PBW-type filtration on quantum groups of type An

Fang, X., Fourier, G. and Reineke, M. (2016) PBW-type filtration on quantum groups of type An. Journal of Algebra, 449, pp. 321-345. (doi: 10.1016/j.jalgebra.2015.09.054)

Abstract

We will introduce an N-filtration on the negative part of a quantum group of type An, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.

• Item Type: Articles
• Additional Information: The work of Xin Fang is supported by the Alexander von Humboldt Foundation. The work of Ghislain Fourier is funded by the DFG priority program 1388 “Representation Theory” (grant FO 867/1-1).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fourier, Dr Ghislain
• Authors: Fang, X., Fourier, G., and Reineke, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Research Group: Algebra
• Journal Name: Journal of Algebra
• Publisher: Elsevier
• ISSN: 0021-8693
• Published Online: 17 November 2015
• Copyright Holders: Copyright © 2015 Elsevier Inc.
• First Published: First published in Journal of Algebra 449: 321-345
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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