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}
}
  • O. Schiffmann
  • Published 15 April 2010
  • Mathematics
  • Journal of Algebraic Combinatorics
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… 
Cusp eigenforms and the hall algebra of an elliptic curve
Abstract We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function
Spherical Hall Algebra of \overline{\text{Spec }(\mathbb{Z})}
We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices
A Categorical Quantum Toroidal Action on Hilbert Schemes
We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot{gl_1})$ acting on the
The spherical Hall algebra of Spec(Z)
We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices
Quantum toroidal algebras and motivic Hall algebras I. Hall algebras for singular elliptic curves
We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and
Addendum to: Olivier Schiffmann, “Drinfeld realization of the elliptic Hall algebra”
In (J. Algebr. Comb. doi:10.1007/s10801-011-0302-8, 2011), O. Schiffmann gave a presentation of the Drinfeld double of the elliptic Hall algebra which is similar in spirit to Drinfeld’s new
Finite Type Modules and Bethe Ansatz for Quantum Toroidal $${\mathfrak{gl}_1}$$gl1
We study highest weight representations of the Borel subalgebra of the quantum toroidal $${\mathfrak{gl}_1}$$gl1 algebra with finite-dimensional weight spaces. In particular, we develop the
Shuffle algebras for quivers as quantum groups
We define a quantum loop group U Q associated to an arbitrary quiver Q = (I, E) and maximal set of deformation parameters, with generators indexed by I × Z and some explicit quadratic and cubic
Equivariant K-theory of Hilbert schemes via shuffle algebra
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
...
...

References

SHOWING 1-10 OF 24 REFERENCES
On the Hall algebra of an elliptic curve, I
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
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 $\mathbb{A}^2$
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
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
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
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
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
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
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
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
...
...