Parafermionic bases of standard modules for affine Lie algebras

Parafermionic bases of standard modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$

“…Next we state the vertex operator formula that will be used later. This is a twisted version of (1.27) in [3]. The proof is completely parallel to [18,Theorem 5.6] or [23,Theorem 6.4].…”

“…spans L(kΛ 0 ). It suffices to check that the arguments used in the proof of [3,Lemma 2.3] also hold for our case. By the vertex operator formula (27), a quasi-particle x ν kαi (m) is expressed by e αi and monomials in U ( ĥ[ν] ± ).…”

“…We have the following lemma by direct computation. Noting that C(β, β) = 1 for a simple root β, the proof is parallel to that of Lemma 3.2 of [3].…”

“…This corresponds to the translation operator of the affine Weyl group of g [ν]. See Section 1.5 of [3] for the untwisted case. Based on the calculations ( 15)-( 18) on V T L , we define the adjoint action on g[ν] of the multiplicative group isomorphic to Q by…”

“…Recently, based on the works on constructing combinatorial bases of the principal subspaces for other affine algebras (see e.g. [1,2]), Butorac, Kožić and Primc obtained fermionic formulas for integrable highest weight modules similar to the above highest weight for all untwisted affine Lie algebras in [3], and thereby proved the so called Kuniba-Nakanishi-Suzuki conjecture [16]. Although the untwisted cases are settled, there are also twisted affine Lie algebras and a similar conjecture was stated in [14].…”