The $(q,t)$-Cartan matrix specialized at $q=1$

TL;DR

This paper explores the $t$-quantized Cartan matrix obtained by specializing the $(q,t)$-Cartan matrix at $q=1$, revealing its connections with quantum unipotent coordinate algebra, root systems, and quantum cluster algebras of skew-symmetrizable type.

Contribution

It introduces and studies the $t$-quantized Cartan matrix and its relations to quantum unipotent coordinate algebra and quantum cluster algebra of skew-symmetrizable type.

Findings

Establishes the properties of the $t$-quantized Cartan matrix.

Links the $t$-quantized Cartan matrix to quantum unipotent coordinate algebra.

Explores the role of the $t$-quantized Cartan matrix in quantum cluster algebra of skew-symmetrizable type.

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 kew-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 quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.