Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$

TL;DR

This paper provides a simplified proof of the Rosso-Yamane Theorem concerning the PBW basis of the quantum group $U_q(A_N)$ using Groebner-Shirshov bases, enhancing understanding of its algebraic structure.

Contribution

It introduces a new, simplified proof of the Rosso-Yamane Theorem for the PBW basis of $U_q(A_N)$ utilizing Groebner-Shirshov bases, which was not previously established.

Findings

Simplified proof of Rosso-Yamane Theorem

Application of Groebner-Shirshov bases to quantum groups

Clarification of PBW basis structure for $U_q(A_N)$

Abstract

Let $U_q(A_N)$ be the Drinfeld-Jimbo quantum group of type $A_N$. In this paper, by using Groebner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$.