A note on principal subspaces of the affine Lie algebras in types Bl(1),Cl(1),F4(1) and G2(1)

Parafermionic bases of standard modules for affine Lie algebras

“…where e µ denote the Weyl group translation operators parametrized by the elements of the coroot lattice Q ∨ of the simple Lie algebra g, the set B U ( h − ) is the Poincaré-Birkhoff-Witt-type basis of the universal enveloping algebra U( h − ) and B ′ W L(Λ) is a certain subset of B W L(Λ) . We verify that set (2) spans L(Λ) by using the relations among quasiparticles and arguing as in [Bu1,Bu2,Bu3,Bu5,BK]. On the other hand, our proof of linear independence relies on generalizing Georgiev's arguments originated in [G1] to standard modules for all untwisted affine Lie algebras.…”

“…Proof. We prove linear independence of the spanning set B L(Λ) by slightly modifying the arguments in [Bu1,Bu2,Bu3,Bu5,BK]. We consider a finite linear combination of vectors in B L(Λ) ,…”

“…Their results were further generalized by G. Georgiev [G1] to g of type A l for all principal subspaces W L(Λ) of the highest weight Λ as in (1). Finally, the quasi-particle bases of W L(Λ) for all Λ as in (1) were constructed by the first author for g of types B l , C l , F 4 , G 2 [Bu1,Bu2,Bu3,Bu5] and by the first and the second author for g of types D l , E 6 , E 7 , E 8 [BK].…”

“…The major step towards finding the parafermionic bases is the construction of the suitable bases of the standard modules. This construction relies on the quasi-particle bases of the corresponding principal subspaces from [Bu1,Bu2,Bu3,Bu5,BK,FS,G1]. Consider the subalgebras h ± = h ⊗ t ± C [t] of g, where h is a Cartan subalgebra of g. We express the bases for L(Λ) as…”

“…[FS]). In [Bu1,Bu2,Bu3,Bu4,Bu5,BK,G1] the bases of the principal subspaces W L(Λ) for the affine Lie algebras g of different types were described in terms of quasi-particles, which we introduce in the following subsection.…”