Abstract. The affine Yangian of gl 1 has recently appeared simultaneously in the work of Maulik-Okounkov [MO] and in connection with the AldayGaiotto-Tachikawa conjecture. While the presentation from [MO] is purely geometric, the algebraic presentation in [SV2] is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an "additivization" of the quantum toroidal algebra of gl 1 in the same way as the Yangian Y h (g) is an "additivization" of the quantum loop algebra Uq(Lg) for a simple Lie algebra g. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl 1 by generalizing the main result of [GTL] to the current settings.

In this paper we construct actions of Ding-Iohara and shuffle algebras on the sum of localized equivariant K-groups of Hilbert schemes of points on C 2 . We show that commutative elements K i of shuffle algebra act through vertex operators over the positive part {h i } i>0 of the Heisenberg algebra in these K-groups. This provides an action of the Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space C[h 1 , h 2 , . . .].

No abstract

We introduce super Yangians of gl(V ), sl(V ) (in the new Drinfeld realization) associated to all Dynkin diagrams of gl(V ). We show that all of them are isomorphic to the super Yangians introduced by M. Nazarov in [Na], by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations for all of those in spirit of [T]. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halves" of type A quantum loop superalgebras associated to arbitrary Dynkin diagrams.

We define an integral form of shifted quantum affine algebras of type A and construct Poincaré-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results (proved earlier in [KMWY, KTWWY] via different techniques).