On the theory of $q$-characters for quantum affine superalgebras of type $A$

Abstract

We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of the associated generalized quantum group, from which one can read off a 2-parameter deformation of Cartan matrices of super type $A$. We also propose a Frenkel-Mukhin-type algorithm for $q$-characters of finite-dimensional simple modules with integral highest $\ell$-weights.

61 pages, v2: this is the version to be submitted for publication where revisions are made throughout the article; large part of the introduction and Section 3.5 rewritten, Section 5.2 on the application to the universal coefficient and the denominator of R-matrices and Appendix A newly added, a number of errors and typos are fixed