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V. Chari and A. Pressley, Quantum affine algebras
1991
Earlier work this paper cites.
J. Ding and K. Iohara, Generalization of Drinfeld quantum affine algebras
1997
Earlier work this paper cites.
H. Nakajima, Quiver varieties and Kac-Moody algebras
1998
Earlier work this paper cites.
H. Nakajima, Lectures on Hilbert schemes of points on surfaces
1999
Earlier work this paper cites.
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4 4 -dimensional pure gauge theory
2005
Earlier work this paper cites.
K. Miki, A ( q , γ ) (q,\gamma) analog of the W 1 + ∞ W_{1+\infty} algebra
2007
Earlier work this paper cites.

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2014

The affine Yangian of $\mathfrak{gl}_1$ revisited

Tsymbaliuk, Alexander

Understand

The affine Yangian of $\mathfrak{gl}_1$ has recently appeared simultaneously in the work of Maulik-Okounkov and Schiffmann-Vasserot in connection with the Alday-Gaiotto-Tachikawa conjecture.

While the former presentation is purely geometric, the latter algebraic presentation 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 $\mathfrak{gl}_1$ in the same way as the Yangian $Y_h(\mathfrak{g})$ is an "additivization" of the quantum loop algebra $U_q(L\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$.
We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of $\mathfrak{gl}_1$ by generalizing the milestone result of Gautam and Toledano Laredo to the current settings.

Built on

V. Chari and A. Pressley, Quantum affine algebras
1991
Earlier work this paper cites.
J. Ding and K. Iohara, Generalization of Drinfeld quantum affine algebras
1997
Earlier work this paper cites.
H. Nakajima, Quiver varieties and Kac-Moody algebras
1998
Earlier work this paper cites.
H. Nakajima, Lectures on Hilbert schemes of points on surfaces
1999
Earlier work this paper cites.
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4 4 -dimensional pure gauge theory
2005
Earlier work this paper cites.
K. Miki, A ( q , γ ) (q,\gamma) analog of the W 1 + ∞ W_{1+\infty} algebra
2007
Earlier work this paper cites.