Abstract
Let \( \mathfrak{g} \) be a simple Lie algebra over ℂ and q ∈ ℂ× transcendental. We consider the category \( {\mathcal{C}}_{\mathcal{P}} \) of finite-dimensional representations of the quantum loop algebra Uq\( \left(\mathrm{\mathcal{L}}\mathfrak{g}\right) \) in which the poles of all l-weights belong to specified finite sets \( \mathcal{P} \). Given the data (\( \mathfrak{g} \); q;\( \mathcal{P} \)), we define an algebra \( \mathcal{A} \) whose raising/lowering operators are constructed to act with definite l-weight (unlike those of Uq\( \left(\mathrm{\mathcal{L}}\mathfrak{g}\right) \) itself). It is shown that there is a homomorphism Uq\( \left(\mathrm{\mathcal{L}}\mathfrak{g}\right) \) → \( \mathcal{A} \) such that every representation V in CP is the pull-back of a representation of \( \mathcal{A} \).
Highlights
Quantum loop algebras and their finite-dimensional representations have been a topic of interest for two decades at least: for a recent review see [CH10]
Besides their original setting in integrable quantum- and statistical-mechanical models, they appear in the contexts of algebraic geometry [GV93, Nak01, VV02, Nak04], combinatorics [JS10, LSS10], and cluster algebras [HL10, Nak11]
The eigenvalues are known as l-weights, and the q-character of V is by definition the formal sum of its l-weights [FR98, Kni95]. It is usually encoded as a Laurent polynomial χq(V ) in formal variables Yi,a, where a ∈ C× and where i runs over the set I of nodes of the Dynkin diagram of g
Summary
Quantum loop algebras and their finite-dimensional representations have been a topic of interest for two decades at least: for a recent review see [CH10]. Just as χ(V (ω)) ∈ eωZ[e−αi ]i∈I , so it is known that χq(L(γ)) ∈ γ Z[A−i,a1]i∈I,a∈C× [FM01] At this stage the analogy with the usual weight theory breaks down, in the following important sense. One can only obtain “x±Ai,a” by “evaluating at z = a” the formal series of generators x±i (z) := r∈Z z−rx±i,r Such an infinite sum x±i (a) = r∈Z a−rx±i,r is ill-defined in a sense that is not merely technical: its would-be matrix representatives have singular entries (c.f. equation (1.1) below). The first key result, namely the motivating observation – (1.1), above – concerning the action of the raising/lowering operators on l-weight modules is proved in §3, which gives the definition of the categories CP.
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