )Λ +(2 k − k − k j )Λ j ) of the affine Lie algebra of type A (1) l − . In the case of F (1)4 , monomial vectors in (3.34) can berealized as elements in W L ( k Λ + k Λ ) of the affine Lie algebra of type C (1)3 , and in the caseof G (1)2 monomial vectors in (3.34) we realize as elements of W L ((3 k +2 k )Λ +(3 k − k − k )Λ ) of the affine Lie algebra of type A (1)1 . For all of these cases we can use Georgiev argument n linear independence from [G1], so by proceeding inductively on charge-type (and on l for the case of B (1) l ) we get c a = 0 and the desired theorem follows.4. Characters of principal subspaces
Character ch W L (Λ) of the principal subspace W L (Λ) is defined bych W L (Λ) = X m,n ,...,n l > dim( W L (Λ) ) − mδ +Λ+ n α + ... + n l α l q m y n · · · y n l l , where q, y , . . . , y l are formal variables and ( W L (Λ) ) − mδ +Λ+ n α + ... + n l α l denote the weightsubspaces of W L (Λ) of weight − mδ + Λ + n α + · · · + n l α l with respect to the Cartansubalgebra e h = h ⊕ C c ⊕ C d of e g .As in [Bu1,Bu2,Bu3,BK,G1], to determine the character of W L (Λ) , we write conditionson energies of quasi-particles of the set B W L (Λ) in terms of dual-charge-type elements r ( s ) i .For a fixed color-type ( n l ; . . . ; n ), charge-type C = (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)1 , , . . . , n , (cid:17) and dual-charge-type D = (cid:16) r (1) l , . . . , r ( k αl ) l ; . . . ; r (1)1 , . . . , r ( k α )1 (cid:17) , a straightforward calculation shows r (1) i X p =1 (2( p − n p,i + n p,i ) = k αi X t =1 r ( t ) i for i = 1 , . . . , l, (4.1)and r (1) i X p =1 r (1) i − X q =1 min { k α i k α i − n q,i − , n p,i } = k X t =1 ν i − X p =0 r ( t ) i − r ( ν i t − p ) i for i = 2 , . . . , l, (4.2)(cf. [Bu1, Bu2, Bu3, BK, G1]). We also have r (1) i X p =1 n p,i X t =1 δ i,j t = k αi X t =1 r ( t ) i δ i,j t = k αj X t = ν j k +( ν j − k j +1 r ( t ) j . (4.3)The last three identities, difference conditions (2.23)–(2.24) and the formula1( q ) r = X n > p r ( n ) q n , where p r ( n ) denotes the number of partitions of n with at most r parts, therefore imply Theorem 4.1.
Set n i = P k αi t =1 r ( t ) i for i = 1 , . . . , l . For any rectangular weight Λ = k Λ + k j Λ j of level k = k + k j we have ch W L (Λ) = X r (1)1 > ··· > r ( kα > ... r (1) l > ··· > r ( kαl ) l > q P li =1 P kαit =1 r ( t )2 i − P li =2 P kt =1 P νi − p =0 r ( t ) i − r ( νit − p ) i + P kαjt = νjk νj − kj +1 r ( t ) j Q li =1 ( q ; q ) r (1) i − r (2) i · · · ( q ; q ) r ( kαi ) i l Y i =1 y n i i , where ( a ; q ) r = Q ri =1 (1 − aq i − ) for r > . cknowledgement I am very grateful to Slaven Koˇzi´c and Mirko Primc for their help, support and valuablecomments during the preparation of this work.This work is partially supported by the QuantiXLie Centre of Excellence, a projectcofinanced by the Croatian Government and European Union through the EuropeanRegional Development Fund - the Competitiveness and Cohesion Operational Programme(Grant KK.01.1.1.01.0004) and by Croatian Science Foundation under the project 8488.
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