# The affine Yangian of gl1 revisited

@article{Tsymbaliuk2017TheAY,
title={The affine Yangian of gl1 revisited},
author={Alexander Tsymbaliuk},
year={2017},
volume={304},
pages={583-645}
}
• A. Tsymbaliuk
• Published 21 April 2014
• Mathematics
• Advances in Mathematics

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## References

SHOWING 1-10 OF 38 REFERENCES

### Limits of quantum toroidal and affine Yangian

In this short note, we compute the classical limits of the quantum toroidal and the affine Yangian algebras of sl(n) by generalizing our arguments for the case of gl(1) from arXiv:1404.5240. These

### Drinfeld realization of the elliptic Hall algebra

We give a new presentation of the Drinfeld double $\boldsymbol{\mathcal {E}}$ of the (spherical) elliptic Hall algebra $\boldsymbol{\mathcal{E}}^{+}$ introduced in our previous work (Burban and

### Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2

• Mathematics
• 2012
We construct a representation of the affine W-algebra of ${\mathfrak{g}}{\mathfrak{l}}_{r}$ on the equivariant homology space of the moduli space of Ur-instantons, and we identify the corresponding

### Quantum Groups and Quantum Cohomology

• Mathematics
• 2012
In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q,

### Equivariant K-theory of Hilbert schemes via shuffle algebra

• Mathematics
• 2011
In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of

### Quantum continuous gl ∞ : Semiinﬁnite construction of representations

• Mathematics
• 2011
We begin a study of the representation theory of quantum continuous gl ∞ , whichwedenoteby E .Thisalgebradependsontwoparametersandisadeformedversion of the enveloping algebra of the Lie algebra of

### A commutative algebra on degenerate CP^1 and Macdonald polynomials

• Mathematics
• 2009
We introduce a unital associative algebra A over degenerate CP^1. We show that A is a commutative algebra and whose Poincar'e series is given by the number of partitions. Thereby we can regard A as a

### On the Hall algebra of an elliptic curve, I

• Mathematics
• 2005
This paper is a sequel to math.AG/0505148, where the Hall algebra U^+_E of the category of coherent sheaves on an elliptic curve E defined over a finite field was explicitly described, and shown to

### Quantum affine algebras

• Mathematics
• 1991
AbstractWe classify the finite-dimensional irreducible representations of the quantum affine algebra $$U_q (\hat sl_2 )$$ in terms of highest weights (this result has a straightforward