Vol. 292, No. 2, 2018

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ISSN: 0030-8730
Noncommutative geometry of homogenized quantum $\mathfrak{sl}(2,\mathbb{C})$

Alex Chirvasitu, S. Paul Smith and Liang Ze Wong

Vol. 292 (2018), No. 2, 305–354
Abstract

We examine the relationship between certain noncommutative analogues of projective 3-space, 3, and the quantized enveloping algebras Uq(sl2). The relationship is mediated by certain noncommutative graded algebras S, one for each q ×, having a degree-two central element c such that S[c1]0Uq(sl2). The noncommutative analogues of 3 are the spaces Projnc(S). We show how the points, fat points, lines, and quadrics, in Projnc(S), and their incidence relations, correspond to finite-dimensional irreducible representations of Uq(sl2), Verma modules, annihilators of Verma modules, and homomorphisms between them.

Keywords
noncommutative algebraic geometry, quantum groups, quantum $\mathfrak{sl}_2$
Mathematical Subject Classification 2010
Primary: 14A22, 16S38, 16W50, 17B37
Milestones
Received: 12 December 2016
Revised: 14 July 2017
Accepted: 24 July 2017
Published: 17 October 2017
Authors
Alex Chirvasitu
Department of Mathematics
University at Buffalo
Buffalo, NY
United States
Department of Mathematics
University of Washington
Seattle, WA
United States
S. Paul Smith
Department of Mathematics
University of Washington
Seattle, WA
United States
Liang Ze Wong
Department of Mathematics
University of Washington
Seattle, WA
United States