The 𝑞-characters of representations of quantum affine algebras and deformations of 𝒲-algebras

Abstract

We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed Walgebras.

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

“…In the limit ε 2 → 0 X w,ν (x) reduces to the Yangian q-characters of finite-dimensional representations of the Yangian Y (g γ ), constructed for finite γ in [60]. In [38] the qcharacters for the quantum affine algebras U q (g γ ) for finite γ's and in [49] for affine γ's are constructed. These correspond to the K-theoretic version of our story in the limit q 2 → 1, q 1 = q finite, which was discussed in [86].…”

“…It would be interesting to apply these ideas to topological strings and to topological gravity. On a more mathematical note, let us discuss the relation of our qq-characters to the tdeformation of q-characters of [38], introduced by H. Nakajima in [75,[78][79][80]. His definition is basically the weighted sum of the Poincare polynomials of the H w,γ -fixed loci on M(w, v).…”

Vertex Algebras and Algebraic Curves

“…For example, because of lack of space, I have not discussed such important topics as the theory of conformal algebras [K2, K3] and their chiral counterpart, Lie * algebras [BD2]; quantum deformations of vertex algebras [B3,EK,FR]; and the connection between vertex algebras and integrable systems.…”

The Solution of the Quantum A 1 T-System for Arbitrary Boundary

“…These relations are called T -systems. In the context of representation theory, these relations are the equations satisfied by the qcharacters [16] of Kirillov-Reshetikhin modules of the Yangians, or the associated quantum affine algebra.…”

“…The T -systems [18,21] satisfied by the transfer matrices of the generalized Heisenberg model or the q-characters of quantum affine algebras [16] can be considered as discrete dynamical systems with special initial conditions. More generally, the equations of these systems can be shown [6] to be mutations in an infinite-rank cluster algebra [13].…”