Non-commutative Geometry of Homogenized Quantum $\mathfrak{sl}(2,\mathbb{C})$

Abstract

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

34 pages + references; minor modifications to address referee comments

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