2015/01/07 by Hironori Oya, Oya, Hironori · 1 citation
Mathematics · #17B37 #20G42 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B37 #msc:20G42
paper · pdf · doi:10.48550/arxiv.1501.01416
47 pages, v2:added Cor 3.11, 3.12 Prop 4.14, 4.17 and Appendix. The proof of Theorem 5.15 (in this version) was simplified by Proposition 4.17. The definitions of the PBW bases of U_q(n^-) and \ket{m} were changed, v3:modified Abstract and Introduction. added references and Remark A.8. Some proofs were shortened
openalex publication_date 2015/01/07 · arxiv created 2015/07/03 · arxiv updated 2015/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected simply-connected simple complex algebraic group and \mathfrakg the corresponding simple Lie algebra. In the first half of the present paper, we study the relation between the positive part Uq(\mathfrakn+) of the quantized enveloping algebra Uq(\mathfrakg) and the specific irreducible representations of the quantized function algebra ℚq[G], taking into account the right Uq(\mathfrakg)-algebra structure of ℚq[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 Uq(\mathfrakn+) using the above relation. Consequently, we show that the constants arising from our calculation are described by the structure constants for the comultiplication of Uq(\mathfrakg). In particular, when \mathfrakg 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 Uq(\mathfrakn±). We explain this coincidence in Appendix.