Spectral characters of finite-dimensional representations of affine algebras

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

“…In eight dimensions, or for the theories in four dimensions with adjoint matter, it will be useful to have four parameters 20) which sum to zero:…”

Finite-Dimensional Representations of Hyper Loop Algebras over Non-algebraically Closed Fields

“…By using our results on ample-forms, this is reduced to proving the claim when K = F in which case it can be reformulated as follows: the ℓ-weights of an indecomposable finite-dimensional U (g) F -module differ by an element of the ℓ-root lattice Q F . In characteristic zero this follows from [3]. The positive characteristic case follows from the characteristic zero one using a corollary of the conjecture of [13].…”

“…In order to do that we first review the argument used in [3,13] to prove this in the case K = F. The first important step is the next proposition (see [3,Proposition 3.4] in characteristic zero and [13,Proposition 4.16] in positive characteristic).…”

“…The overall scheme of the study is the same as that used in [8,4], [3], and [13] to study the blocks of the categories of finite-dimensional representations of quantum, classical, and hyper loop algebras, respectively (see also [2]). Namely, the study is split in two parts -the first one consists in showing that the category in question is a direct sum of certain abelian subcategories and the second one in showing that these subcategories are indecomposable.…”

“…The positive characteristic case follows from the characteristic zero one using a corollary of the conjecture of [13]. For the second step, the tensor product results were the main tools used in [8,3,4,13] providing a way of constructing a sequence of indecomposable modules linking two irreducible ones having the same spectral character. The same technique does not work when K is not algebraically closed since tensor products are not as well-behaved.…”