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

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}^\time...

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Published in:arXiv.org 2017-07
Main Authors: Chirvasitu, Alex, Smith, S Paul, Liang Ze Wong
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description 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.
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title Non-commutative Geometry of Homogenized Quantum \(\mathfrak{sl}(2,\mathbb{C})\)
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