We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}. As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest-weights iΛ_1+(k-i)Λ_2, where 1 \leq i \leq k and Λ_1 and Λ_2 are fundamental weights of \hat{\goth{sl}(3)}.
We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.
We study the principal subspaces of higher level standard $A_2^{(2)}$-modules, extending earlier work in the level one case, by Calinescu, Lepowsky, and Milas. We prove natural presentations of principal subspaces and also of certain related spaces.
Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard $\widehat{\mathfrak{sl}(n)}$-modules, $n \ge 3$. As a consequence, we obtain the multigraded dimensions of the principal subspaces $W(k_1Λ_1 + k_2 Λ_2)$ and $W(k_{n-2}Λ_{n-2} + k_{n-1} Λ_{n-1})$. This generalizes earlier work by Calinescu o…
We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.
We construct quasi-particle bases of principal subspaces of standard modules $L(Λ)$, where $Λ=k_0Λ_0+k_jΛ_j$, and $Λ_j$ denotes the fundamental weight of affine Lie algebras of type $B_l^{(1)}$, $C_l^{(1)}$, $F_4^{(1)}$ or $G_2^{(1)}$ of level one. From the given bases we find characters of principal subspaces.
Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard $\widehat{\goth{sl}(3)}$-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard $\widehat{\goth{sl}(3)}$-modules. We then provide a conjecture for the case of $\widehat{\goth{sl}(n+1)}$.
Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac-Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra A_2^{(2)}, which we expect can be extended to higher rank algebras. As a "test case," we consider the principal subspace of the basic A_2^{(2)}-module and explore its structure in depth.
The notion of {\it free} generalized vertex algebras is introduced. It is equivalent to the notion of {\it generalized principal subspaces} associated with lattices which are not necessarily integral. Combinatorial bases and the characters of the free generalized vertex algebras are given. As an application, the commutants of principal subspaces are described by using generalized principal subspaces.
This is the first in a series of papers in which we study vertex-algebraic structure of Feigin-Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even ``small'' spanning sets of these spaces. In this paper we prove presentations of the principal subspaces of the basic A_1^(1)-modules. These convenient presentations were previously used in work of Capparelli-Lepowsky-Milas for the purpose of obtaining the classical Rogers-Ramanujan recurs…
This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices $L$ whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra $V_L$. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recur…
We give an a priori proof of the known presentations of (that is, completeness of families of relations for) the principal subspaces of all the standard A_1^(1)-modules. These presentations had been used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers-Selberg recursions for the graded dimensions of the principal subspaces. This paper generalizes our previous paper.
We consider principal subspace W(Λ) of integrable highest weight module L(Λ) for quantum affine algebra $U_q(\hat{\mathfrak{sl}}_{n+1})$. We introduce quantum analogues of the quasi-particles associated with the principal subspaces for $\hat{\mathfrak{sl}}_{n+1}$ and discover certain relations among them. By using these relations we find, for certain highest weight Λ, combinatorial bases of principal subspace W(Λ) in terms of monomials of quantum quasi-particles.
The aim of this work is to construct the quasi-particle basis of principal subspace of standard module of highest weight $kΛ_0$ of level $k\geq 1$ of affine Lie algebra of type $G_2^{(1)}$ by means of which we obtain the basis of principal subspace of generalized Verma module.
By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type iden…
We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.
We consider principal subspaces $W_{L(kΛ_0)}$ and $W_{N(kΛ_0)}$ of standard module $L(kΛ_0)$ and generalized Verma module $N(kΛ_0)$ at level $k\geq 1$ for affine Lie algebra of type $B_2^{(1)}$. By using the theory of vertex operator algebras, we find combinatorial bases of principal ubspaces in terms of quasi-particles. From quasi-particle bases, we obtain character formulas for $W_{L(kΛ_0)}$ and $W_{N(kΛ_0)}$.
Generalizing our earlier work, we construct quasi-particle bases of principal subspaces of standard module $L_{X_l^{(1)}}(kΛ_0)$ and generalized Verma module $N_{X_l^{(1)}}(kΛ_0)$ at level $k\geq 1$ in the case of affine Lie algebras of types $B_l^{(1)}$ and $C_l^{(1)}$. As a consequence, from quasi-particle bases, we obtain the graded dimensions of these subspaces.
Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions (q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.
The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.
We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.
We study the principal subspaces, introduced by B. Feigin and A. Stoyanovsky, of the level 1 standard modules for $\hat{\goth{sl}(l+1)}$ with $l \geq 2$. In this paper we construct exact sequences which give us a complete set of recursions that characterize the graded dimensions of the principal subspaces of these representations. This problem can be viewed as a continuation of a new program to obtain Rogers-Ramanujan-type recursions, which was initiated by S. Capparelli, J. Lepowsky and A. Milas. In order to prove the exactness of the sequences we use intertwining vertex operators and we supp…
We consider two different methods of associating vertex algebraic structures with the level $1$ principal subspaces for $U_q (\widehat{\mathfrak{sl}}_2)$. In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B. L. Feigin and A. V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal $\underline{\mathsf{q}}$-vertex algebras, investigate their properties and construct the nonlocal $\underl…
In this paper we study a quantum analogue of a degenerate principal series of $U_q \mathfrak{su}_{n,n}$-modules ($0<q<1$) related to the Shilov boundary of the quantum $n \times n$-matrix unit ball. We give necessary and sufficient conditions for the modules to be simple and unitarizable and investigate their equivalence. These results are q-analogues of known classical results on reducibility and unitarizability of SU(n,n)-modules obtained by Johnson, Sahi, Zhang, Howe and Tan.
We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand-Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fe…
We present complete realization of irreducible $A_1 ^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1) ^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring ${\Bbb C}[α(-1/2), α(-3/2), ...]$. Explicit combinatorial bases…
Let $\tilde{\mathfrak g}$ be an affine Lie algebra of the type $A_\ell^{(1)}$. We find a combinatorial basis of Feigin-Stoyanovsky's type subspace $W(Λ)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(Λ)$ is obtained as an "inductive limit" of the basis of $W(Λ)$.
We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include "out-of-bounds" values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
We construct particle basis for Feigin-Stoyanovsky's type subspaces of level $1$ standard $\tilde{\mathfrak{sl}}_{\ell+1}(\mathbb{C})$-modules. From the description we obtain character formulas.
BRST-resolution is studied for the principally graded Wakimoto module of {\hat sl_2} recently found in math.QA/0005203. The submodule structure is completely determined and irreducible representations can be obtained as the zero-th cohomology group.
Let $\tilde{\mathfrak g}$ be an affine Lie algebra of type $A_\ell^{(1)}$. Suppose we're given a $\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus {\mathfrak g}_1$; then we also have the induced $\mathbb Z$-gradation of the affine Lie algebra $$\tilde{\mathfrak g}=\tilde{\mathfrak g}_{-1} \oplus \tilde{\mathfrak g}_0 \oplus \tilde{\mathfrak g}_1.$$ Let $L(Λ)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(Λ)$ is the $\tilde{\mathfrak g}_1$-submodule of $L(Λ)$ generated by the hi…
This is the first of a series of papers studying combinatorial (with no ``subtractions'') bases and characters of standard modules for affine Lie algebras, as well as various subspaces and ``coset spaces'' of these modules. In part I we consider certain standard modules for the affine Lie algebra $\ga,\;\g := sl(n+1,\C),\;n\geq 1,$ at any positive integral level $k$ and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky [FS]). The bases are given in terms of partitions: a color $i,\;1\leq i \leq n,$ and a charge $s,\; 1\leq s \leq k,$ are a…
Feigin-Stoyanovsky's type subspace $W(Λ)$ of a standard $\tilde{\mathfrak g}$-module $L(Λ)$ is a $\tilde{\mathfrak g}_1$-submodule of $L(Λ)$ generated by the highest-weight vector $v_Λ$, where $\tilde{\mathfrak g}_1$ is a certain commutative subalgebra of $\tilde{\mathfrak g}$. Based on the description of basis of $W(Λ)$ for $\tilde{\mathfrak g}$ of type $C_\ell^{(1)}$, we give a presentation of this subspace in terms of generators and relations $$W(Λ)\simeq U(\tilde{\mathfrak g}_1^-)/J.$$
Let $\gtl$ be an affine Lie algebra of type $D_{\ell}^{(1)}$ and $L(Λ)$ its standard module with a highest weight vector $v_Λ$. For a given $\Z$-gradation $\gtl = \gtl_{-1} + \gtl_0 + \gtl_1$, we define Feigin-Stoyanovsky's type subspace as $$W(Λ) = U(\gtl_1) \cdot v_Λ.$$ By using vertex operator relations for standard modules we reduce the Ponicaré-Brikhoff-Witt spanning set of $W(Λ)$ to a basis and prove its linear independence by using Dong-Lepowsky intertwining operators.
We use combinatorial description of bases of Feigin-Stoyanovsky's type subspaces of standard modules of level 1 for affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$ to obtain character formulas. These descriptions naturally lead to systems of recurrence relations for which we also find solutions.
Feigin-Stoyanovsky's type subspaces for affine Lie algebras of type $C_\ell^{(1)}$ have monomial bases with a nice combinatorial description. We describe bases of whole standard modules in terms of semi-infinite monomials obtained as "a limit of translations" of bases for Feigin-Stoyanovsky's type subspaces.
In this paper we construct combinatorial bases of Feigin-Stoyanovsky's type subspaces of standard modules for level $k$ affine Lie algebra $C_\ell^{(1)}$. We prove spanning by using annihilating field $x_θ(z)^{k+1}$ of standard modules. In the proof of linear independence we use simple currents and intertwinining operators whose existence is given by fusion rules.
