[Paper Review] Drinfeld realization of the elliptic Hall algebra

Olivier Schiffmann|arXiv (Cornell University)|Apr 15, 2010

Algebraic structures and combinatorial models7 references4 citations

TL;DR

This paper provides a Drinfeld-type presentation of the elliptic Hall algebra, analogous to Drinfeld's new realization of quantum affine algebras. It establishes that the Drinfeld double of the spherical elliptic Hall algebra is governed by quadratic relations and a single set of cubic relations, resolving Kapranov's question on Eisenstein series functional equations and verifying recent conjectures by Feigin, Feigin, Jimbo, Miwa, and Mukhin.

ABSTRACT

We give a new presentation of the Drinfeld double of the elliptic Hall algebra introduced in a previous work with I. Burban. 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) Eisenstein series for the groups GL(n) over a function field. It also provides proofs of some recent conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin.

Motivation & Objective

• To provide a Drinfeld-type presentation of the Drinfeld double of the spherical elliptic Hall algebra, analogous to Drinfeld's new realization of quantum affine algebras.
• To answer Kapranov's question regarding functional relations satisfied by unramified Eisenstein series for GL(n) over function fields in the case of elliptic curves.
• To prove that the elliptic Hall algebra admits a presentation generated by Drinfeld generators subject to standard quadratic relations and one set of cubic relations.
• To verify recent conjectures by Feigin, Feigin, Jimbo, Miwa, and Mukhin on the algebra's structure and representation theory.

Proposed method

• The method relies on a combinatorial realization of the elliptic Hall algebra via lattice paths in ℤ², developed in prior work [BS].
• The proof proceeds by induction on the degree of the path, using the minimality of paths and the absence of interior lattice points in associated triangles.
• The algebra is presented using Drinfeld generators 𝕋₀⁺(z), 𝕋₀⁻(z), 𝕋₁(z), and 𝕋₋₁(z), with coproduct formulas given in terms of these generators.
• Relations are derived by analyzing the structure of shuffle algebras and the action of the algebra on K-theory of Hilbert schemes.
• The proof establishes that certain commutators vanish modulo an ideal J, using induction and the injectivity of the map from the universal algebra to the Hall algebra.
• The Hopf algebra structure is described explicitly via coproduct formulas involving the Drinfeld generators, showing that the positive and negative parts are sub-bialgebras.

Experimental results

Research questions

• RQ1 What functional relations govern the unramified Eisenstein series for GL(n) over function fields when the underlying curve is an elliptic curve?
• RQ2 How can the Drinfeld double of the spherical elliptic Hall algebra be presented in a manner analogous to Drinfeld's new realization of quantum affine algebras?
• RQ3 Do the cubic relations derived in this work resolve the conjectures of Feigin, Feigin, Jimbo, Miwa, and Mukhin on the structure of the elliptic Hall algebra?
• RQ4 Is there a Hopf algebra structure on the elliptic Hall algebra compatible with the Drinfeld realization?
• RQ5 Does the SL(2,ℤ)-symmetry persist when considering the coproduct in the Drinfeld presentation?

Key findings

• The Drinfeld double of the spherical elliptic Hall algebra admits a presentation via Drinfeld generators satisfying standard quadratic relations and one additional set of cubic relations, as stated in Theorem 4.
• The proof confirms that the algebra is isomorphic to the universal enveloping algebra of the Drinfeld realization, with the isomorphism induced by the lattice path model.
• The coproduct in the Drinfeld realization is explicitly computed, showing that the positive and negative parts are sub-bialgebras.
• The SL(2,ℤ)-symmetry is broken at the level of the coproduct, though the unipotent subgroup acts as a Hopf algebra automorphism.
• The results verify several conjectures from [FFJMM1], particularly those concerning the algebraic structure and representation theory of the elliptic Hall algebra.
• The algebra is shown to be isomorphic to the stable limit of spherical Cherednik algebras and to act on the K-theory of Hilbert schemes of points in ℂ².