Weber, Thomas (2019) Braided Commutative Geometry and Drinfel'd Twist Deformations. [Tesi di dottorato]


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Item Type: Tesi di dottorato
Lingua: English
Title: Braided Commutative Geometry and Drinfel'd Twist Deformations
Date: 9 December 2019
Number of Pages: 152
Institution: Università degli Studi di Napoli Federico II
Department: Matematica e Applicazioni "Renato Caccioppoli"
Dottorato: Scienze matematiche e informatiche
Ciclo di dottorato: 32
Coordinatore del Corso di dottorato:
De Giovanni,
D'Andrea, FrancescoUNSPECIFIED
Date: 9 December 2019
Number of Pages: 152
Uncontrolled Keywords: Quantum Algebra, Mathematical Physics, Deformation Quantization, Drinfel'd Twists.
Settori scientifico-disciplinari del MIUR: Area 01 - Scienze matematiche e informatiche > MAT/03 - Geometria
Area 01 - Scienze matematiche e informatiche > MAT/07 - Fisica matematica
Date Deposited: 13 Jan 2020 13:05
Last Modified: 17 Nov 2021 12:18


This thesis revolves around the notion of twist star product, which is a certain type of deformation quantization induced by quantizations of a symmetry of the system. On one hand we discuss obstructions of twist star products, while on the other hand we provide a recipe to obtain new examples as projections from known ones. Furthermore, we construct a noncommutative Cartan calculus on braided commutative algebras, generalizing the calculus on twist star product algebras. The starting point is the observation that Drinfel'd twists not only deform the algebraic structure of quasi-triangular Hopf algebras and their representations but also induce star products on Poisson manifolds with symmetry. We further investigate the correspondence of Drinfel'd twists and classical r-matrices as well as twist deformation of Morita equivalence bimodules. It turns out that connected compact symplectic manifolds are homogeneous spaces if they admit a twist star product and that we can assume the corresponding classical r-matrix to be non-degenerate. Furthermore, invariant line bundles with non-trivial Chern class and twists star products cannot coexist if they are based on the same symmetry. In particular, the symplectic 2-sphere and the connected orientable symplectic Riemann surfaces of genus g > 1 do not admit a star product induced by a Drinfel'd twist, while the complex projective spaces cannot be endowed with a twist star product based on a matrix Lie algebra. Another goal of the thesis is to study braided commutative algebras and provide a noncommutative Cartan calculus on them, in complete analogy to differential geometry. Notably, this recovers the calculus on twist star product algebras. We further discuss equivariant covariant derivatives and metrics, resulting in the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. We also verify that the constructions are compatible with Drinfel'd twist gauge equivalences. Under certain conditions, the braided geometry projects to submanifold algebras and twist deformation commutes with the projection. As a consequence, twisted products can be projected to submanifold algebras if the latter are respected by the symmetry.


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