A Brief Introduction to Quantum Geometry

Table of Contents

Introduction
From Geometry to Algebra: 1
Part 2
Non-commutative Generalizations
Examples
Spaces With Indistinguishable Points
Quantum Groups and Quantum Bundles
Quantum Tori and Matrix Spaces
Supermanifolds
Quantization of Symplectic Manifolds
C*-Algebraic Extensions and BDF-Theory
Literature

Reformulating Basic Geometrical Concepts [2/2]

Vector Bundles

Let us now assume that X is a compact smooth manifold, and let vector-bundle be a smooth complex vector bundle over X. Let vector-bundle-sections be the space of smooth sections of E. This space is a finite and projective module over the *-algebra c-infinity of smooth functions on X.

Conversely, if Gamma is an arbitrary finite and projective module over smooth-functions then there exists a unique (up to the isomorphisms) smooth vector bundle E over X such that sections-interpretation.

Vector Fields

Let xi be a smooth vector field on X and let lie-derivative be the corresponding Lie derivative (the derivation along xi). The map D satisfies the Leibniz rule

d-leibniz-rule

in other words it is a derivation on cal-A. Moreover, D is hermitian in the sense that D*=*D, as long as xi is a real vector field.

Conversely, if Derivation is an arbitrary hermitian derivation on smooth-A then there exists a unique (real) vector field xi on X such that d-recovery. In other words, there exists a natural bijection between vector fields on X and hermitian derivations on smooth-A. If we relax the hermicity assumption, then we obtain a correspondence between all derivations on smooth-A and complex vector fields on X.

Differential Forms

The algebra differential-forms of smooth differential forms over a compact smooth manifold X can be constructed as follows.

Let us first consider the Lie algebra smooth-derivations of all derivations over smooth-A (vector fields, in accordance with the previous paragraph). This algebra is naturally acting in smooth-A, so we can construct the Chevalley complex chevalley-complex. This space possesses a natural graded-differential *-algebra structure. By definition, the elements of n-th degree bzi are all possible n-linear antisymmetric maps

definition-omegas

The differential in C is given by

definition-of-diferential

where, by definition, the symbols within the hats are omitted from the expression. The *-algebra structure is defined by

definition-of-*

Here eta-in-complex and shuffle-subsystem is a subset cosisting of all (k+l)-permutations preserving the orders of the first k, and the last l elements.

By definition, we put smooth-functions-id. The algebra of differential forms diff-forms can be viewed as a differential subalgebra of C generated by smooth-A. In other words, the spaces forms-k are additively spanned by the elements of the form

w
where ai-in-smooth-A. We can summarize our discussion so far in the following table, which is a kind of a mini dictionary between geometry and algebra:

Elementary Geometry-Algebra Dictionary
Compact topological spaces X Unital commutative C*-algebras A
X=compact topological space A=C(X)={complex continuous
functions on X}
Points x-in-X Characters kappa-chars
Continuous maps between compacts X and Y-corresponds-2B Unital *-homomorphisms from B to A
The direct product X×Y The C*-tensor product tensor-product-*
Symmetries of X Automorphisms of A
Group structure on X Coproduct map Phi
Probabilty measures on a metrizable compact X Positive normalized linear functionals on A
Locally-compact noncompact topological spaces X Non-unital commutative C*-algebras A
Pure measure theory Commutative von Neumann algebras
X=compact smooth manifold smooth-A-X ={complex smooth functions on X}
Vector bundles over X Finite projective modules over smooth-A
Vector fields on X Hermitian derivations on smooth-A
Differential forms on X Graded-Differential algebra differential-forms-incl

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