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

DOI: 10.2140/pjm.2018.292.305

Abstract

We examine the relationship between certain noncommutative analogues of projective 3-space, ${\mathbb{P}}^3$, and the quantized enveloping algebras $U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$. The relationship is mediated by certain noncommutative graded algebras $S$, one for each $q\in {ℂ}^{\times }$, having a degree-two central element $c$ such that $S{\left[c^{-1}\right]}_0\cong U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$. The noncommutative analogues of ${\mathbb{P}}^3$ are the spaces ${Proj}_{nc}\left(S\right)$. We show how the points, fat points, lines, and quadrics, in ${Proj}_{nc}\left(S\right)$, and their incidence relations, correspond to finite-dimensional irreducible representations of $U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$, 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