Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases

TL;DR

This paper explores the relationship between quantized function algebras and bases of quantum groups, calculating transition matrices and demonstrating positivity in type A-D-E cases, extending Lusztig and Kato's results.

Contribution

It provides explicit calculations of transition matrices from canonical to PBW bases and links these to structure constants, confirming positivity for certain Lie types.

Findings

Transition matrices are described by structure constants of the quantum group.

Positivity of transition matrices is established for type A-D-E.

Constants match those from bilinear form calculations on quantum groups.

Abstract

Let $G$ be a connected simply-connected simple complex algebraic group and $\mathfrak{g}$ the corresponding simple Lie algebra. In the first half of the present paper, we study the relation between the positive part $U_q(\mathfrak{n^+})$ of the quantized enveloping algebra $U_q(\mathfrak{g})$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q}_q[G]$, taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. This work is motivated by Kuniba, Okado and Yamada's result together with Tanisaki and Saito's results. In the latter half, we calculate the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n^+})$ using the above relation. Consequently, we show that the constants arising from our calculation are described by the structure constants for the comultiplication of $U_q(\mathfrak{g})$. In particular, when $\mathfrak{g}$ is of type $ADE$, this result implies the positivity of the transition matrices, which was originally proved by Lusztig in the case when the PBW bases are associated with the adapted reduced words of the longest element of the Weyl group, and by Kato in arbitrary cases. In fact, the constants in our calculation coincide with ones arising from the calculation using the bilinear form on $U_q(\mathfrak{n}^{\pm})$. We explain this coincidence in Appendix.