We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that par
abolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.
The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algeb
ra $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$-modules.
We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is g
iven by an action of $mathrm{SL}_{2}(mathbb{Z})$ on the toroidal Lie algebra. As a byproduct, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.
We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been writ
ten down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra $hat{mathfrak{gl}}_N$ by the evaluation map.
We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $hat{mathfrak{gl}}_N$.
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic ration
al Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $mathfrak{gl}(1)$.
We construct actions of the affine Yangian of type A on higher level Fock spaces by extending known actions of the Yangian of finite type A due to Uglov. This is a degenerate analog of a result by Takemura-Uglov, which constructed actions of the quan
tum toroidal algebra on higher level $q$-deformed Fock spaces.
The localized equivariant homology of the quiver variety of type $A_{N-1}^{(1)}$ can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack($mathfrak{gl}_N$) symmetric functi
ons introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for $mathfrak{sl}_N$ and the affine Lie algebra $hat{mathfrak{sl}}_N$, on both sides. Consequently we obtain affine Yangian action on the Fock space.
We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
