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

TL;DR

This paper explores the non-commutative geometric structures associated with quantum groups, specifically relating non-commutative projective spaces to representations of $U_q(sl_2)$, revealing geometric- algebraic correspondences.

Contribution

It establishes a novel connection between non-commutative projective geometry and the representation theory of quantum groups $U_q(sl_2)$.

Findings

Points, lines, and quadrics in non-commutative projective space correspond to quantum group representations.

Finite dimensional irreducible representations relate to geometric objects in the non-commutative space.

The incidence relations mirror homomorphisms and module structures in the quantum group context.

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 $\operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $\operatorname{Proj}_{nc}(S)$, 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.