Monomial bases and canonical bases for quantum affine algebras

TL;DR

This paper constructs a monomial basis for quantum affine algebras of simply-laced type, providing an algorithm to compute the canonical basis and relating it to Lusztig's geometric construction.

Contribution

It introduces a new monomial basis linked to PBW bases and presents an algorithm for canonical basis computation in quantum affine algebras.

Findings

Constructed a monomial basis associated with PBW basis

Developed a simple algorithm for canonical basis computation

Connected the canonical basis with Lusztig's geometric basis

Abstract

We construct a monomial basis of a quantum affine algebra of simply-laced type, associated to the PBW basis of Beck-Nakajima. We show that there exists a simple algorithm of computing canonical basis in terms of the monomial basis. We dsicuss the relations of the canonical basis obtained from this PBW basis with Lusztig's canonical basis constructed by using the geometry of quivers.