Quantum loop algebras, quiver varieties, and cluster algebras

Abstract

These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima's geometric description of the irreducible q-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finitedimensional representations by means of cluster algebras. This takes the form of a general conjecture depending on a level ℓ ∈ N. The conjecture for ℓ = 1 is now proved thanks to some joint work with Hernandez, and a subsequent paper of Nakajima. The general case is still open.

A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules

“…with the structure of a standard U q ( g)-module, with highest weight encoded by W. It was proved by Lusztig (in the ungraded case), and by Savage and Tingley (in the graded case), that L • (V, W) is homeomorphic to a Grassmannian of submodules of an injective module over the graded preprojective algebra (see [Le2,§2.8]). Therefore, using the description of K (i) 1,r given in Example 4.7, we see that the varieties involved in our geometric q-character formula for standard modules in the simply laced case are homeomorphic to certain Nakajima varieties L • (V, W).…”

Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

“…Compared to the intense works on affine quantum groups (see the reviews [13,40]), the representation theory of U q ( g) is still less understood as the super case poses one essential difficulty, the smallness of Weyl group symmetry.…”

Representations of Uq(Lsl2) at roots of unity and generalised cl

“…Let M q be the set of Laurent monomials in the Y i,a , and let M + q be the subset of dominant monomials in M q . If S is a simple object of C q such that the highest monomial of χ q (S) is m ∈ M + q , then S will be denoted by L(m) [16]. For i ∈ I and a ∈ C(q), the simple modules L(Y i,a ) are called fundamental modules.…”