Quasi-particle Bases of Principal Subspaces of Affine Lie Algebras

Abstract

Generalizing our earlier work, we construct quasi-particle bases of principal subspaces of standard module L X (1) l (kΛ 0 ) and generalized Verma module N X (1) l (kΛ 0 ) at level k ≥ 1 in the case of affine Lie algebras of types B(1) l and C (1) l . As a consequence, from quasi-particle bases, we obtain the graded dimensions of these subspaces. IntorductionLet g be a simple complex Lie algebra of type X l , with a Cartan subalgebra h, the set of simple roots Π = {α 1 , . . . , α l } and the triangular decomposition g = n − ⊕ h ⊕ n + , where n + is a direct sum of its one dimensional subalgebras corresponding to the positive roots. Denote by L(n + ) a subalgebra of untwisted affine Lie algebra g of type X(1) lLet V be a highest g-module with highest weight Λ and highest weight vector v Λ . We define the principal subspace W V of V asIn this paper we study principal subspaces of the generalized Verma module N X (1) l (kΛ 0 ) and its irreducible quotient L X (1) l (kΛ 0 ) at level k ≥ 1, defined over the affine Lie algebras of type B(1) l and C(1) l . The study of principal subspaces of standard (i.e., integrable highest weight) modules of the simply laced affine Lie algebras and its connection to Rogers-Ramanujan identities was initiated in the work of B. L. Feigin and A. V. Stoyanovsky [FS] and has been further developed in [AKS]

Vertex-algebraic structure of principal subspaces of the basic modules for twisted affine Lie algebras of type A2n−1(2),Penn1, Sadowski2 2018Journal of Algebra122320View full textAdd to dashboardCite

“…These principal subspaces were also student in [AKS], [FFJMM], and in other works by these authors. Georgiev used the theory of vertex operator algebras and intertwining operators to construct combinatorial bases for principal subspaces of certain higher level standard modules for untwisted affine Lie algebras of type A, D, E in [G], and this work has since been extended to untwisted affine Lie algebras of type B, C by Butorac in [Bu1]- [Bu2] and to the quantum group case by Kožić in [Ko]. Principal subspaces of standard modules for more general lattice vertex operator algebras have been studied in [MPe], [P], and [Ka1]- [Ka2].…”

Quasi-particle bases of principal subspaces of the affine Lie algebra of type G_2^{(1)}

“…We start this section with introducing all necessary notions and facts needed in the construction of quasi-particle bases of principal subspaces. Some terms and labels which we use, but are not mentioned, are the same as in our previous work, therefore, for more details we refer to [4,5] and also to [18].…”

“…), we will say it is of charge-type [4,5,18]) if for every color R and R ′ are mutually conjugate partitions of r i (cf. [1]).…”

“…where W L(kΛ0) (m,r1,r2) is a weight subspace spanned by monomial vectors of weight −m and color-type (r 1 , r 2 ) (see [4,5,18]).…”

“…[16]), from which were easily obtained the Rogers-Ramanujan type character formulas. In [4] and [5] we extended Georgiev's construction of quasiparticle bases for principal subspaces of standard module L(kΛ 0 ) and generalized Verma module N (kΛ 0 ) of highest weight kΛ 0 , k ∈ N for affine Lie algebras of type B…”