Elliptic quantum groups and their finite-dimensional representations

Abstract

Let g be a complex semisimple Lie algebra, τ a point in the upper half-plane, and ∈ C a deformation parameter such that the image of in the elliptic curve C/(Z + τ Z) is of infinite order. In this paper, we give an intrinsic definition of the category of finite-dimensional representations of the elliptic quantum group E ,τ (g) associated to g. The definition is given in terms of Drinfeld half-currents, and extends that given by Enriquez-Felder for g = sl 2 [14]. When g = sln, it reproduces Felder's RLL definition [24] via the Gauß decomposition obtained in [14] for n = 2 and [31] for n ≥ 3. We classify the irreducible representations of E ,τ (g) in terms of elliptic Drinfeld polynomials, in close analogy to the case of the Yangian Y (g) and quantum loop algebra Uq(Lg) of g. A crucial ingredient in the classification is a functor Θ from finitedimensional representations of Uq(Lg) to those of E ,τ (g), which is an elliptic analogue of the monodromy functor constructed in [33], and circumvents the fact that E ,τ (g) does not admit Verma modules. Our classification is new even for g = sl 2 . It holds more generally when g is a symmetrisable Kac-Moody algebra, provided finite-dimensionality is replaced by an integrability and category O conditions.

Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

“…For a of general type, a category of E τ, (a)-modules was studied in [30] with wellbehaved q-character theory, although its tensor product structure is unclear.…”

Elliptic quantum groups and Baxter relations

“…In this paper we study representations of E τ, (sl N ) via the RLL presentation [17] so as to bypass affine quantum groups, yet along the way we borrow ideas from the affine case. Compared to other works [7,12,19,32,44,45,51,52], our approach emphasizes more on the Grothendieck ring structure of representation category. It is a higher rank extension of a recent joint work with G. Felder [21].…”

“…Meanwhile, Nekrasov-Pestun-Shatashvili [49] from the 6d quiver gauge theory predicted the elliptic quantum group associated to an arbitrary Kac-Moody algebra, the precise definition of which (as an associative algebra) was proposed by Gautam-Toledano Laredo [32]. See also [52] in the context of quiver geometry.…”

“…In other works[7,12,19,32,44,45,51,52]: a module V is an h-graded C-vector space; morphisms of modules depend on the dynamical parameter λ, so do their kernel and cokernel; the abelian category structure is non trivial. The scalar extension gives a module V ⊗ C M in the present situation.…”

Yangians and Baxter's relations

“…We expect the asymptotic modules as well as their three-term relations to exist for elliptic quantum groups outside of type A [12,21,40], one of the main technical difficulties being the lack of coproduct. As an intermediate step, it is interesting to study quantum toroidal algebras and affine Yangians whose Drinfeld-Jimbo type coproduct has been partly established [22,29].…”