#### Vol. 292, No. 2, 2018

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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, ${ℙ}^{3}$, 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 $q\in {ℂ}^{×}$, 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 ${ℙ}^{3}$ 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