Convex Bases of PBW type for Quantum Affine Algebras

TL;DR

This paper proves an isomorphism for certain sequences and constructs a convex PBW basis for affine quantum groups, extending braid group actions with outer automorphisms.

Contribution

It establishes an isomorphism for admissible sequences and introduces a convex PBW basis for affine quantum algebras, expanding the algebraic framework.

Findings

Proves the isomorphism for certain admissible sequences.

Constructs a convex PBW basis for affine quantum groups.

Extends braid group actions with outer automorphisms.

Abstract

This note has two purposes. First we establish that the map defined in [L, $\S 40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for $\bup$, an affine untwisted quantized enveloping algebra of Drinfel$'$d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra.