Quasi-particle bases of principal subspaces for the affine Lie algebras of types B (1) l and C (1) l
aa r X i v : . [ m a t h . QA ] M a y QUASI-PARTICLE BASES OF PRINCIPAL SUBSPACES FOR THEAFFINE LIE ALGEBRAS OF TYPES B (1) l AND C (1) l MARIJANA BUTORAC
Abstract.
Generalizing our earlier work, we construct quasi-particle bases of principalsubspaces of standard module L X (1) l ( k Λ ) and generalized Verma module N X (1) l ( k Λ ) atlevel k ≥ B (1) l and C (1) l . As a consequence,from quasi-particle bases, we obtain the graded dimensions of these subspaces. Intorduction
Let g be a simple complex Lie algebra of type X l , with a Cartan subalgebra h , theset of simple roots Π = { α , . . . , α l } and the triangular decomposition g = n − ⊕ h ⊕ n + ,where n + is a direct sum of its one dimensional subalgebras corresponding to the positiveroots. Denote by L ( n + ) a subalgebra of untwisted affine Lie algebra b g of type X (1) l L ( n + ) = n + ⊗ C [ t, t − ] . Let V be a highest b g -module with highest weight Λ and highest weight vector v Λ . Wedefine the principal subspace W V of V as W V = U ( L ( n + )) v Λ . In this paper we study principal subspaces of the generalized Verma module N X (1) l ( k Λ )and its irreducible quotient L X (1) l ( k Λ ) at level k ≥
1, defined over the affine Lie algebrasof type B (1) l and C (1) l .The study of principal subspaces of standard (i.e., integrable highest weight) modulesof the simply laced affine Lie algebras and its connection to Rogers-Ramanujan identitieswas initiated in the work of B. L. Feigin and A. V. Stoyanovsky [FS] and has been furtherdeveloped in [AKS], [Cal1]–[Cal2], [CalLM1]–[CalLM4], [CLM1]–[CLM2], [G], [Ko], [KP],[MiP], [S1]–[S2].Quasi-particle descriptions of principal subspaces of standard modules for untwistedaffine Kac-Moody algebras originate from the work of Feigin and Stoyanovsky [FS] andG. Georgiev [G]. In order to compute the character formulas of standard A (1)1 -modules,Feigin and Stoyanovsky have constructed monomial bases of principal subspaces of thestandard modules in terms of the expansion coefficients of a certain vertex operators(cf. [DL], [LL]). These monomial bases have an interesting physical interpretation,as quasi-particles, whose energies comply the difference-two condition (see in particular[DKKMM], [FS], [G]).Later on, Georgiev in [G] generalized the construction of quasi-particle bases to prin-cipal subspaces of certain standard modules in the ADE type. His bases were built of
Mathematics Subject Classification.
Primary 17B67; Secondary 17B69, 05A19.
Key words and phrases. affine Lie algebras, vertex operator algebras, principal subspaces, quasi-particle bases. uasi-particles x rα i ( m ) of color i (1 ≤ i ≤ l ), charge r ≥ − mx rα i ( m ) = Res z z m + r − x α i ( z ) · · · x α i ( z ) | {z } r factors , where x α i ( z ) = P j ∈ Z x α i ( j ) z − j − are vertex operators associated to elements x α i ∈ N A (1) l ( k Λ ). From this bases Georgiev obtained character formulas, which are in thecase of A (1)1 character formulas first obtained by J. Lepowsky and M. Primc in [LP].In our previous paper [Bu] we have used ideas of Georgiev to construct quasi-particlebases of principal subspaces of level k ≥ L B (1)2 ( k Λ ) and generalizedVerma module N B (1)2 ( k Λ ) of an affine Lie algebra of type B (1)2 . From the graded dimen-sions (characters) of principal subspaces of generalized Verma module we obtained a newidentity of Rogers-Ramanujan’s type.Our present work is a generalization of [Bu] to the case of B (1) l , l ≥
3, and to the caseof C (1) l , l ≥
3. Our methods for these cases are the same as the methods that we usedin [Bu]. First, using relations among vertex operators associated with the simple roots α i ∈ Π, we find spanning sets of principal subspaces, which are built of quasi-particles ofcolors i , 1 ≤ i ≤ l , and charges r ≥ B (1) l these quasi-particle monomials are of form b ( α l ) b ( α l − ) · · · b ( α ) , where b ( α i ) is a product of quasi-particles corresponding to simple root α i ∈ Π. Fromcombinatorial point of view, difference conditions for energies of quasi-particles of colors i , 1 ≤ i ≤ l −
2, are identical with the difference conditions for energies of Georgiev’squasi-particles in the case of standard A (1) l − -modules of level k and difference conditionsfor energies of quasi-particles of colors l − l are the same as difference conditionsfor energies for level k given in [Bu].In the case of C (1) l , quasi-particle monomials in the spanning sets are of form b ( α ) · · · b ( α l − ) b ( α l ) , where difference conditions for energies of quasi-particles colored with colors l and l − k B (1)2 -modules,and difference conditions for energies of quasi-particles of colors i , 1 ≤ i ≤ l −
2, areidentical with difference conditions for energies of quasi-particles in the case of standard A (1) l − -modules of level 2 k .For the purpose of proving the linear independence of spanning sets, we use a projectionof principal subspaces on the tensor product of h -weight subspaces of standard modulesdefined in [Bu]. The projection enables the usage of certain coefficients of intertwiningoperators, simple current operators and “Weyl group translation” operator defined onthe level one standard modules. We prove linear independence by induction on theorder on quasi-particle monomials. Important argument in the proof will be the linearindependence of quasi-particle vectors from [Bu] for the B (1)2 case and linear independenceof A (1) l − monomial vectors obtained in [G].The main results of this paper are character formulas for principal subspaces of standardmodules L X (1) l ( k Λ ) (Theorem 4 and Theorem 9) and principal subspaces of generalizedVerma modules N X (1) l ( k Λ ) (Theorem 6 and Theorem 11). As a consequence, we also btained two new identities, which are generalization of identity from [Bu]. The first onewas obtained from character formulas of principal subspace of N B (1) l ( k Λ ) in the B (1) l case Theorem 1. Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )= X r (1)1 ≥ ... ≥ r ( u ≥ u ≥ q r (1)21 + ··· + r ( u ( q ) r (1)1 − r (2)1 · · · ( q ) r ( u y r X r (1)2 ≥ ... ≥ r ( u ≥ u ≥ q r (1)22 + ··· + r ( u − r (1)1 r (1)2 −···− r ( u r ( u ( q ) r (1)2 − r (2)2 · · · ( q ) r ( u y r · · · X r (1) l − ≥ ... ≥ r ( ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r ( ul − l − − r (1) l − r (1) l − −···− r ( ul − l − r ( ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r ( ul − l − y r l − l − X r (1) l ≥ ... ≥ r (2 ul ) l ≥ u l ≥ q r (1)2 l + ··· + r (2 ul )2 l − r (1) l − ( r (1) l + r (2) l ) −···− r ( ul ) l − ( r (2 ul − l + r (2 ul ) l ) ( q ) r (1) l − r (2) l · · · ( q ) r (2 ul ) l y r l l . In the C (1) l case we get the following identity Theorem 2. Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) 1(1 − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )= X r (1)1 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)21 + ··· + r (2 u ( q ) r (1)1 − r (2)1 · · · ( q ) r (2 u y r r (1)2 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)22 + ··· + r (2 u − r (1)1 r (1)2 −···− r (2 u r (2 u ( q ) r (1)2 − r (2)2 · · · ( q ) r (2 u y r · · · X r (1) l − ≥ ... ≥ r (2 ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r (2 ul − l − − r (1) l − r (1) l − −···− r (2 ul − l − r (2 ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r (2 ul − l − y r l − l − X r (1) l ≥ ... ≥ r ( ul ) l ≥ u l ≥ q r (1)2 l + ··· + r ( ul )2 l − r (1) l ( r (1) l − + r (2) l − ) −···− r ( ul ) l ( r (2 ul − l − + r (2 ul ) l − ) ( q ) r (1) l − r (2) l · · · ( q ) r ( ul ) l y r l l . The plan of the paper is as follows. In Section 1 we recall some fundamental resultsconcerning affine Lie algebras and their modules. Next, we introduce a notion of a quasi-particle and relations among quasi-particles of the same color. In Section 2, we recallthe definition of the principal subspace. In Section 3 we construct quasi-particle basesof principal subspaces of standard module L B (1) l ( k Λ ) and genrealized Verma module N B (1) l ( k Λ ) of B (1) l . We will start with finding relations among quasi-particles of differentcolors. Using these relations along with relations among quasi-particles of the samecolor we will construct the spanning sets of principal subspaces. Then we will introduceoperators which we use in the proof of linear independance. At the end of this section wewill find character formulas. Section 4 is devoted to the construction of bases of principalsubspaces in the case of C (1) l . 1. Preliminaries
In this paper we are interested in principal subspaces of two different types of affine Liealgebras, so it will be convenient to introduce principal subspace (and latter quasi-particlemonomials) for modules of a general untwisted affine Lie algebra.1.1.
Modules of affine Lie algebra.
Let g be a complex simple Lie algebra of type X l , h a Cartan subalgebra of g and R the corresponding root system. Let Π = { α , ..., α l } bea set of simple roots and let θ denote the maximal root. Denote with R + ( R − ) the set ofpositive (negative) roots. Then we have the triangular decomposition g = n − ⊕ h ⊕ n + . We use h· , ·i to denote the standard symmetric invariant nondegenerate bilinear formon g which enables us to identify h with its dual h ∗ . We normalize this form so that h α, α i = 2 for every long root α ∈ R . For α ∈ R let α ∨ = α h α,α i denote the correspondingcoroot. Denote by Q = P li =1 Z α i and P = P li =1 Z ω i the root and weight lattices, where ω , . . . , ω l are the fundamental weights of g , that is (cid:10) ω i , α ∨ j (cid:11) = δ i,j , i, j = 1 , . . . , l . Forlater use, we set ω = 0.The associated affine Lie algebra of type X (1) l is the Lie algebra b g = g ⊗ C [ t, t − ] ⊕ C c, where c is a non-zero central element (cf. [K]). For every x ∈ g and j ∈ Z , we write x ( j )for elements x ⊗ t j . Commutation relations are then given by[ c, x ( j )] = 0 , [ x ( j ) , y ( j )] = [ x, y ] ( j + j ) + h x, y i j δ j + j , c, or any x, y ∈ g , j, j , j ∈ Z . We introduce the following subalgebras of b g b g ≥ = M n ≥ g ⊗ t n ⊕ C c, b g < = M n< g ⊗ t n , L ( n + ) = n + ⊗ C [ t, t − ] , L ( n + ) ≥ = L ( n + ) ⊗ C [ t ] and L ( n + ) < = L ( n + ) ⊗ t − C [ t ] . By adjoining the degree operator d to the Lie algebra b g , such that(1.1) [ d, x ( j )] = jx ( j ) , [ d, c ] = 0 , one obtains the affine Kac-Moody algebra e g = b g ⊕ C d, (cf. [K]).Set e h = h ⊕ C c ⊕ C d . The form h· , ·i on h extends naturally to e h . We shall identify e h with its dual space e h ∗ via this form. We define δ ∈ e h ∗ by δ ( d ) = 1, δ ( c ) = 0 and δ ( h ) = 0,for every h ∈ h . Set α = δ − θ and α ∨ = c − θ ∨ . Then { α ∨ , α ∨ , . . . , α ∨ l } is a set of simplecoroots of e g .Define fundamental weights of e g by (cid:10) Λ i , α ∨ j (cid:11) = δ i,j for i, j = 0 , , . . . , l and Λ i ( d ) =0. Denote by L (Λ ), L (Λ ), . . . , L (Λ l ) standard e g -modules, that is integrable highestweight e g -modules with highest weights Λ , Λ , . . . , Λ l and with highest weight vectors v Λ , v Λ , . . . , v Λ l .The object of our study is b g -module N X (1) l ( k Λ ) and its irreducible quotient L X (1) l ( k Λ ),where level k is a positive integer. The generalized Verma module N X (1) l ( k Λ ) is definedas the induced b g -module N X (1) l ( k Λ ) = U ( b g ) ⊗ U ( b g ≥ ) C v k Λ , where C v k Λ is 1-dimensional b g ≥ -module, such that cv k Λ = kv k Λ and ( g ⊗ t j ) v k Λ = 0 , for every j ≥
0. From the Poincar´e-Birkhoff-Witt theorem, we have N X (1) l ( k Λ ) ∼ = U ( b g < ) ⊗ C C v k Λ as vector spaces. Set v N X (1) l ( k Λ ) = 1 ⊗ v k Λ . We view b g -modules N X (1) l ( k Λ ) and L X (1) l ( k Λ ) as e g -modules, where d acts as(1.2) dv N X (1) l ( k Λ ) = 0(see [LL]).Throughout this paper, we will write x ( m ) for the action of x ⊗ t m on any b g -module,where x ∈ g and j ∈ Z . .2. Definition of quasi-particles.
For every positive integer k , the generalized Vermamodule N X (1) l ( k Λ ) has a structure of vertex operator algebra (see [LL], [Li1], [MP]), where v N X (1) l ( k Λ ) is the vacuum vector. For x ∈ g Y ( x ( − v N X (1) l ( k Λ ) , z ) = x ( z ) = X m ∈ Z x ( m ) z − m − is vertex operator associated with the vector x ( − v N X (1) l ( k Λ ) ∈ N X (1) l ( k Λ ). In addition,on the irreducible b g module L X (1) l ( k Λ ) we have the structure of a simple vertex operatoralgebra and all the level k standard modules are modules for this vertex operator algebra(cf. [LL], [MP]). Remark 1.2.1.
Later, we will realize standard modules of level k > as submodules oftensor products of standard modules of level . Vertex operators x ( z ) , where x ∈ g , acton the tensor product of standard modules of level as Lie algebra elements (cf. [LL] ). For α i ∈ Π and r >
0, we have x rα i ( z ) := x α i ( z ) r = Y (( x α i ( − r v N X (1) l ( k Λ ) , z ) . For given i ∈ { , . . . , l } , r ∈ N and m ∈ Z define a quasi-particle of color i , charge r and energy − m by(1.3) x rα i ( m ) = Res z z m + r − x α i ( z ) · · · x α i ( z ) | {z } r factors . We shall say that vertex operator x rα i ( z ) represents the generating function for quasi-particles of color i and charge r .From (1.3) it follows x rα i ( m ) = X m ,...,m r ∈ Z m + ··· + m r = m x α i ( m r ) · · · x α i ( m ) , where the family of operators( x α i ( m r ) · · · x α i ( m )) m ,...,m r ∈ Z m + ··· + m r = m on the highest weight module is a summable family (cf. [LL]).We shall usually denote a product of quasi-particles of color i by b ( α i ). We say thatmonomial b ( α i ) is a monochromatic monomial colored with color-type r i , if the sum ofall quasi-particle charges in monomial b ( α i ) is r i . We say that a monochromatic quasi-particle monomial b ( α i ) = x n r (1) i ,i α i ( m r (1) i ,i ) · · · x n ,i α i ( m ,i ) , is of color-type r i , charge-type(1.4) (cid:16) n r (1) i ,i , . . . , n ,i (cid:17) where 0 ≤ n r (1) i ,i ≤ · · · ≤ n ,i , and dual-charge-type(1.5) (cid:16) r (1) i , r (2) i , . . . , r ( s ) i (cid:17) , here r (1) i ≥ r (2) i ≥ · · · ≥ r ( s ) i ≥ s ≥ , if (1.4) and (1.5) are conjugate partitions of r i (cf. [Bu], [G]).Since quasi-particles of the same color commute, we arrange quasi-particles of the samecolor and the same charge so that the values m p,i , for 1 ≤ p ≤ r (1) i , form a decreasingsequence of integers from right to left.1.2.1. Relations among quasi-particles of the same color . Relations among par-ticles of the same color, that is, expressions for the products of the form x nα ( m ) x n ′ α ( m ′ ),where α = α i , n, n ′ ∈ N and m, m ′ ∈ Z , can be divided into two sets. The first set ofrelations is described by the following proposition. Proposition 1.2.1 (cf. [LL], [Li1], [MP]) . Let k ∈ N . Then we have the followingrelations on the standard module L X (1) l ( k Λ ) : x α ( z ) k +1 = 0 , (1.6) x β ( z ) k +1 = 0 , (1.7) where α ∈ R is a long root and β ∈ R is a short root. The second set of relations was proved in [F], [FS], [G] and [JP]:
Lemma 1.2.1.
For fixed
M, j ∈ Z and ≤ n ≤ n ′ any of n monomials from the set A = { x nα ( j ) x n ′ α ( M − j ) , x nα ( j + 1) x n ′ α ( M − j − , . . . ,. . . , x nα ( j + 2 n − x n ′ α ( M − j − n + 1) } can be expressed as a linear combination of monomials from the set { x nα ( m ) x n ′ α ( m ′ ) : m + m ′ = M } \ A and monomials which have as a factor quasi-particle x ( n ′ +1) α ( j ′ ) , j ′ ∈ Z . Corollary 1.2.1.
Fix n ∈ N and j ∈ Z . The elements from the set A = { x nα ( m ) x nα ( m ′ ) : m ′ − n < m ≤ m ′ } can be expressed as linear combinations of monomials x nα ( m ) x nα ( m ′ ) , such that m ≤ m ′ − n and monomials with quasi-particle x ( n +1) α i ( j ′ ) , j ′ ∈ Z . In order to find relations among quasi-particles, which are differently colored, we willuse the commutator formula among vertex operators:[ Y ( x α ( − v N ( k Λ ) , z ) , Y ( x β ( − r v N ( k Λ ) , z )](1.8) = X j ≥ ( − j j ! (cid:18) ddz (cid:19) j z − δ (cid:18) z z (cid:19) Y ( x α ( j ) x β ( − r v N ( k Λ ) , z ) , where α, β ∈ R , (cf. [FHL]). . Principal subspaces and Quasi-particle monomials
Principal subspace.
Let k ∈ N and let Λ = k Λ . Set v L X (1) l ( k Λ ) to be the highestweight vector of the standard module L X (1) l ( k Λ ). As in [FS] and [G], we define theprincipal subspace W L X (1) l ( k Λ ) of the standard module L X (1) l ( k Λ ) as W L X (1) l ( k Λ ) = U ( L ( n + )) v L X (1) l ( k Λ ) , and the principal subspace W N X (1) l ( k Λ ) of the generalized Verma module N X (1) l ( k Λ ) as W N X (1) l ( k Λ ) = U ( L ( n + )) v N X (1) l ( k Λ ) , where U ( L ( n + )) is the universal enveloping algebra of Lie algebra L ( n + ).2.2. Quasi-particle monomials.
In Subsections 3.2 and 4.2, we construct bases of W L X (1) l ( k Λ ) and W N X (1) l ( k Λ ) consisting of vectors of the form bv L X (1) l ( k Λ ) and bv N X (1) l ( k Λ ) ,where monomials b are composed of monochromatic monomials b ( α i ), where i = 1 , . . . , l .Here we extend definitions of monochromatic monomials to polychromatic monomials.We use the same terminology for the products of generating functions.For (polychromatic) monomial b = b ( α l ) · · · b ( α ) , = x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) · · · x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) , we will say it is of charge-type R ′ = (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1) l ,l , . . . , n , (cid:17) , where 0 ≤ n r (1) i ,i ≤ . . . ≤ n ,i , dual-charge-type R = (cid:16) r (1) l , . . . , r ( s l ) l ; . . . ; r (1)1 , . . . , r ( s )1 (cid:17) , where r (1) i ≥ r (2) i ≥ . . . ≥ r ( s i ) i ≥ r l , . . . , r ) , where r i = r (1) i X p =1 n p,i = s i X t =1 r ( t ) i and s i ∈ N , if for every color i , 1 ≤ i ≤ l , (cid:16) n r (1) i ,i , . . . , n ,i (cid:17) and (cid:16) r (1) i , r (2) i , . . . , r ( s ) i (cid:17) are mutually conjugate partitions of r i (cf. [Bu], [G]). emark 2.2.1. In the case of affine Lie algebra of type C (1) l the bases of W L C (1) l ( k Λ ) and W N C (1) l ( k Λ ) will generate polychromatic monomials b ( α ) · · · b ( α l ) whose charge-types, dual-charge types and color-types are defined similarly. We compare the (polychromatic) monomials as in [Bu] and [G]. We state(2.1) b < b if one of the following conditions holds:(1) (cid:16) n r (1) l ,l , . . . , n , (cid:17) < (cid:16) n r (1) l ,l , . . . , n , (cid:17) ,i.e., if there is u ∈ N , such that n ,i = n ,i , n ,i = n ,i , . . . , n u − ,i = n u − ,i , and u = r (1) i + 1 or n u,i < n u,i ;(2) (cid:16) n r (1) l ,l , . . . , n , (cid:17) = (cid:16) n r (1) l ,l , . . . , n , (cid:17) , (cid:16) m r (1) l ,l , . . . , m , (cid:17) < (cid:16) m r (1) l ,l , . . . , m , (cid:17) i.e. if there is u ∈ N , 1 ≤ u ≤ r i , such that m ,i = m ,i , m ,j = m ,j , . . . m u − ,i = m u − ,i and m u,i < m u,i . Remark 2.2.2.
Similarly definition is for the C (1) l case. First we compare the charge-types and if the charge-types are the same, we compare the sequences of energies, startingfrom color i = l . Characters of principal subspaces.
We extend the definition of character of theprincipal subspaces W L X (1) l ( k Λ ) and W N X (1) l ( k Λ ) from [Bu].Denote by ch W L X (1) l ( k Λ ) the characters of W L X (1) l ( k Λ ) :ch W L X (1) l ( k Λ ) = X m,r ,...,r l ≥ dim W L X (1) l ( k Λ )( m,r ,...,r l ) q m y r · · · y r l l , (2.2)where q, y , . . . y l are formal variables and W L X (1) l ( k Λ )( m,r ,...,r l ) = W L X (1) l ( k Λ ) − mδ + r α + ... + r l α l is the e h -weight subspace of weight − mδ + r α + . . . + r l α l .In the same way we define the character of the principal subspace W N X (1) l ( k Λ ) .3. The case B (1) l Principal subspaces for affine Lie algebra of type B (1) l . Let g be of the type B l , l ≥
2. The root system R of g will be identified as a subset R l , where { ǫ , . . . , ǫ l } denotes the usual orthonormal basis of the R l . We have the base of R :Π = { α = ǫ − ǫ , . . . , α l − = ǫ l − − ǫ l , α l = ǫ l } , the set of positive roots: R + = { ǫ i − ǫ j : i < j } ∪ { ǫ i + ǫ j : i = j } ∪ { ǫ i : 1 ≤ i ≤ l } and the highest root θ = ǫ + ǫ = α + 2 α + · · · + 2 α l . or each root α ∈ R + we have a root vector x α ∈ g . We define a one-dimensionalsubalgebras of g n α = C x α , α ∈ R + , with the corresponding subalgebras of b g L ( n α ) = n α ⊗ C [ t, t − ] . We denote with U B (1) l the vector space U B (1) l = U ( L ( n α l )) · · · U ( L ( n α )) . Using the same argument as Georgiev in [G] we can prove
Lemma 3.1.1.
Let k ≥ . We have W L B (1) l ( k Λ ) = U B (1) l v L B (1) l ( k Λ ) ,W N B (1) l ( k Λ ) = U B (1) l v N B (1) l ( k Λ ) . (cid:3) By extending the construction of bases of principal subspaces in the case of affineLie algebra of type B (1)2 , we shall construct bases for the principal subspaces W L B (1) l ( k Λ ) and W N B (1) l ( k Λ ) , which will be generated by quasi-particles acting on the highest weightvectors. We start with the principal subspaces W L B (1) l ( k Λ ) .3.2. The spanning set of W L B (1) l ( k Λ ) . In order to find a set of quasi-particle monomialswhich generate a basis of W L B (1) l ( k Λ ) , first we complete the list of relations among quasi-particles. Here we find the expressions for the products of the form x n i α i ( m i ) x n ′ j α j ( m ′ j ),where i = 1 , . . . , l − j = i + 1 n i , n ′ j ∈ N and m i , m ′ j ∈ Z .First, notice that as in the case of B (1)2 , we have: Lemma 3.2.1.
Let n l − , n l ∈ N be fixed. One has (cid:18) − z l − z l (cid:19) min { n l , n l − } x n l α l ( z l ) x n l − α l − ( z l − ) v N B (1) l ( k Λ ) (3.1) ∈ z − min { n l , n l − } l W N B (1) l ( k Λ ) [[ z l , z l − ]] . (cid:3) Now, fix color i , 1 ≤ i ≤ l − Lemma 3.2.2.
Let n i +1 , n i ∈ N be fixed. One has a) ( z − z ) n i x n i α i ( z ) x n i +1 α i +1 ( z ) = ( z − z ) n i x n i +1 α i +1 ( z ) x n i α i ( z );b) ( z − z ) n i +1 x n i α i ( z ) x n i +1 α i +1 ( z ) = ( z − z ) n i +1 x n i +1 α i +1 ( z ) x n i α i ( z ) . Proof.
Follows by direct computation employing the commutator formula 1.8 for vertexoperators. (cid:3)
From Lemma 3.2.2 follows:
Lemma 3.2.3.
Let n i +1 , n i ∈ N be fixed. One has (cid:18) − z i z i +1 (cid:19) min { n i +1 ,n i } x n i +1 α i +1 ( z i +1 ) x n i α i ( z i ) v N B (1) l ( k Λ ) (3.2) ∈ z − min { n i +1 ,n i } i +1 W N B (1) l ( k Λ ) [[ z i +1 , z i ]] . roof. (3.2) is immediate from creation property of vertex operators (cf. [LL]) and Lemma3.2.2, i.e.( z i +1 − z i ) min { n i +1 ,n i } x n i +1 α i +1 ( z i +1 ) x n i α i ( z i ) = ( z i +1 − z i ) min { n i +1 ,n i } x n i α i ( z i ) x n i +1 α i +1 ( z i +1 ) . (cid:3) Remark 3.2.1.
The obtained relation in Lemma is generalization of relations ob-tained in [G] for the case of A (1) l − . Later, we will show similar relation for quasi-particlescorresponding to the long roots in the C (1) l case (see Lemma ). Using the above considerations and relations among quasi-particles of the same color,induction on charge-type and total energie of quasi-particle monomials ([Bu], [G]), followsthe proof of the following proposition:
Proposition 3.2.1.
The set B W LB (1) l ( k Λ0) = (cid:26) bv L B (1) l ( k Λ ) : b ∈ B W LB (1) l ( k Λ0) (cid:27) , where (3.3) B W LB (1) l ( k Λ0) = [ n r (1)1 , ≤ ... ≤ n , ≤ k...n r (1) l − ,l − ≤ ... ≤ n ,l − ≤ kn r (1) l ,l ≤ ... ≤ n ,l ≤ k or, equivalently, [ r (1)1 ≥···≥ r ( k )1 ≥ ··· r (1) l − ≥···≥ r ( k ) l − ≥ r (1) l ≥···≥ r (2 k ) l ≥ { b = b ( α l ) · · · b ( α ) = x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) · · · x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m p,i ≤ − n p,i + P r (1) i − q =1 min { n q,i − , n p,i } − P p>p ′ > min { n p,i , n p ′ ,i } , ≤ p ≤ r (1) i , ≤ i ≤ l − m p +1 ,i ≤ m p,i − n p,i if n p +1 ,i = n p,i , ≤ p ≤ r (1) i − , ≤ i ≤ l − m p,l ≤ − n p,l + P r (1) l − q =1 min { n q,l − , n p,l } − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l ≤ m p,l − n p,l if n p,l = n p +1 ,l , ≤ p ≤ r (1) l − , and where r (1)0 = 0 , spans the principal subspace W L B (1) l ( k Λ ) . (cid:3) Proof of linear independence.
Here we introduce operators which we use in ourproof of linear independence of the set B W LB (1) l ( k Λ0) .3.3.1.
Projection π R . We start with a projection π R , which is a generalisation of pro-jection introduced in [Bu]. If we restrict the action of the Cartan subalgebra h = h ⊗ W L B (1) l (Λ ) of level 1 standard modules L B (1) l (Λ ), we get the directsum of vector spaces: W L B (1) l (Λ ) = M u l ,...,u ≥ W L B (1) l (Λ )( u l ,...,u ) , where W L B (1) l (Λ )( u l ,...,u ) = W BL (Λ ) u l α l + ··· + u α s the weight subspace of weight u l α l + · · · + u α ∈ Q. Fix a level k >
1. The principal subspace W L B (1) l ( k Λ ) has a realization as a subspaceof the tensor product of k principal subspaces W L B (1) l (Λ ) of level 1 W L B (1) l ( k Λ ) ⊂ W L B (1) l (Λ ) ⊗ · · · ⊗ W L B (1) l (Λ ) ⊂ L B (1) l (Λ ) ⊗ k , where v L B (1) l ( k Λ ) = v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) | {z } k factors is the highest weight vector of weight k Λ .For a chosen dual-charge-type R = (cid:16) r (1) l , . . . , r (2 k ) l ; r (1) l − , . . . , r ( k ) l − ; . . . ; r (1)1 , . . . , r ( k )1 (cid:17) and the corresponding charge-type R ′ R ′ = (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)1 , , . . . , n , (cid:17) , denote with π R the projection of principal subspace W L B (1) l ( k Λ ) to the subspace W L B (1) l (Λ )( µ ( k ) l ; r ( k ) l − ; ... ; r ( k )1 ) ⊗ · · · ⊗ W L B (1) l (Λ )( µ (1) l ; r (1) l − ; ... ; r (1)1 ) , where µ ( t ) l = r (2 t ) l + r (2 t − l , for every 1 ≤ t ≤ k (cf. Figure 1 and 2). The projection can be in an obvious waygeneralized to the space of formal Laurent series with coefficients in W L B (1) l (Λ ) ⊗ · · · ⊗ W L B (1) l (Λ ) . Let(3.4) x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n ,l α l ( z ,l ) 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 , )be a generating function of the chosen dual-charge-type R and the corresponding charge-type R ′ .From the relation x α i ( z ) = 0, 1 ≤ i ≤ l −
1, on the principal subspace W L B (1) l (Λ ) andthe definition of the action of Lie algebra on the modules, follows that n p,i generatingfunctions x α i ( z p,i ) (1 ≤ p ≤ r (1) i ), whose product generates a quasi-particle of charge n p,i ,“are placed at” the first (from right to left) n p,i tensor factors: x n ( k ) p,i α i ( z p,i ) ⊗ x n ( k − p,i α i ( z p,i ) ⊗ · · · ⊗ x n (2) p,i α i ( z p,i ) ⊗ x n (1) p,i α i ( z p,i ) , where 0 ≤ n ( t ) p,i ≤ , ≤ t ≤ k, n (1) p,i ≥ n (2) p,i ≥ . . . ≥ n ( k − p,i ≥ n ( k ) p,i , n p,i = k X t =1 n ( t ) p,i , for every every p , 1 ≤ p ≤ r (1) i , so that, in the t -tensor factor from the right (1 ≤ t ≤ k ),we have: · · · x n ( t ) r ( t ) i ,i α i ( z r ( t ) i ,i ) · · · x n ( t )1 ,i α i ( z ,i ) · · · v L B (1) l (Λ ) ⊗ · · · , s in the in the example in Figure 1, where each box represents n ( t ) p,i . n r (1) i ,i n r ( k ) i ,i n ,i ... ... ... ... ... ...... r (1) i r (2) i r (3) i v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) r ( k − i r ( k − i r ( k ) i Figure 1.
Sketch of projection π R for color i , 1 ≤ i ≤ l − x α l ( z ) = 0 on the principal subspace W L B (1) l (Λ ) , follows that atmost, two generating functions of color i = l “can be placed at” every tensor factor. If n p,l (1 ≤ p ≤ r (1) l ) is an even number, then two generating functions x α l ( z p,l ) “are placedat” the first n p,l tensor factors (from right to left) and if n p,l is an odd number, then twogenerating functions x α l ( z p,l ) “are placed at” the first n p,l − tensor factors (from right toleft), and the last generating function x α l ( z p,l ) “is placed at” n p,l − + 1 tensor factor: x n ( k ) p,l α ( z p,l ) ⊗ x n ( k − p,l α ( z p,l ) ⊗ · · · ⊗ x n (2) p,l α ( z p,l ) ⊗ x n (1) p,l α ( z p,l ) , where 0 ≤ n ( t ) p,l ≤ , n (1) p,l ≥ n (2) p,l ≥ . . . ≥ n ( k − p,l ≥ n ( k ) p,l , n p,l = k X t =1 n ( t ) p,l , for every every p , 1 ≤ p ≤ r (1) l , so that at most one n ( t ) p,l (1 ≤ t ≤ k ) can be 1 and so that,in every t -tensor factor from the right (1 ≤ t ≤ k ), we have: · · · ⊗ x n ( t ) r (2 t − l ,l α l ( z r (2 t − ,l ) · · · x n ( t ) r (2 t ) l ,l α l ( z r (2 t ) l ,l ) · · · x n ( t )1 ,l α l ( z ,l ) · · · v L B (1) l (Λ ) ⊗ · · · . This situation is shown in the example in Figure 2. n r (1) l ,i n ,l ... ... ... ... ... ...... r (1) l r (2) l r (3) l r (4) l v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) v L B (1) l (Λ ) r (2 k − l r (2 k − l r (2 k − l r (2 k ) l Figure 2.
Sketch of projection π R for color i = l ow, we have the projection of the generating function (3.4) π R x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n , α ( z , ) v L B (1) l ( k Λ ) (3.5)=C x n ( k ) r (2 k − l ,l α l ( z r (2 k − l ,l ) · · · x n ( k ) r (2 k ) l ,l α l ( z r (2 k ) l ,l ) · · · x n ( k )1 ,l α l ( z ,l ) x n ( k ) r (1) l − ,l − α l − ( z r (1) l − ,l − ) · · ·· · · x n ( k )1 ,l − α l − ( z ,l − ) · · · x n ( k ) r ( k )1 , α ( z r ( k )1 , ) · · · x n ( k )1 , α ( z , ) v L B (1) l (Λ ) ⊗ . . . ⊗⊗ x n (1) r (1) l ,l α l ( z r (1) l ,l ) · · · x n (1) r (2) l ,l α l ( z r (2) l ,l ) · · · x n (1)1 ,l α l ( z ,l ) x n (1) r (1) l − ,l − α l − ( z r (1) 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)1 , α ( z , ) v L B (1) l (Λ ) , where C ∈ C ∗ .From the above considerations follows that the projection of the monomial vector bv L B (1) l ( k Λ ) , where b ∈ B W LB (1) l ( k Λ0) is a monomial(3.6) b = x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) · · · x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , )colored with color-type ( r l , . . . , r ) , charge-type R ′ and dual-charge-type R , is a coefficientof the projection of the generating function (3.5) which we denote as π R bv L B (1) l ( k Λ ) . Remark 3.3.1.
Here we note, that if ¯ b ∈ B W LB (1) l ( k Λ0) is monomial such that it is ofcharge-type (¯ n ¯ r (1) l ,l , . . . , ¯ n ,l ; . . . ; ¯ n ¯ r (1)1 , , . . . , ¯ n , ) , dual-charge-type R = (cid:16) ¯ r (1) l , . . . , ¯ r (2 k ) l ; . . . ; ¯ r (1)1 , . . . , ¯ r ( k )1 (cid:17) and such that R ′ < R ′ , then, from the definition of projection, follows that π R ¯ bv L B (1) l ( k Λ ) = 0 . This argument we will use in our proof of linear independance.
A coefficient of an intertwining operator . Denote by Y ( · , z ) the vertex op-erator which determines the structure of L B (1) l (Λ )-module L B (1) l (Λ ). We shall use thecoefficient of intertwining operator I ( · , z ) of type (cid:18) L B (1) l (Λ ) L B (1) l (Λ ) L B (1) l (Λ ) (cid:19) , defined by I ( w, z ) v = exp( zL ( − Y ( v, − z ) w, w ∈ L B (1) l (Λ ) , v ∈ L B (1) l (Λ )(3.7)(cf. [FHL]). If we use the commutator formula (cid:20) x ( m ) , I ( v L B (1) l (Λ ) , z ) (cid:21) = X j ≥ (cid:18) mj (cid:19) z m − j I ( x ( j ) v L B (1) l (Λ ) , z ) cf. (2.13) in [Li2]), where x α i ( m ) ∈ b g for α i ∈ Π, we have: (cid:20) x α i ( m ) , I ( v L B (1) l (Λ ) , z ) (cid:21) = 0 . We define the following coefficient of an intertwining operator A ω = Res z z − I ( v L B (1) l (Λ ) , z )and by (3.7), we have(3.8) A ω v L B (1) l (Λ ) = v L B (1) l (Λ ) . Let s ≤ k . We consider the operator on L B (1) l (Λ ) ⊗ · · · ⊗ L B (1) l (Λ ) defined as(3.9) A s = 1 ⊗ · · · ⊗ A ω ⊗ ⊗ · · · ⊗ | {z } s − . If we act with this operator on the vector v L B (1) l ( k Λ ) = v L B (1) l (Λ ) ⊗· · ·⊗ v L B (1) l (Λ ) , it followsfrom (3.8):(3.10) A s ( v L B (1) l ( k Λ ) ) = v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) ⊗ v L B (1) l (Λ ) ⊗ v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) | {z } s − . Set b ∈ B W LB (1) l ( k Λ0) as in (3.6). It follows that A s π R bv L B (1) l ( k Λ ) is the coefficient of A s π R x n r (1)2 , α ( z r (1)2 , ) · · · x sα ( z , ) v L B (1) l ( k Λ ) . From (3.10), it follows that operator A ω acts only on the s -th tensor factor from theright: ⊗ x n ( s ) r (2 s − l ,l α l ( z r (2 s − l ,l ) · · · x n ( s ) r (2 s ) l ,l α l ( z r (2 s ) l ,l ) · · · x n ( s )1 ,l α l ( z ,l ) x n ( s ) r ( s )1 , α ( z r ( s )1 , ) · · · x α ( z , ) v L B (1) l (Λ ) ⊗ , where 0 ≤ n ( s ) p,i ≤
1, for 1 ≤ p ≤ r ( s ) i and 0 ≤ n ( s ) p,l ≤
2, for 1 ≤ p ≤ r (2 s − l (see (3.5)).Since A ω commutes with the generating functions, in the s -th tensor facor from the right,we have · · · ⊗ x n ( s ) r (2 s − l ,l α l ( z r (2 s − l ,l ) · · · x n ( s ) r (2 s ) l ,l α l ( z r (2 s ) l ,l ) · · · x n ( s )1 ,l α l ( z ,l )(3.11) x n ( s ) r ( s )1 , α ( z r ( s )1 , ) · · · x α ( z , ) v L B (1) l (Λ ) ⊗ · · · . .3.3. Simple current operator e ω . In the same way as in [Bu] in the proof of linearindependence, we use simple current operators e ω on level 1 standard modules for B (1) l , l ≥ e ω : L B (1) l (Λ ) → L B (1) l (Λ ) , associated to ω ∈ h , which are uniquely characterized by their action on the highestweight vector(3.12) e ω v L B (1) l (Λ ) = v L B (1) l (Λ ) and by their commutation relations(3.13) x α ( z ) e ω = e ω z α ( ω ) x α ( z ) , for all α ∈ R , or, written by components,(3.14) x α ( m ) e ω = e ω x α ( m + α ( ω )) , for all α ∈ R and m ∈ Z (cf. [DLM], [Li2]).Let s ≤ k . We define the linear bijection(3.15) B s = 1 ⊗ . . . ⊗ ⊗ e ω ⊗ ⊗ . . . ⊗ | {z } s − . If we act with this operator (3.15) on the vector v L B (1) l ( k Λ ) = v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) ,we get B s ( v L B (1) l ( k Λ ) ) = v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) ⊗ v L B (1) l (Λ ) ⊗ v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) | {z } s − . Now it follows that in (3.11) we can commute B s to the left and obtain · · · ⊗ x n ( s ) r (2 s ) l ,l α l ( z r (2 s − l ,l ) · · · x n ( s ) r (2 s ) l ,l α l ( z r (2 s ) l ,l ) · · · x n ( s )1 ,l α l ( z ,l ) · · · x n ( s ) r ( k )1 , α ( z r ( k )1 , ) z r ( k )1 , · · · x α ( z , ) z , v L B (1) l (Λ ) ⊗ · · · . By taking the corresponding coefficients, we have A s π R bv L B (1) l ( k Λ ) = B s π R b + v L B (1) l ( k Λ ) where the monomial b + : b + = b + ( α l ) · · · b + ( α ) , is such that b + ( α i ) = b ( α i ) , ≤ i ≤ lb + ( α ) = x n r (1)1 , α ( m r (1)1 , + 1) · · · x sα ( m , + 1)= x n r (1)1 , α ( m + r (1)1 , ) · · · x sα ( m +1 , ) . .3.4. Operator e α i . For every simple root α i ∈ Π, 1 ≤ i ≤ l , we define on the level 1standard module L B (1) l (Λ ), the “Weyl group translation” operator e α i by(3.16) e α i = exp x − α i (1) exp ( − x α i ( − x − α i (1) exp x α i (0) exp ( − x − α i (0)) exp x α i (0) , (cf. [K]). Using (3.16) we see that Lemma 3.3.1.
Let i , ( ≤ i ≤ l − ) be fixed. For every i ′ = i, i + 1 , we have: a) e α v B (1) l (Λ ) = − x α i ( − v B (1) l (Λ ) ; b) x α i ( z ) e α i = z e α i x α i ( z ) ; c) x α i +1 ( z ) e α i = z − e α i x α i +1 ( z );d) x α i ′ ( z ) e α i = e α i x α i ′ ( z ) . (cid:3) Set 1 ⊗ · · · ⊗ ⊗ e α i ⊗ e α i ⊗ · · · ⊗ e α i | {z } s factors where s ≤ k . From Lemma 3.3.1 a), it now follows(1 ⊗ · · · ⊗ ⊗ e α i ⊗ e α i ⊗ · · · ⊗ e α i ) v L B (1) l ( k Λ ) =( − s v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) ⊗ x α i ( − v L B (1) l (Λ ) ⊗ x α i ( − v L B (1) l (Λ ) ⊗ · · · ⊗ x α i ( − v L B (1) l (Λ ) | {z } s factors . Let 1 ≤ i ≤ l − b be a monomial b = b ( α i +1 ) b ( α i ) x sα i ( − s )(3.17) = x n r (1) i +1 ,i +1 α i +1 ( m r (1) i +1 ,i +1 ) · · · x n ,i +1 α i +1 ( m ,i +1 ) x n r (1) i ,i α i ( m r (1) i ,i ) · · · x n ,i α i ( m ,i ) x sα i ( − s ) , of dual-charge-type R = (cid:16) r (1) i +1 , . . . , r ( p ) i +1 ; r (1) i , . . . , r ( s ) i , . . . , (cid:17) , where p = k if i + 1 < l or p = 2 k if i + 1 = l .Assume that i < l −
1. The situation of i = l − π R be the projection of principal subspace W L B (1) l (Λ ) ⊗ · · · ⊗ W L B (1) l (Λ ) onthe vector space W L B (1) l (Λ )( r ( k ) i +1 ;0) ⊗ · · · ⊗ W L B (1) l (Λ )( r ( s ) i +1 ; r ( s ) i ) ⊗ · · · ⊗ W L B (1) l (Λ )( r (1) i +1 ; r (1) i ) . The projection π R b (cid:18) v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) (cid:19) of the monomial vector b (cid:18) v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) (cid:19) is a coefficient of the generatingfunction π R x n r (1) i +1 ,i +1 α i +1 ( z r (1) i +1 ,i +1 ) · · · x n ,i +1 α i +1 ( z ,i +1 ) x n r (1) i ,i α i ( z r (1) i ,i ) · · · x n ,i α i ( z ,i ) v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) ⊗ x α i ( − v L B (1) l (Λ ) ⊗ · · · ⊗ x α i ( − v L B (1) l (Λ ) (cid:19) = Cx n ( k ) r ( k ) i +1 ,i +1 α i +1 ( z r ( k ) i +1 ,i +1 ) · · · x n ( k )1 ,i +1 α i +1 ( z ,i +1 ) v L B (1) l (Λ ) ⊗ · · · ⊗⊗ x n ( s ) r (2 s − i +1 ,i +1 α i +1 ( z r (2 s − i +1 ,i +1 ) · · · x n ( s ) r (2 s ) i +1 ,i +1 α i +1 ( z r (2 s ) i +1 ,i +1 ) · · · x n ( s )1 ,i +1 α i +1 ( z ,i +1 ) x n ( s ) r ( s ) i ,i α i ( z r ( s ) i ,i ) · · · x n ( s )2 ,i α i ( z ,i ) e α i v L B (1) l (Λ ) ⊗ · · · ⊗⊗ x n (1) r (1) i +1 ,i +1 α i +1 ( z r (1)2 ,i +1 ) · · · x n (1) r (2) i +1 ,i +1 α i +1 ( z r (2) i +1 ,i +1 ) · · · x n (1)2 ,i +1 α ( z ,i +1 ) x n (1)1 ,i +1 α i +1 ( z ,i +1 ) x n (1) r (1) i ,i α i ( z r (1) i ,i ) · · · x n (1)2 ,i α i ( z ,i ) e α i v L B (1) l (Λ ) , where C ∈ C ∗ (see (3.5)). We shift operator 1 ⊗ . . . ⊗ e α i ⊗ e α i ⊗ . . . ⊗ e α i all the way tothe left using commutation relations b), c) in Lemma 3.3.1(1 ⊗ · · · ⊗ ⊗ e α i ⊗ e α i ⊗ · · · ⊗ e α i ) π R ′ b ′ (cid:18) v L B (1) l (Λ ) ⊗ · · · ⊗ v L B (1) l (Λ ) (cid:19) , where R ′ = (cid:16) r (1) i +1 , . . . , r ( s ) i +1 ; r (1) i − , . . . , r ( s ) i − (cid:17) and b ′ = b ′ ( α i +1 ) b ′ ( α i )= · · · x n ,i +1 α i +1 ( m ,i +1 − n (1)1 ,i +1 − · · · − n ( s )1 ,i +1 ) x n r (1) i αi ( m r (1) i ,i + 2 n r (1) i ) · · · x n ,i α i ( m ,i + 2 n ,i )= x n r (1) i +1 ,i +1 α i +1 ( m ′ r (1) i +1 ,i +1 ) · · · x n ,i +1 α i +1 ( m ′ ,i +1 ) x n r (1) i αi ( m ′ r (1)1 ,i ) · · · x n ,i α i ( m ′ ,i ) . In the proof of linear independence, we use the following proposition:
Proposition 3.3.1.
Let b ( ) be an element of the set B W LB (1) l ( k Λ0) . Then the monomial b ′ is an element of the set B W LB (1) l ( k Λ0) .Proof.
The proposition follows by considering the possible situation for n p,i , 2 ≤ p ≤ r (1) i ,and n p,i +1 , 1 ≤ p ≤ r (1) i +1 , from which it follows that m p,i comply the defining conditionsof the set B W LB (1) l ( k Λ0) . As before we will assume that i < l −
1, since for the i = l − B (1)2 (see [Bu]).(1) For n p,i = ¯ s ≤ s , we have m ′ p,i = m p,i + 2¯ s ≤ − ¯ s − p − s + 2¯ s = − ¯ s − p − s nd m ′ p +1 ,i = m p +1 ,i + 2¯ s ≤ − s + m p,i + 2¯ s = m ′ p,i − s for n p +1 ,i = n p,i . (2) For n p,i +1 ≤ s , we have m ′ p,i +1 = m p,i +1 − n p,i +1 ≤ − n p,i +1 − X p>p ′ > { n p,i +1 , n p ′ ,i +1 } + r (1) i X q =1 min { n p,i +1 , n q,i } − n p,i +1 = − n p,i +1 − X p>p ′ > { n p,i +1 , n p ′ ,i +1 } + r (1) i X q =2 min { n p,i +1 , n q,i } and m ′ p +1 ,i +1 = m p +1 ,i +1 − n p,i +1 ≤ m p,i +1 − n p,i +1 − n p,i +1 = m ′ p,i +1 − n p,i +1 for n p +1 ,i +1 = n p,i +1 . (3) For n p,i +1 > s , we have: m ′ p,i +1 = m p,i +1 − s ≤ − n p,i +1 − X p>p ′ > { n p,i +1 , n p ′ ,i +1 } + r (1) i X q =1 min { n p,i +1 , n q,i } − s = − n p,i +1 − X p>p ′ > { n p,i +1 , n p ′ ,i +1 } + r (1) i X q =2 min { n p,i +1 , n q,i } and m ′ p +1 ,i +1 = m p +1 ,i +1 − s ≤ m p,i +1 − n p,i +1 − s = m ′ p,i +1 − n p,i +1 for n p +1 ,i +1 = n p,i +1 . (cid:3) The proof of linear independence . By Proposition 3.2.1 the set B W LB (1) l ( k Λ0) ofmonomial vectors bv L B (1) l ( k Λ ) spans W L B (1) l ( k Λ ) . We prove linear independence of this setby induction on l and charge-type R ′ of monomials b ∈ B W LB (1) l ( k Λ0) . Linear independencefor the case l = 2 is proved in [Bu]. Remark 3.3.2.
The idea of proof is that for the “minimum” quasi-monomial vector ofdual-charge-type R in a given subset of B W LB (1) l ( k Λ0) , we define the projection π R , which“kills” all monomial vectors higher in the linear lexicographic ordering “ < ” (see Remark ). e fix 1 < i ≤ l and the dual-charge-type(3.18) R = (cid:16) r (1) l , . . . , r (2 k ) l ; r (1) l − , . . . , r ( k ) l − ; . . . ; r (1) i , . . . , r ( k ) i (cid:17) ,r (1) l ≥ . . . ≥ r (2 k ) l ,r (1) l − ≥ . . . ≥ r ( k ) l − , · · · r (1) i ≥ . . . ≥ r ( k ) i . Denote by A R ⊂ B W LB (1) l ( k Λ0) the set of monomial vectors bv L B (1) l ( k Λ ) , where monomials b are of dual-charge-type (3.18) and the corresponding charge-type R ′ = (cid:16) n r (1) l ,l , . . . , n ,l ; n r (1) l − ,l − , . . . , n ,l − ; . . . ; n r (1) i ,i , . . . , n ,i (cid:17) ,n r (1) i ,l ≤ . . . ≤ n ,l ≤ k,n r (1) l − ,l − ≤ . . . ≤ n ,l − ≤ k, · · · n r (1) i ,i ≤ . . . ≤ n ,i ≤ k. Note, that monomials b ∈ B W LB (1) l ( k Λ0) b = b ( α l ) b ( α l − ) · · · b ( α i ) =(3.19) = x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) x n r (1) l − ,l − α l − ( m r (1) l − ,l − ) · · · x n ,l − α l − ( m ,l − ) · · ·· · · x n r (1) i ,i α i ( m r (1) i ,i ) · · · x n ,i α i ( m ,i ) , of the charge-type R ′ and dual-charge type R can be realised as elements of the principalsubspace in the case of the affine Lie algebra of type B (1) l − i +1 .Under consideration at the subsection 3.3.1, the default dual-charge-type R determinesthe projection π R on the vector space W L B (1) l (Λ )( µ ( k ) l ; r ( k ) l − ; ... ; r ( k ) i ) ⊗ · · · ⊗ W L B (1) l (Λ )( µ (1) l ; r (1) l − ; ... ; r (1) i ) ⊂ W L B (1) l (Λ ) ⊗ · · · ⊗ W L B (1) l (Λ ) . Since the restriction of B (1) l -module L (Λ ) on the subalgebra B (1) l − i +1 is a direct sum of thelevel one B (1) l − i +1 -modules L B (1) l − i +1 (Λ ), with a highest weight vector v L B (1) l (Λ ) = v L B (1) l − i +1 (Λ ) ,it follows that(3.20) π R bv L B (1) l ( k Λ ) ∈ W L B (1) l − i +1 (Λ ) ⊗ · · · ⊗ W L B (1) l − i +1 (Λ ) ⊂ W L B (1) l (Λ ) ⊗ · · · ⊗ W L B (1) l (Λ ) , where W L B (1) l − i +1 (Λ ) = W L (Λ )0 α + ··· +0 α i − is a principal subspace of standard B (1) l − i +1 -module L B (1) l − i +1 (Λ ) ⊂ L B (1) l (Λ ).On (3.20) we can act with operators A n ,i , B n ,i and e α i defined for vertex operatoralgebra L B (1) l − i +1 (Λ ), whose properties are described in subsections 3.3.2, 3.3.3 and 3.3.4.With these operators we “move” monomial vectors π R bv L B (1) l ( k Λ ) from one space to an-other until we get vectors of the form π R b ( α l ) b ( α l − ) v L B (1) l ( k Λ ) ∈ π R A . In [Bu] has beenproven that the set π R A of vectors π R b ( α l ) b ( α l − ) v L B (1) l ( k Λ ) is a linearly independent set.By using the previous observations, we can prove: heorem 3. The set B W LB (1) l ( k Λ0) forms a basis for the principal subspace W L B (1) l ( k Λ ) ⊂ L B (1) l ( k Λ ) .Proof. Assume that we have(3.21) X a ∈ A c a b a v L B (1) l ( k Λ ) = 0 , where A is a finite non-empty set and b a ∈ B W LB (1) l ( k Λ0) . Assume that all b a are the same color-type ( r l , . . . , r ). Let b be the smallest monomialin the linear lexicographic ordering “ < ” b = b ( α l ) · · · b ( α ) b ( α )= x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) · · · x n r (1)2 , α ( m r (1)2 , ) · · · x n , α ( − m , ) x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( − j ) , of dual-charge-type R = (cid:16) r (1) l , . . . , r (2 k ) l ; . . . ; r (1)2 , . . . , r ( k )2 ; r (1)1 , . . . , r ( n , )1 (cid:17) , and charge-type(3.22) R ′ = (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)2 , , . . . , n , ; n r (1)1 , , . . . , n , (cid:17) , such that c a = 0. Then for every other monomial in (3.21) we have m , ≥ − j. Dual-charge-type R determines projection π R of W L B (1) l (Λ ) ⊗ . . . ⊗ W L B (1) l (Λ ) | {z } k factors on the vec-tor space W L B (1) l (Λ )( µ ( k ) l ; ... ; r ( k )2 ;0) ⊗ . . . ⊗ W L B (1) l (Λ )( µ ( n , l ; ... ; r ( n , ;0) ⊗⊗ W L B (1) l (Λ )( µ ( n , l ; ... ; r ( n , ; r ( n , ) ⊗ · · · W L B (1) l (Λ )( µ (1) l ; ... ; r (1)2 ; r (1)1 ) , where µ ( t ) l = r (2 t ) l + r (2 t − l . By Remark 3.3.1, π R maps to zero all monomial vectors b a v L B (1) l ( k Λ ) such that b a has alarger charge-type in the linear lexicographic ordering “ < ” than (3.22). So, in (3.23)(3.23) X a c a π R b a v L B (1) l ( k Λ ) = 0 , we have a projection of b a v L B (1) l ( k Λ ) , where b a are of charge-type (3.22). On (3.23), weact with A n , = 1 ⊗ . . . ⊗ A ω ⊗ ⊗ . . . ⊗ | {z } n , − , then, from 3.3.2 and 3.3.3 follows A n , X a ∈ A c a π R b a v L B (1) l ( k Λ ) ! = e n , X a ∈ A c a π R b + a v L B (1) l ( k Λ ) ! , here e n , = 1 ⊗ . . . ⊗ e ω ⊗ ⊗ . . . ⊗ | {z } n , − . After leaving out the invertible operator e n , , we get X a c a π R b + a v L B (1) l ( k Λ ) = 0 , where b + a ∈ A R ⊂ B W LB (1) l ( k Λ0) are the same charge-type as b a in (3.21). We act with A n , and e n , until j becomes − n , . Assume that after n , , − j steps we got X a c a π R b a ( α l ) · · · b a ( α l ) · · · b + a ( α ) x n , α ( − n , ) v L B (1) l ( k Λ ) = 0 , where monomial b + a ( α ) x n , α ( − n , ) is of color i = 1 and b R ( α l ) · · · b + R ( α ) x n , α ( − n , ) v L B (1) l ( k Λ ) ∈ A R . Now, from the subsection 3.3.4 follows π R b ( α l ) · · · b ( α ) b + ( α ) x n , α ( − n , ) v L B (1) l ( k Λ ) = (1 ⊗ · · · ⊗ e α ⊗ e α · · · ⊗ e α ) b ′ ( α ) b ′ ( α ) v L B (1) l ( k Λ ) , where b ( α l ) · · · b ′ ( α ) b ′ ( α ) does not have a quasi-particle of charge n , . Monomial b ( α l ) · · · b ′ ( α ) b ′ ( α ) is of dual-charge-type R − = (cid:16) r (1) l , . . . , r (2 k ) l ; . . . ; r (1)2 , . . . , r ( k )2 ; r (1)1 − , . . . , r ( n , )1 − (cid:17) , and charge-type (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)2 , , . . . , n , ; n r (1)1 , , . . . , n , (cid:17) , such that (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)2 , , . . . , n , ; n r (1)1 , , . . . , n , (cid:17) << (cid:16) n r (1) l ,l , . . . , n ,l ; . . . ; n r (1)2 , , . . . , n , ; n r (1)1 , , . . . , n , , n , (cid:17) . From Proposition 3.3.1, it follows that with the described process, we get elements fromthe set B W LB (1) l ( k Λ0) . We continue with the described algorithm, until we get monomial“colored” only with colors i = l and i = l −
1. Thus, under the consideration at thebeginning of this subsection, it follows c a = 0. (cid:3) Characters of the principal subspace W L B (1) l ( k Λ ) . We use the following expres-sions (3.24),(3.25), (3.26), and (3.27) to determine the character of W L B (1) l ( k Λ ) . Theseexpressions can be easy proved by using induction on the level k ∈ N of the standardmodule L B (1) l ( k Λ ). Lemma 3.4.1.
For the given color-type ( r l , r l − , . . . , r , r ) , charge-type (cid:16) n r (1) l ,l , . . . , n ,l ; n r (1) l − ,l − , . . . , n ,l − ; . . . ; n r (1)2 , , . . . , n , ; n r (1)1 , , . . . , n , (cid:17) and dual-charge-type (cid:16) r (1) l , r (2) l , . . . , r (2 k ) l ; r (1) l − , r (2) l − , . . . , r ( k ) l − ; . . . ; r (1)2 , r (2)2 , . . . , r ( k )2 ; r (1)1 , r (2)1 , . . . , r ( k )1 (cid:17) , e have: r (1) l X p =1 r (1) l − X q =1 min { n p,l , n q,l − } = k X s =1 r ( s ) l − ( r (2 s − l + r (2 s ) l ) , (3.24) r (1) i X p =1 r (1) i − X q =1 min { n p,i , n q,i − } = k X s =1 r ( s ) i r ( s ) i − , ≤ i ≤ l − , (3.25) r (1) i X p =1 ( X p>p ′ > { n p,i , n p ′ ,i } + n p,i ) = k X s =1 r ( s ) i , ≤ i ≤ l − , (3.26) r (1) l X p =1 ( X p>p ′ > { n p,l , n p ′ ,l } + n p,l ) = k X s =1 r ( s ) l . (3.27) (cid:3) We also need the combinatorial identity1( q ) r = X j ≥ p r ( j ) q j , (3.28)where 1( q ) r = 1(1 − q )(1 − q ) · · · (1 − q r ) ,r > p r ( j ) is the number of partition of j with most r parts (cf. [A]).Now, from the definition of the set B W LB (1) l ( k Λ0) and (3.24), (3.25), (3.26), (3.27), (3.28)follows the character formula:
Theorem 4. ch W L B (1) l ( k Λ ) = X r (1)1 ≥ ... ≥ r ( k )1 ≥ q r (1)21 + ··· + r ( k )21 ( q ) r (1)1 − r (2)1 · · · ( q ) r ( k )1 y r X r (1)2 ≥ ... ≥ r ( k )2 ≥ q r (1)22 + ··· + r ( k )22 − r (1)1 r (1)2 −···− r ( k )1 r ( k )2 ( q ) r (1)2 − r (2)2 · · · ( q ) r ( k )2 y r · · · X r (1) l − ≥ ... ≥ r ( k ) l − ≥ q r (1)2 l − + ··· + r ( k )2 l − − r (1) l − r (1) l − −···− r ( k ) l − r ( k ) l − ( q ) r (1) l − − r (2) l − · · · ( q ) r ( k ) l − y r l − l − X r (1) l ≥ ... ≥ r (2 k ) l ≥ q r (1)2 l + ··· + r (2 k )2 l − r (1) l − ( r (1) l + r (2) l ) −···− r ( k ) l − ( r (2 k − l + r (2 k ) l ) ( q ) r (1) l − r (2) l · · · ( q ) r (2 k ) l y r l l . (cid:3) .5. The basis of the W N B (1) l ( k Λ ) . Using the relations among quasi-particles of the sameand different colors, using the proof of the theorem 3 with the same arguments as in [Bu],we can prove:
Theorem 5.
The set B W NB (1) l ( k Λ0) = (cid:26) bv N B (1) l ( k Λ ) : b ∈ B W NB (1) l ( k Λ0) (cid:27) , where (3.29) B W NB (1) l ( k Λ0) = [ n r (1)1 , ≤ ... ≤ n , ...n r (1) l − ,l − ≤ ... ≤ n ,l − n r (1) l ,l ≤ ... ≤ n ,l or, equivalently, [ r (1)1 ≥···≥ ··· r (1) l − ≥···≥ r (1) l ≥···≥ { b = b ( α l ) · · · b ( α ) = x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) · · · x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m p,i ≤ − n p,i + P r (1) i − q =1 min { n q,i − , n p,i } − P p>p ′ > min { n p,i , n p ′ ,i } , ≤ p ≤ r (1) i , ≤ i ≤ l − m p +1 ,i ≤ m p,i − n p,i if n p +1 ,i = n p,i , ≤ p ≤ r (1) i − , ≤ i ≤ l − m p,l ≤ − n p,l + P r (1) l − q =1 min { n q,l − , n p,l } − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l ≤ m p,l − n p,l if n p,l = n p +1 ,l , ≤ p ≤ r (1) l − , where r (1)0 = 0 , is the base of the principal subspace W N B (1) l ( k Λ ) . (cid:3) Characters of the principal subspace W N B (1) l ( k Λ ) . From the above theorem and(3.24)-(3.28) we can write the character formulas of principal subspace W N B (1) l ( k Λ ) : Theorem 6. ch W N B (1) l ( k Λ ) (3.30) = X r (1)1 ≥ ... ≥ r ( u ≥ u ≥ q r (1)21 + ··· + r ( u ( q ) r (1)1 − r (2)1 · · · ( q ) r ( u y r X r (1)2 ≥ ... ≥ r ( u ≥ u ≥ q r (1)22 + ··· + r ( u − r (1)1 r (1)2 −···− r ( u r ( u ( q ) r (1)2 − r (2)2 · · · ( q ) r ( u y r · · · X r (1) l − ≥ ... ≥ r ( ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r ( ul − l − − r (1) l − r (1) l − −···− r ( ul − l − r ( ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r ( ul − l − y r l − l − r (1) l ≥ ... ≥ r (2 ul ) l ≥ u l ≥ q r (1)2 l + ··· + r (2 ul )2 l − r (1) l − ( r (1) l + r (2) l ) −···− r ( ul ) l − ( r (2 ul − l + r (2 ul ) l ) ( q ) r (1) l − r (2) l · · · ( q ) r (2 ul ) l y r l l . (cid:3) We can determine the character of principal subspace W N B (1) l ( k Λ ) using the Poincar´e-Birkhoff-Witt theorem, since we have W N B (1) l ( k Λ ) ∼ = U ( L ( n + ) < ) . Set { x α ( m ) : α ∈ R + , m < } a basis of the Lie algebra L ( n + ) < with a total order onthis set: x ( m ) ≤ y ( m ′ ) ⇔ x < y or x = y and m < m ′ . Now, we can write a basis of U ( L ( n + ) < ): x α ( m ) · · · x α ( m s ) x α + α ( m ) · · · x α + α ( m s ) · · · x α + α + ··· + α l ( m l ) · · · (3.31) · · · x α + α + ··· + α l ( m s l l ) x α +2 α + ··· +2 α l ( m l +1 ) · · · x α +2 α + ··· +2 α l ( m s l +1 l +1 ) · · ·· · · x α + α + ··· +2 α l ( m l − ) · · · x α + α + ··· +2 α l ( m s l − l − ) · · ·· · · x α l − ( m l − ) · · · x α l − ( m s l − l − ) · · · x α l − +2 α l ( m l − ) · · · x α l − +2 α l ( m s l − l − ) x α l ( m l ) · · · x α l ( m s l l ) , with m i ≤ · · · ≤ m s i i , s i ∈ N for i = 1 , . . . , l .It follows that the subspace U ( L ( n + ) < ) ( m,r ,...,r l ) has basis (3.31), where( m, r , . . . , r l − , r l ) = ( l X i =1 s i X j =1 m ji , s + s + · · · + s l − , . . . , s l − + · · · + s l − , s l +2 s l +1 + · · · + s l ) . The bijection map U ( L ( n + ) < ) → W N B (1) l ( k Λ ) b bv N B (1) l ( k Λ ) maps weighted subspace U ( L ( n + ) < ) ( m,r ,...,r n ) on W N B (1) l ( k Λ )( m,r ,...,r l ) . Thus, we also have(3.32) ch W BN ( k Λ ) = Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l )1(1 − q m y y · · · y l ) · · · − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )Now from (3.30) and (3.32) follows a new identity of Rogers-Ramanujan’s type: heorem 7. Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )= X r (1)1 ≥ ... ≥ r ( u ≥ u ≥ q r (1)21 + ··· + r ( u ( q ) r (1)1 − r (2)1 · · · ( q ) r ( u y r X r (1)2 ≥ ... ≥ r ( u ≥ u ≥ q r (1)22 + ··· + r ( u − r (1)1 r (1)2 −···− r ( u r ( u ( q ) r (1)2 − r (2)2 · · · ( q ) r ( u y r · · · X r (1) l − ≥ ... ≥ r ( ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r ( ul − l − − r (1) l − r (1) l − −···− r ( ul − l − r ( ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r ( ul − l − y r l − l − X r (1) l ≥ ... ≥ r (2 ul ) l ≥ u l ≥ q r (1)2 l + ··· + r (2 ul )2 l − r (1) l − ( r (1) l + r (2) l ) −···− r ( ul ) l − ( r (2 ul − l + r (2 ul ) l ) ( q ) r (1) l − r (2) l · · · ( q ) r (2 ul ) l y r l l . (cid:3) The case C (1) l Principal subspaces for affine Lie algebra of type C (1) l . Let g be of the type C l , l ≥
3. We have the following base of the root system R :Π = (cid:26) α = 1 √ ǫ − ǫ ) , . . . , α l − = 1 √ ǫ l − − ǫ l ) , α l = √ ǫ l (cid:27) , (where { ǫ , . . . , ǫ l } is as before orthonormal basis of the R l ), the set of positive roots: R + = (cid:26) √ ǫ i − ǫ j ) : i < j (cid:27) ∪ (cid:26) √ ǫ i + ǫ j ) : i = j (cid:27) ∪ n √ ǫ i : 1 ≤ i ≤ l o and the highest root θ = √ ǫ = 2 α + · · · + 2 α l − + α l (cf: [K]). We denote the vector space U C (1) l = U ( L ( n α )) · · · U ( L ( n α l )) , where L ( n α ) = n α ⊗ C [ t, t − ] , n α = C x α , α ∈ R + . Now, we have: emma 4.1.1. Let k ≥ . We have W L C (1) l ( k Λ ) = U C (1) l v L C (1) l ( k Λ ) ,W N C (1) l ( k Λ ) = U C (1) l v N C (1) l ( k Λ ) . (cid:3) The spanning set of W L C (1) l ( k Λ ) . Here we establish relations among quasi-particlesof colors i = l − i = l and relations among quasi-particles of colors i = 1 , . . . , l and j = i ± C (1)2 , we have: Lemma 4.2.1.
Let n l − , n l ∈ N be fixed. One has (cid:18) − z l z l − (cid:19) min { n l − , n l } x n l − α l − ( z l − ) x n l α l ( z l ) v N C (1) l ( k Λ ) ∈ z − min { n l − , n l } l − W N C (1) l ( k Λ ) [[ z l − , z l ]] . (4.1) (cid:3) Using the commutator formula for vertex operators we can prove:
Lemma 4.2.2.
Let ≤ i ≤ l − , n i +1 , n i ∈ N be fixed. One has a) ( z − z ) n i x n i α i ( z l ) x n i +1 α i +1 ( z ) = ( z − z ) n i x n i +1 α i +1 ( z ) x n i α i ( z ) . b) ( z − z ) n i +1 x n i α i ( z ) x n i +1 α i +1 ( z ) = ( z − z ) n i +1 x n i +1 α i +1 ( z i +1 ) x n i α i ( z ) . (cid:3) Using the same arguments as in proof of Lemma 3.2.3 follows:
Lemma 4.2.3. (cid:18) − z i z i +1 (cid:19) min { n i +1 ,n i } x n i α i ( z i ) x n i +1 α i +1 ( z i +1 ) v N C (1) l ( k Λ ) ∈ z − min { n i +1 ,n i } p,i +1 W N C (1) l ( k Λ ) [[ z i , z i +1 ]] . (4.2) (cid:3) Now, as in Subsection 3.2 follows
Proposition 4.2.1.
The set B W LC (1) l ( k Λ0) = (cid:26) bv L C (1) l ( k Λ ) : b ∈ B W LC (1) l ( k Λ0) (cid:27) , where (4.3) B W LC (1) l ( k Λ0) = [ n r (1)1 , ≤ ... ≤ n , ≤ k...n r (1) l − ,l − ≤ ... ≤ n ,l − ≤ kn r (1) l ,l ≤ ... ≤ n ,l ≤ k or, equivalently, [ r (1)1 ≥···≥ r (2 k )1 ≥ ··· r (1) l − ≥···≥ r (2 k ) l − ≥ r (1) l ≥···≥ r ( k ) l ≥ { b = b ( α ) · · · b ( α l ) = x n r (1)1 , α ( m r (1)1 , ) · · · x n ,l α l ( m ,l ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m p,l ≤ − n p,l − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l ≤ m p,l − n p,l if n p,l = n p +1 ,l , ≤ p ≤ r (1) l − m p,l − ≤ − n p,l − + P r (1) l q =1 min { n q,l , n p,l − } − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l − ≤ m p,l − − n p,l − if n p +1 ,l − = n p,l − , ≤ p ≤ r (1) l − − m p,i ≤ − n p,i + P r (1) i +1 q =1 min { n q,i +1 , n p,i } − P p>p ′ > min { n p,i , n p ′ ,i } , ≤ p ≤ r (1) i , ≤ i ≤ l − m p +1 ,i ≤ m p,i − n p,i if n p +1 ,i = n p,i , ≤ p ≤ r (1) i − , ≤ i ≤ l − , spans the principal subspace W L C (1) l ( k Λ ) . (cid:3) Proof of linear independence.
As we mentioned in the Introduction, we prove thelinear independence of the monomial vectors from Proposition 4.2.1 using the coefficient ofan intertwining operator, the simple current operator and the “Weyl group translation”operator. So, first we describe their properties, which we use in the proof of linearindependence of quasi-particle bases.4.3.1.
Projection π R . Fix a level k >
1. Consider the direct sum decomposition oftensor product of k principal subspaces W L C (1) l (Λ ) of level 1 standard modules L C (1) l (Λ ) W L C (1) l (Λ ) ⊗· · ·⊗ W L C (1) l (Λ ) = [ u (1)1 ≥···≥ u ( k )1 ≥ ··· u (1) l − ≥···≥ u ( k ) l − ≥ u (1) l ≥···≥ u ( k ) l ≥ W L C (1) l (Λ )( u ( k )1 ; ... ; u ( k ) l − ; u ( k ) l ) ⊗· · ·⊗ W L C (1) l (Λ )( u (1)1 ; ... ; u (1) l − ; u (1) l ) , where W L C (1) l (Λ )( u ( j )1 ; ... ; u ( j ) l − ; u ( j ) l ) is a h -weight subspace of weight P li =1 u ( j ) i α i , 1 ≤ j ≤ k ,and where v L C (1) l ( k Λ ) = v L C (1) l (Λ ) ⊗ · · · ⊗ v L C (1) l (Λ ) | {z } k factors is the highest weight vector of weight k Λ .For a chosen dual-charge-type R = (cid:16) r (1)1 , . . . , r (2 k )1 ; . . . ; r (1) l − , . . . , r (2 k ) l − ; r (1) l , . . . , r ( k ) l (cid:17) , set the projection π R of principal subspace W L C (1) l ( k Λ ) to the subspace W L C (1) l (Λ )( µ ( k )1 ; ... ; µ ( k ) l − ; r ( k ) l ) ⊗ · · · ⊗ W L C (1) l (Λ )( µ (1)1 ; ... ; µ (1) l − ; r (1) l ) , where µ ( t ) i = r (2 t ) i + r (2 t − i , for every 1 ≤ t ≤ k and 1 ≤ i ≤ l −
1. If we denote by the same symbol π R the generalization of this projection to the space of formal series with coefficients in W L C (1) l (Λ ) ⊗ · · · ⊗ W L C (1) l (Λ ) , then for a generating function(4.4) x n r (1)1 , α ( z r (1)1 , ) · · · x n , α ( z , ) · · · x n r (1) l − ,l − α l − ( z r (1) l − ,l − ) · · · x n ,l − α l − ( z ,l − ) x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n ,l α l ( z ,l ) , e have π R x n r (1)1 , α ( z r (1)1 , ) · · · x n ,l α l ( z ,l ) v L C (1) l ( k Λ ) (4.5) =C x n ( k ) r (2 k − , α ( z r (2 k − , ) · · · x n ( k ) r (2 k )1 , α ( z r (2 k )1 , ) · · · x n ( k )1 , α ( z , ) · · ·· · · x n ( k ) r (2 k − l − ,l − α l − ( z r (2 k − ,l − ) · · · x n ( k ) r (2 k ) l − ,l − α l − ( z r (2 k ) l − ,l − ) · · · x n ( k )1 ,l − α l − ( z ,l − ) x n ( k ) r ( k ) l ,l α l ( z r ( k ) l ,l ) · · · · · · x n ( k )1 ,l α l ( z ,l ) v L C (1) l (Λ ) ⊗⊗ . . . ⊗⊗ x n (1) r (1)1 , α ( z r (1)1 , ) · · · x n (1) r (2)1 , α ( z r (2)1 , ) · · · x n (1)1 , α ( z , ) · · · x n (1) r (1) l − ,l − α l − ( z r (1)1 ,l − ) · · · x n (1) r (2) l − ,l − α l − ( z r (2) l − ,l − ) · · · x n (1)1 ,l − α l − ( z ,l − ) x n (1) r (1) l ,l α l ( z r (1) l ,l ) · · · x n (1)1 ,l α l ( z ,l ) v L C (1) l (Λ ) , where C ∈ C ∗ ,0 ≤ n ( t ) p,l ≤ , n (1) p,l ≥ n (2) p,l ≥ . . . ≥ n ( k − p,l ≥ n ( k ) p,l , n p,l = n (1) p,l + n (2) p,l + · · · + n ( k − p,l + n ( k ) p,l and0 ≤ n ( t ) p,i ≤ , n (1) p,i ≥ n (2) p,i ≥ . . . ≥ n ( k − p,i ≥ n ( k ) p,i , n p,i = n (1) p,i + n (2) p,i + · · · + n ( k − p,i + n ( k ) p,i , for every t , 1 ≤ t ≤ k , and every p , 1 ≤ p ≤ r (1) i , 1 ≤ i ≤ l .In the projection (4.5), n p,l generating functions x α l ( z p,l ) (1 ≤ p ≤ r (1) l ), whose productgenerates a quasi-particle of charge n p,l , “are placed at” the first (from right to left) n p,l tensor factors. This can be shown as in the example in Figure 3, where each box represents n ( t ) p,l . The situation for the genarationg functions of colors 1 ≤ i ≤ l − n r (1) l ,l n r ( k ) l ,l n ,l ... ... ... ... ... ...... r (1) l r (2) l r (3) l v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) r ( k − l r ( k − l r ( k ) l Figure 3.
Sketch of projection π R for color i = l as in the example in Figure 4, where two generating functions x α i ( z p,i ) (1 ≤ p ≤ r (1) i )“are placed at” the first n p,i tensor factors (from right to left) if n p,i is an even numberand if n p,i is an odd number, then two generating functions x α i ( z p,i ) “are placed at” thefirst n p,i − tensor factors (from right to left), and the last generating function x α i ( z p,i ) “isplaced at” n p,i − + 1 tensor factor. Therefore, for a given monomial b ∈ B W LC (1) l ( k Λ0) (4.6) b = x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) · · · x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α ( m ,l ) r (1) i ,i n ,i ... ... ... ... ... ...... r (1) i r (2) i r (3) i r (4) i v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) v L C (1) l (Λ ) r (2 k − i r (2 k − i r (2 k − i r (2 k ) i Figure 4.
Sketch of projection π R for color 1 ≤ i ≤ l − r , . . . , r l ) , charge-type R ′ and dual-charge-type R , the projectionis a coefficient of the projection of the generating function (4.5) which we denote as π R bv L C (1) l ( k Λ ) . A coefficient of an intertwining operator . With A ω l , A ω l = Res z z − I ( v L C (1) l (Λ l ) , z ) , we denote the coefficient of an intertwining operator I ( · , z ) of type (cid:18) L C (1) l (Λ l ) L C (1) l (Λ l ) L C (1) l (Λ ) (cid:19) , defined by I ( w, z ) v = exp( zL ( − Y ( v, − z ) w, w ∈ L C (1) l (Λ l ) , v ∈ L C (1) l (Λ ) , (4.7)which commutes with the quasi-particles (see 3.3.2). From definition (4.7), we have(4.8) A ω l v L C (1) l (Λ ) = v L C (1) l (Λ l ) . Let s ≤ k . As in the case B (1) l we consider the operator on L C (1) l (Λ ) ⊗ · · · ⊗ L C (1) l (Λ ),which we denote by the same symbol as in 3.3.2 A s = 1 ⊗ · · · ⊗ A ω l ⊗ ⊗ · · · ⊗ | {z } s − . Set b ∈ B W LC (1) l ( k Λ0) as in (4.6). From the consideration in Subsection 4.3.1, it followsthat A s π R bv L C (1) l ( k Λ ) is the coefficient of A s π R x n r (1)1 , α ( z r (1)1 , ) · · · x sα l ( z ,l ) v L C (1) l ( k Λ ) , where operator A ω l acts only on the s -th tensor factor from the right · · · ⊗ x n ( s ) r (2 s − , α ( z r (2 s − , ) · · · x n ( s ) r (2 s )1 , α ( z r (2 s )1 , ) · · · x n ( s )1 , α ( z , ) · · · x n ( s ) r ( s ) l ,l α l ( z r ( s ) l ,l ) · · · x α ( z ,l ) v L C (1) l (Λ ) ⊗ · · · , here 0 ≤ n ( s ) p,l ≤
1, for 1 ≤ p ≤ r ( s ) l and 0 ≤ n ( s ) p,i ≤
2, for 1 ≤ p ≤ r (2 s − i , 1 ≤ i ≤ l − s -th tensor factor from the right, we have · · · ⊗ x n ( s ) r (2 s − , α ( z r (2 s − , ) · · · x n ( s ) r (2 s )1 , α ( z r (2 s )1 , ) · · · x n ( s )1 , α ( z , )(4.9) · · · x n ( s ) r ( s ) l ,l α l ( z r ( s ) l ,l ) · · · x α l ( z ,l ) v L C (1) l (Λ ) ⊗ · · · . Simple current operator e ω l . For ω l ∈ h we denoted by e ω l a simple currentoperator (cf. [Li2]) between the level 1 standard modules e ω l : L C (1) l (Λ ) → L C (1) l (Λ l ) , such that e ω l v L C (1) l (Λ ) = v L C (1) l (Λ l ) and(4.10) x α ( z ) e ω l = e ω l z α ( ω l ) x α ( z ) , for all α ∈ R .We can rewrite (4.9) as · · · ⊗ x n ( s ) r (2 s − , α ( z r (2 s − , ) · · · x n ( s ) r (2 s )1 , α ( z r (2 s )1 , ) · · · x n ( s )1 , α ( z , ) x n ( s ) r (2 s − l − ,l − α l − ( z r (2 s − l − ,l − ) · · · x n ( s ) r (2 s ) l − ,l − α l − ( z r (2 s ) l − ,l − ) · · · x n ( s )1 ,l − α ( z ,l − ) x n ( s ) r ( k ) l ,l α l ( z r ( k ) l ,l ) z r ( k ) l ,l · · · x α l ( z ,l ) z ,l e ω l v L C (1) l (Λ ) ⊗ · · · . By taking the corresponding coefficients, we have A s π R bv L C (1) l ( k Λ ) = B s π R b + v L C (1) l (Λ ) where B s = 1 ⊗ · · · ⊗ ⊗ e ω l ⊗ ⊗ · · · ⊗ | {z } s − and where the monomial b + : b + = b + ( α ) · · · b + ( α l ) , is such that b + ( α i ) = b ( α i ) , ≤ i ≤ l − b + ( α l ) = x n r (1) l ,l α l ( m r (1) l ,l + 1) · · · x sα l ( m , + 1)= x n r (1) l ,l α l ( m + r (1) l ,l ) · · · x sα l ( m +1 ,l ) . .3.4. Operator e α l . On the level 1 standard module L C (1) l (Λ ) we define the “Weylgroup translation” operator e α l e α l = exp x − α l (1) exp ( − x α l ( − x − α l (1) exp x α l (0) exp ( − x − α l (0)) exp x α l (0) . The properties of e α l , which we use in the proof of linear independance are described inthe following lemma. Lemma 4.3.1. a) e α l v L C (1) l (Λ ) = − x α l ( − v L C (1) l (Λ ) b) x α l ( z ) e α l = z e α l x α l ( z )c) x α l − ( z ) e α l = z − e α l x α l − ( z )d) x α i ( z ) e α l = e α l x α i ( z ) , ≤ i ≤ l − . (cid:3) Let b be a monomial b = b ( α ) · · · b ( α l − ) b ( α l ) x sα l ( − s )(4.11)= x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) · · · x n r (1) l − ,l − α l − ( m r (1) l − ,l − ) · · · x n ,l − α l − ( m ,l − ) x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) x sα l ( − s ) , of dual-charge-type R = (cid:16) r (1)1 , . . . , r (2 k )1 ; . . . ; r (1) l − , . . . , r (2 k ) l − ; r (1) l , . . . , r ( s ) l , . . . , (cid:17) . As in Subsection 4.3.1, let π R be the projection of principal subspace W L C (1) l (Λ ) ⊗ · · · ⊗ W L C (1) l (Λ ) on the vector space W L C (1) l (Λ )( µ ( k )1 ; ... ; µ ( k ) l − ;0) ⊗ · · · ⊗ W L C (1) l (Λ )( µ ( s +1)1 ; ... ; µ ( s +1) l − ;0) ⊗⊗ W L C (1) l (Λ )( µ ( s )1 ; ... ; µ ( s ) l − ; r ( s )1 ) ⊗ · · · ⊗ W L C (1) l (Λ )( µ (1)1 ; ... ; µ (1)1 ; r (1)1 ) . The projection π R b (cid:18) v L C (1) l (Λ ) ⊗ · · · ⊗ v L C (1) l (Λ ) (cid:19) of the monomial vector b (cid:18) v L C (1) l (Λ ) ⊗ · · · ⊗ v L C (1) l (Λ ) (cid:19) is a coefficient of the generatingfunction π R x n r (1)1 , α ( z r (1)1 , ) · · · x n , α ( z , ) · · · x n r (1) l ,l α l ( z r (1) l ,l ) · · · x n ,l α l ( z ,l ) (cid:18) v L C (1) l (Λ ) ⊗ · · · ⊗ v L C (1) l (Λ ) ⊗ x α l ( − v L C (1) l (Λ ) ⊗ · · · ⊗ x α l ( − v L C (1) l (Λ ) (cid:19) = C · · · x n ( k ) r (2 k − , α ( z r (2 k − , ) · · · x n ( k ) r (2 k )1 , α ( z r (2 k )1 , ) · · · x n ( k )1 , α ( z , ) · · · x n ( k ) r (2 k − l − ,l − α l − ( z r (2 k − l − ,l − ) · · · x n ( k ) r (2 k ) l − ,l − α l − ( z r (2 k ) l − ,l − ) · · · x n ( k )1 ,l α l ( z ,l ) v L C (1) l (Λ ) ⊗ · · · ⊗⊗ x n ( s ) r (2 s − , α ( z r (2 s − , ) · · · x n ( s ) r (2 s )1 , α ( z r (2 s )1 , ) · · · x n ( s )1 , α ( z , ) · · · x n ( s ) r (2 s − l − ,l − α l − ( z r (2 s − l − ,l − ) · · · x n ( s ) r (2 s ) l − ,l − α l − ( z r (2 s ) l − ,l − ) · · · x n ( s )1 ,l − α l − ( z ,l − ) n ( s ) r ( s ) l ,l α l ( z r ( s ) l ,l ) · · · x n ( s )2 ,l α l ( z ,l ) e α l v L C (1) l (Λ ) ⊗ · · · ⊗⊗ x n (1) r (1)1 , α ( z r (1)1 , ) · · · x n (1) r (2)1 , α ( z r (2)1 , ) · · · x n (1)2 , α ( z , ) x n (1)1 , α ( z , ) · · · x n (1) r (1) l − ,l − α l − ( z r (1) l − ,l − ) · · · x n (1) r (2) l − ,l − α l − ( z r (2) l − ,l − ) · · · x n (1)1 ,l − α l − ( z ,l − ) x n (1) r (1) l ,l α l ( z r (1) l ,l ) · · · x n (1)2 ,l α l ( z ,l ) e α l v L C (1) l (Λ ) , (see (4.5)). Now if we shift operator 1 ⊗ · · · ⊗ e α l ⊗ e α l ⊗ · · · ⊗ e α l all the way to the leftwe get (1 ⊗ · · · ⊗ e α l ⊗ e α l ⊗ · · · ⊗ e α l ) π R ′ b ′ (cid:18) v L C (1) l (Λ ) ⊗ . . . ⊗ v L C (1) l (Λ ) (cid:19) , where R ′ = (cid:16) r (1)1 , . . . , r (2 s )1 ; · · · ; r (1) l − , . . . , r ( s ) l − (cid:17) and b ′ = b ′ ( α ) · · · b ′ ( α l − ) b ′ ( α l )= x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , )= x n r (1) l − ,l − α l − ( m r (1) l − ,l − − n (1) r (1) l − ,l − − · · · − n ( s ) r (1)1 , ) · · · x n ,l − α l − ( m ,l − − n (1)1 ,l − − · · · − n ( s )1 ,l − ) x n r (1) l αl ( m r (1) l ,l + 2 n r (1) l ) · · · x n ,l α l ( m ,l + 2 n ,l )= x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) x n r (1) l − ,l − α l − ( m ′ r (1) l − ,l − ) · · · x n ,l − α l − ( m ′ ,l − ) x n r (1) l αl ( m ′ r (1) l ,l ) · · · x n ,l α l ( m ′ ,l ) . Lemma 4.3.2. If b ( ) is an element of the set B W LC (1) l ( k Λ0) , then the monomial b ′ ,from the above consideration, is an element of the set B W LC (1) l ( k Λ0) .Proof.
This Lemma easy follows by considering the possible situations for n p,l , 2 ≤ p ≤ r (1) l and n p,l − , 1 ≤ p ≤ r (1) l − from which follows that m ′ p,i , l − ≤ i ≤ l comply thedefining conditions of the set B W LC (1) l ( k Λ0) . (cid:3) The proof of linear independence of the set B W LC (1) l ( k Λ0) . In this section,we prove the following theorem:
Theorem 8.
The set B W LC (1) l ( k Λ0) forms a basis for the principal subspace W L C (1) l ( k Λ ) of L C (1) l ( k Λ ) .Proof. Assume that we have(4.12) X a ∈ A c a b a v L C (1) l ( k Λ ) = 0 , where A is a finite non-empty set and b a ∈ B W LC (1) l ( k Λ0) . ssume that all b a are the same color-type ( r , . . . , r l ).Let b be the smallest monomial in the linear lexicographic ordering “ < ” b = b ( α ) · · · b ( α l − ) b ( α l )= x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) · · ·· · · x n r (1) l − ,l − α l − ( m r (1) l − ,l − ) · · · x n ,l − α l − ( m ,l − ) x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( − j ) , of charge-type(4.13) (cid:16) n r (1)1 , , . . . , n , ; · · · ; n r (1) l − ,l − , . . . , n ,l − ; n r (1) l ,l , . . . , n ,l (cid:17) such that c a = 0 and such that, for every b a in (4.12), we have m ,l ≥ − j. Denote by R = (cid:16) r (1)1 , . . . , r (2 k )1 ; . . . ; r (1) l − , . . . , r ( n ,l − ) l − ; r (1) l , . . . , r ( n ,l ) l (cid:17) , the dual-charge-type of b . For every 1 ≤ t ≤ k such that µ ( t ) i = r (2 t ) i + r (2 t − i where 1 ≤ i ≤ l − π R be the projection of W L C (1) l (Λ ) ⊗ . . . ⊗ W L C (1) l (Λ ) | {z } k factors on the vectorspace W L C (1) l (Λ )( µ ( k )1 ; ... ;0) ⊗ . . . ⊗ W CL (Λ )( µ ( n ,l +1)1 ; ... ;0) ⊗ W L C (1) l (Λ )( µ ( n ,l )1 ; ... ; r ( n ,l )1 ) ⊗· · · W L C (1) l (Λ )( µ (1)1 ; ... ; r (1) l ) . It is not hard to see that the projection π R maps to zero all monomial vectors b a v L C (1) l ( k Λ ) such that b a has a larger charge-type in the linear lexicographic ordering “ < ” than (4.13).So, in (4.12) we have a projection of b a v L C (1) l ( k Λ ) , where b a are of charge-type (4.13)(4.14) X a ∈ A c a π R b a v L C (1) l ( k Λ ) = 0 . On (4.14), we act with operators A n ,l = 1 ⊗ · · · ⊗ A ω l ⊗ ⊗ · · · ⊗ | {z } n ,l − and B n ,l = 1 ⊗ · · · ⊗ e ω l ⊗ ⊗ · · · ⊗ | {z } n ,l − until j becomes − n ,l . In that case, we get X a c a π R b a ( α ) · · · b a ( α l − ) b + a ( α l ) x n ,l α l ( − n ,l ) v L C (1) l ( k Λ ) = 0 , where b + a ( α l ) x n ,l α l ( − n ,l ) is of color i = l and b a ( α ) · · · b a ( α l − ) b + a ( α l ) x n ,l α l ( − n ,l ) v L C (1) l ( k Λ ) ∈ B W LC (1) l ( k Λ0) . From the subsection 4.3.4 follows π R b a ( α ) · · · b a ( α l − ) b + a ( α l ) x n ,l α l ( − n ,l ) v L C (1) l ( k Λ ) = (1 ⊗ · · · ⊗ e α l ⊗ e α l · · · ⊗ e α l ) b a ( α ) · · · b ′ ( α l − ) b ′ ( α l ) v L C (1) l ( k Λ ) , here b a ( α ) · · · b ′ a ( α l ) does not have a quasi-particle of charge n ,l . b ( α ) · · · b ′ ( α l − ) b ′ ( α l )is of dual-charge-type (cid:16) r (1)1 , . . . , r (2 k )1 ; . . . ; r (1) l − , . . . , r (2 k ) l − ; r (1) l − , . . . , r ( n ,l ) l − (cid:17) , and charge-type (cid:16) n r (1)1 , , . . . , n , ; . . . ; n r (1) l − ,l − , . . . , n ,l − ; n r (1) l ,l , . . . , n ,l (cid:17) , such that (cid:16) n r (1) i , , . . . , n , ; . . . ; n r (1) l ,l , . . . , n ,l (cid:17) < (cid:16) n r (1)2 , , . . . , n , ; . . . ; n r (1) l ,l , . . . , n ,l , n ,l (cid:17) . From Remark 4.3.2, it follows that we get elements from the set B W LC (1) l ( k Λ0) .We apply the described processes on 4.14, until we get monomial “colored” with colors1 ≤ i ≤ l −
1. Assume that after a finite number of steps we get(4.15) X a ∈ A c a π R b a ( α ) · · · b a ( α l − ) v L C (1) l ( k Λ ) = 0 , where b a ( α ) · · · b a ( α l − ) ∈ A R − ⊂ B W LC (1) l ( k Λ0) . By A R − we denote the set of monomial vectors of dual-charge type R − = (cid:16) r (1)1 , . . . , r (2 k )1 ; . . . ; r (1) l − , . . . , r (2 k ) l − (cid:17) . From the condition x α i ( z ) = 0, (1 ≤ i ≤ l − W L A (1) l − (2Λ ) ⊗ . . . ⊗ W L A (1) l − (2Λ ) | {z } k factors , where W L A (1) l − (2Λ ) = W L C (1) l (Λ )0 · α l is the principal subspace of the standard module L A (1) l − (2Λ ) ⊂ L A (1) l − (Λ ) of the affine Lie algebra A (1) l − , with the highest weight vector v L A (1) l − (Λ ) = v L C (1) l (Λ ) . Denote by W L A (1) l − (2Λ )( u ,...,u l − ) = W L C (1) l (Λ ) u α + ··· + u l − α l − , h -weighted subspace of W L A (1) l − (2Λ ) . On every factor in the tensor product W L A (1) l − (2Λ ) ⊗· · · ⊗ W L A (1) l − (2Λ ) of k principal subspaces W L A (1) l − (2Λ ) , we have embedding W L A (1) l − (2Λ )( µ ( p )1 ; ... ; µ ( p ) l − ) ֒ → X u ,...,ul − ,v ,...,vl − ∈ N µ ( p ) i = u i + v i , ≤ i ≤ l − W L A (1) l − (Λ )( u ; ... ; u l − ) ⊗ W L A (1) l − (Λ )(( v ; ... ; v l − ) , for 1 ≤ p ≤ k .Denote by π ′ R − the projection of the vector space W L A (1) l − (2Λ ) ⊗ · · · ⊗ W L A (1) l − (2Λ ) on subspace W L A (1) l − (Λ )( r (2 k )1 ; ... ; r (2 k ) l − ) ⊗ W L A (1) l − (Λ )( r (2 k − ; ... ; r (2 k − l − ) ⊗· · ·⊗ W L A (1) l − (Λ )( r (2)1 ; ... ; r (2) l − ) ⊗ W L A (1) l − (Λ )( r (1)1 ; ... ; r (1) l − ) . n particular, extending the above projection on the space of formal series with coefficientsin W L A (1) l − (Λ ) ⊗ · · · ⊗ W L A (1) l − (Λ ) | {z } k factors from the condition x α i ( z ) = 0 (1 ≤ i ≤ l −
1) follows π ′ R − (cid:18) π R b a ( α ) · · · b a ( α l − ) v L C (1) l k Λ (cid:19) ∈ W L A (1) l − (Λ )( r (2 k )1 ; ... ; r (2 k ) l − ) ⊗ W L A (1) l − (Λ )( r (2 k − ; ... ; r (2 k − l − ) ⊗ · · · ⊗⊗ W L A (1) l − (Λ )( r (2)1 ; ... ; r (2) l − ) ⊗ W L A (1) l − (Λ )( r (1)1 ; ... ; r (1) l − ) . Georgiev showed that π ′ R − ◦ π R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) WLA (1) l − k Λ0) A R − is a linearly independent set. Thus, it follows that the set c a = 0 and the desired theoremfollows. (cid:3) Characters of the principal subspace W L C (1) l ( k Λ ) . In determining the characterformulas of W L C (1) l ( k Λ ) we will use the expressions in Lemma 4.4.1. Lemma 4.4.1.
For the given color-type ( r , . . . , r l ) , charge-type (cid:16) n r (1)1 , , . . . , n , ; . . . n r (1) l ,l , . . . , n ,l (cid:17) and dual-charge-type (cid:16) r (1)1 , r (2)1 , . . . , r (2 k )1 ; . . . ; r (1) l , r (2) l , . . . , r ( k ) l (cid:17) , we have: r (1) l − X p =1 r (1) l X q =1 min { n p,l − , n q,l } = k X s =1 r ( s ) l ( r (2 s − l − + r (2 s ) l − ) , (4.16) r (1) i X p =1 r (1) i +1 X q =1 min { n p,i , n q,i +1 } = k X s =1 r ( s ) i r ( s ) i +1 , (4.17) r (1) l X p =1 ( X p>p ′ > { n p,l , n p ′ ,l } + n p,l ) = k X s =1 r ( s ) l , (4.18) r (1) i X p =1 ( X p>p ′ > { n p,i , n p ′ ,i } + n p,i ) = k X s =1 r ( s ) i , ≤ i ≤ l − . (4.19) (cid:3) Now, from the definition of the set B W LC (1) l ( k Λ0) and (4.16- 4.19), (3.28) follows thecharacter formula of W L C (1) l ( k Λ ) : heorem 9. ch W L C (1) l ( k Λ ) = X r (1)1 ≥ ... ≥ r (2 k )1 ≥ q r (1)21 + ··· + r (2 k )21 ( q ) r (1)1 − r (2)1 · · · ( q ) r (2 k )1 y r X r (1)2 ≥ ... ≥ r (2 u ≥ q r (1)22 + ··· + r (2 k )22 − r (1)1 r (1)2 −···− r (2 k )1 r (2 k )2 ( q ) r (1)2 − r (2)2 · · · ( q ) r (2 k )2 y r · · · X r (1) l − ≥ ... ≥ r (2 ul − l − ≥ q r (1)2 l − + ··· + r (2 k )2 l − − r (1) l − r (1) l − −···− r (2 k ) l − r (2 k ) l − ( q ) r (1) l − − r (2) l − · · · ( q ) r (2 k ) l − y r l − l − X r (1) l ≥ ... ≥ r ( ul ) l ≥ q r (1)2 l + ··· + r ( k )2 l − r (1) l ( r (1) l − + r (2) l − ) −···− r ( k ) l ( r (2 kl − + r (2 k ) l − ) ( q ) r (1) l − r (2) l · · · ( q ) r ( k ) l y r l l . (cid:3) The basis of the W N C (1) l ( k Λ ) . As in the case of B (1) l we can prove: Theorem 10.
The set B W NC (1) l ( k Λ0) = (cid:26) bv N C (1) l ( k Λ ) : b ∈ B W NC (1) l ( k Λ0) (cid:27) , where (4.20) B W NC (1) l ( k Λ0) = [ n r (1)1 , ≤ ... ≤ n , ...n r (1) l − ,l − ≤ ... ≤ n ,l − n r (1) l ,l ≤ ... ≤ n ,l or, equivalently [ r (1)1 ≥···≥ ··· r (1) l − ≥···≥ r (1) l ≥···≥ { b = b ( α ) · · · b ( α l ) = x n r (1)1 , α ( m r (1)1 , ) · · · x n , α ( m , ) · · · x n r (1) l ,l α l ( m r (1) l ,l ) · · · x n ,l α l ( m ,l ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m p,l ≤ − n p,l − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l ≤ m p,l − n p,l if n p,l = n p +1 ,l , ≤ p ≤ r (1) l − m p,l − ≤ − n p,l − + P r (1) l q =1 min { n q,l , n p,l − } − P p>p ′ > min { n p,l , n p ′ ,l } , ≤ p ≤ r (1) l ; m p +1 ,l − ≤ m p,l − − n p,l − if n p +1 ,l − = n p,l − , ≤ p ≤ r (1) l − − m p,i ≤ − n p,i + P r (1) i +1 q =1 min { n q,i +1 , n p,i } − P p>p ′ > min { n p,i , n p ′ ,i } , ≤ p ≤ r (1) i , ≤ i ≤ l − m p +1 ,i ≤ m p,i − n p,i if n p +1 ,i = n p,i , ≤ p ≤ r (1) i − , ≤ i ≤ l − , is the base of the principal subspace W N C (1) l ( k Λ ) . (cid:3) .6. Characters of the principal subspace W N C (1) l ( k Λ ) . From the definition of theset B W NC (1) l ( k Λ0) and (4.16- 4.19), (3.28) follows the character formula of the principalsubspace W N C (1) l ( k Λ ) : Theorem 11. ch W N C (1) l ( k Λ ) (4.21) = X r (1)1 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)21 + ··· + r (2 u ( q ) r (1)1 − r (2)1 · · · ( q ) r (2 u y r X r (1)2 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)22 + ··· + r (2 u − r (1)1 r (1)2 −···− r (2 u r (2 u ( q ) r (1)2 − r (2)2 · · · ( q ) r (2 u y r · · · X r (1) l − ≥ ... ≥ r (2 ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r (2 ul − l − − r (1) l − r (1) l − −···− r (2 ul − l − r (2 ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r (2 ul − l − y r l − l − X r (1) l ≥ ... ≥ r ( ul ) l ≥ u l ≥ q r (1)2 l + ··· + r ( ul )2 l − r (1) l ( r (1) l − + r (2) l − ) −···− r ( ul ) l ( r (2 ul − l − + r (2 ul ) l − ) ( q ) r (1) l − r (2) l · · · ( q ) r ( ul ) l y r l l . (cid:3) By Poincar´e-Birkhoff-Witt theorem, we obtain the base of the universal enveloping alge-bra U ( L ( n + ) < ): x α ( m ) · · · x α ( m s ) x α + α ( m ) · · · x α + α ( m s ) · · · x α + α + ··· + α l − ( m l − ) · · · (4.22) · · · x α + α + ··· + α l ( m s l − l − ) x α +2 α + ··· +2 α l ( m l ) · · · x α +2 α + ··· +2 α l ( m s l l ) · · ·· · · x α + α + ··· +2 α l ( m l − ) · · · x α + α + ··· +2 α l ( m s l − l − ) · · · x α +2 α + ··· + α l ( m l − ) · · · x α +2 α + ··· + α l ( m s l − l − ) · · ·· · · x α l − ( m l − ) · · · x α l − ( m s l − l − ) · · · x α l − + α l ( m l − ) · · · x α l − + α l ( m s l − l − ) x α l ( m l ) · · · x α l ( m s l l ) , with m i ≤ · · · ≤ m s i i , s i ∈ N for i = 1 , . . . , l . Now, we also have a following characterformula(4.23) ch W N C (1) l ( k Λ ) = Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) 1(1 − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · − q m y y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )From (4.21) and (4.23) now follows: Theorem 12. Y m> − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) 1(1 − q m y y · · · y l )1(1 − q m y ) 1(1 − q m y y ) · · · − q m y · · · y l − ) 1(1 − q m y y · · · y l ) · · · − q m y y · · · y l ) 1(1 − q m y y · · · y l ) · · · − q l − ) 1(1 − q m y l − y l ) 1(1 − q m y l − y l ) 1(1 − q m y l )= X r (1)1 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)21 + ··· + r (2 u ( q ) r (1)1 − r (2)1 · · · ( q ) r (2 u y r X r (1)2 ≥ ... ≥ r (2 u ≥ u ≥ q r (1)22 + ··· + r (2 u − r (1)1 r (1)2 −···− r (2 u r (2 u ( q ) r (1)2 − r (2)2 · · · ( q ) r (2 u y r · · · X r (1) l − ≥ ... ≥ r (2 ul − l − ≥ u l − ≥ q r (1)2 l − + ··· + r (2 ul − l − − r (1) l − r (1) l − −···− r (2 ul − l − r (2 ul − l − ( q ) r (1) l − − r (2) l − · · · ( q ) r (2 ul − l − y r l − l − X r (1) l ≥ ... ≥ r ( ul ) l ≥ u l ≥ q r (1)2 l + ··· + r ( ul )2 l − r (1) l ( r (1) l − + r (2) l − ) −···− r ( ul ) l ( r (2 ul − l − + r (2 ul ) l − ) ( q ) r (1) l − r (2) l · · · ( q ) r ( ul ) l y r l l . (cid:3) Acknowledgement
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