Modules over quantized coordinate algebras and PBW-bases

2014/09/29 by Toshiyuki Tanisaki, T. Tanisaki, Tanisaki, T. · 2 citations
Mathematics · #17B37 #20G05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B37 #msc:20G05

paper · pdf · doi:10.48550/arxiv.1409.7973

47pages

openalex publication_date 2014/09/29 · arxiv created 2015/10/21 · arxiv updated 2015/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Around 1990 Soibelman constructed certain irreducible modules over the quantized coordinate algebra. A. Kuniba, M. Okado, Y. Yamada recently found that the relation among natural bases of Soibelman's irreducible module can be described using the relation among the PBW-type bases of the positive part of the quantized enveloping algebra, and proved this fact using case-by-case analysis in rank two cases. In this paper we will give a realization of Soibelman's module as an induced module, and give a unified proof of the above result of Kuniba, Okado, Yamada. We also verify Conjecture 1 of Kuniba, Okado, Yamada about certain operators on Soibelman's module.

Citations

• Quantized coordinate rings, PBW-type bases and q-boson algebras
• Some quantum analogues of solvable Lie groups

Cited by

• Symmetric bilinear form on a Lie algebra
• Partially compactified quantum cluster structures on simple algebraic groups and the full Berenstein--Zelevinsky conjecture

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