The (q, t)-Cartan matrix specialized at q=1 and its applications

Abstract

The (q, t)-Cartan matrix specialized at t = 1, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type. In this paper, we study the (q, t)-Cartan matrix specialized at q = 1, called the t -quantized Cartan matrix, and investigate the relations with (ii') its corresponding unipotent quantum coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.