Braided Commutative Geometry and Drinfel'd Twist Deformations

TL;DR

This thesis explores obstructions to Drinfel'd twist deformation quantization on symplectic manifolds and develops a unified noncommutative Cartan calculus applicable to braided commutative algebras, extending classical differential geometry.

Contribution

It introduces a noncommutative Cartan calculus and equivariant Levi-Civita derivative within braided commutative geometry, unifying classical and quantum geometric frameworks.

Findings

Identifies obstructions to twist deformation quantization on symplectic manifolds.

Constructs a noncommutative Cartan calculus applicable to braided commutative algebras.

Shows the Drinfel'd functor induces equivalence classes in braided geometry.

Abstract

In this thesis we give obstructions for Drinfel'd twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfel'd functor leads to equivalence classes in braided commutative geometry and commutes with submanifold algebra projection.