Plane curves which are quantum homogenous spaces

Brown, K. and Tabiri, A. A. (2022) Plane curves which are quantum homogenous spaces. Algebras and Representation Theory, 25(4), pp. 889-922. (doi: 10.1007/s10468-021-10052-y)

Abstract

Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring O(C) of C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Brown, Professor Ken and Tabiri, Miss Angela
• Authors: Brown, K., and Tabiri, A. A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Algebras and Representation Theory
• Publisher: Springer
• ISSN: 1386-923X
• ISSN (Online): 1572-9079
• Published Online: 26 April 2021
• Copyright Holders: Copyright © 2021 The Authors
• First Published: First published in Algebras and Representation Theory 25(4): 889-922
• Publisher Policy: Reproduced under a Creative Commons licence

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Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
300872		Aspects of noncommutative geometry and noncommutative algebra	Kenneth Brown	Leverhulme Trust (LEVERHUL)	EM-2017-081\9	M&S - Mathematics