Two-parameter twisted quantum affine algebras

Abstract

ABSTRACT. We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.

Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization

“…An important new development in the present paper is the introduction of the structures of quantum N -toroidal algebra for all types uniformly, which is a natural generalization of classical quantum toroidal algebra, just like the relation between 2-toroidal Lie algebra and N -toroidal Lie algebra. In [JZ1] and [JZ2], we formulate a simplified set of Drinfeld generators for quantum affine algebras and the quantum toroidal algebra for type A, respectively, and we prove that this subset can replace the original set of generators. In the same way, the quantum N -toroidal algebra can be identified with its subalgebra generated by a set of simplified generators.…”

Two-parameter quantum affine algebra of type Cn(1), Drinfeld realization and vertex representation

“…Also for type G (1) 2 , the validness of our definition for the two-parameter Drinfeld realization can be wellchecked in the level of its two-parameter quantum vertex representation, see [GHZ]. As two generalizations of [HZ] to the two-parameter quantum affine algebras of twisted types X (r) for r = 2, 3, the readers can consult [JZ1], to the two-parameter quantum toroidal algebras, please refer to [JZ2], where the authors gave a McKay correspondence formalism of the vertex representations obtained in [HZ].…”

“…It was known that the theory of two-parameter quantum affine algebras has been developed with some analogous stories as in the one-parameter counterpart such as Drinfeld realization theorem with a different argument approach [HRZ,HZ,JZ1], fermionic realization [JZ3] and a finite-dimensional representation theory [JZ4]. The quantum vertex representations of one-parameter quantum affine algebras for the untwisted types were first constructed by Frenkel-Jing [FJ] that confirmed the Drinfeld's celebrated conjectural "new realization" [Dr2] in the level of vertex representations, even though it was not proved until Beck gave his rigorous proof for the untwisted types by generalizing the Lusztig's braid automorphisms suitable for the quantum affine algebras based on the work of Damiani [D] and Levendorskii-Soibelḿan-Stukopin [LSS].…”

PBWD bases and shuffle algebra realizations for $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ and their integral forms

“…However, we provide technical details as they are needed both for Section 3.2.3 and for the generalizations to the two-parameter and super-cases. At that point, we should note that while the two-parameter quantum affine algebras have been extensively studied since the original work [HRZ], see [JL,JZ1,JZ2] for a partial list of references (see also [Ta,BW1,BW2,JMY] for the case of two-parameter quantum finite groups), not many results have been established for them. In particular, it is still an open question whether these are flat deformations of the corresponding universal enveloping algebras.…”

“…In particular, it is still an open question whether these are flat deformations of the corresponding universal enveloping algebras. In [JZ2], an isomorphism between the Drinfeld-Jimbo and the new Drinfeld realizations of these algebras has been established (generalizing [HRZ,Theorem 3.12] for type A), see Remark 4.8, but it is not known how the construction of the PBW basis of [B1, B2] can be generalized to the former realization.…”