Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Abstract

We introduce and study the quantum toroidal algebra E m|n (q 1 , q 2 , q 3 ) associated with the superalgebra gl m|n with m = n, where the parameters satisfy q 1 q 2 q 3 = 1.We give an evaluation map. The evaluation map is a surjective homomorphism of algebras E m|n (q 1 , q 2 , q 3 ) → U q gl m|n to the quantum affine algebra associated with the superalgebra gl m|n at level c completed with respect to the homogeneous grading, where q 2 = q 2 and q m−n 3 = c 2 . We also give a bosonic realization of level one E m|n (q 1 , q 2 , q 3 )-modules. Quantum toroidal gl m|nAssume m, n ≥ 1 and m = n. In this section we introduce the quantum toroidal algebra associated with gl m|n , denoted by E m|n , and collect a few properties.2.1. Definition of E m|n . Fix d, q ∈ C × and define

Quiver Yangian from crystal melting

“…This coincides with the same function for the affine Yangian for the Lie superalgebra gl m|n , which in turn arises from the rational reduction of the quantum toroidal gl m|n algebra constructed recently in [66,67] 36 (which generalizes the case of quantum toroidal gl n constructed earlier in [63]). In particular, the symmetric matrix A is nothing but the Dynkin diagram for the Lie superalgebra gl m|n .…”

“…Namely, the Serre relation for the fermionic generators e (a) involve the e (a) , e (a+1) , e (a) , e (a+1) from left to right in that order, and this seems to correspond to the superpotential term Tr(Φ a,a+1 Φ a+1,a Φ a,a−1 Φ a−1,a ) in (8.83). Similarly, we have a cubic Serre relation for the bosonic generators e (a) for the affine Yangian for gl m|n [66] (recall also the cubic Serre relation for the C 3 geometry in (5.51)), and this corresponds naturally to the cubic superpotential term Tr(Φ a,a Φ a,a+1 Φ a+1,a ) in (8.83). It is tempting to speculate that this is a general pattern and that the Serre relations can be identified from the data of the superpotential.…”

“…Here we adopted the Serre relation from [66]. 38 Note that e (a) and f (a) are fermionic generators, and hence we have both commutators [−, −] and anti-commutators {−, −} in the Serre relations.…”

“…107) 36 In the notations of [66], the quantum-toroidal counterpart of the function (8.105) is given by…”

“…The paper[66] strictly speaking does not deal with affine Yangians of gl 1|1 , or more generally gl n|n .…”