# Drinfeld realization of the elliptic Hall algebra

@article{Schiffmann2010DrinfeldRO, title={Drinfeld realization of the elliptic Hall algebra}, author={Olivier Schiffmann}, journal={Journal of Algebraic Combinatorics}, year={2010}, volume={35}, pages={237-262} }

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 Schiffmann in Duke Math. J. preprint math.AG/0505148, 2005). This presentation is similar in spirit to Drinfeld’s ‘new realization’ of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relations satisfied by (principal, unramified…

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## References

SHOWING 1-10 OF 24 REFERENCES

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…

The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

- MathematicsCompositio Mathematica
- 2010

Abstract We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine…

The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$

- Mathematics
- 2009

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA…

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…

Kernel function and quantum algebras

- Mathematics
- 2010

We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the…

Eisenstein series and quantum affine algebras

- Mathematics
- 1996

Let X be a smooth projectibe curve over a finite field. We consider the Hall algebra H whose basis is formed by isomorphism classes of coherent sheaves on X and whose typical structure constant is…

Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations

- Mathematics
- 2011

We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the…

Hall algebras and quantum groups

- Mathematics
- 1990

The Hall algebra of a finitary category encodes its extension structure. The story starts from the work of Steinitz on the module category of an abelian p-group, where the Hall algebra is the algebra…

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…

Vector bundles on elliptic curve and Sklyanin algebras

- Mathematics
- 1995

In [4] we introduce the associative algebras $Q_{n,k}(\CE,\tau)$. Recall the definition. These algebras are labeled by discrete parameters $n,k$; $n,k$ are integers $n>k>0$ and $n$ and $k$ have not…