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Symmetry, Integrability and Geometry: Methods and Applications, 2013, Volume 9, 049, 23 pp.
DOI: https://doi.org/10.3842/SIGMA.2013.049
(Mi sigma832)
 

This article is cited in 12 scientific papers (total in 12 papers)

A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions

Atsuo Kunibaa, Masato Okadob, Yasuhiko Yamadac

a Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
c Department of Mathematics, Faculty of Science, Kobe University, Hyogo 657-8501, Japan
References:
Abstract: For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U^+_q(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.
Keywords: quantized enveloping algebra; PBW bases; quantized algebra of functions; tetrahedron equation.
Received: March 19, 2013; in final form July 10, 2013; Published online July 19, 2013
Bibliographic databases:
Document Type: Article
Language: English
Citation: Atsuo Kuniba, Masato Okado, Yasuhiko Yamada, “A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions”, SIGMA, 9 (2013), 049, 23 pp.
Citation in format AMSBIB
\Bibitem{KunOkaYam13}
\by Atsuo~Kuniba, Masato~Okado, Yasuhiko~Yamada
\paper A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
\jour SIGMA
\yr 2013
\vol 9
\papernumber 049
\totalpages 23
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\crossref{https://doi.org/10.3842/SIGMA.2013.049}
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  • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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