Noncommutative geometry and theoretical physics
Robert Coquereaux1989, Journal of Geometry and Physics
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Abstract
The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.
Key takeaways
- First, the algebra of functions on a manifold is replaced by an arbitrary associativebut not necessarily commutativealgebra A ; then we introduce in Section 2 the universal differential algebra L (A) and in Section 3 the Hochshild cohomology H* which plays the analogue of the algebra of differential forms on a manifold.
- In the last subsection, we saw the Hochshild homology of the commutative algebra C°°(X) was related to the set of differential forms A(X) ; but this set, endowed with the laws of addition and exterior multiplication is itself a noncommutative (and Z 2 -graded) algebra.
- 2) If A is an algebra, then the set of m x m matrices with elements in A is also an algebra (all the algebra we are dealing with are supposed to be associative) and it is natural to ask what the Hochshild cohomology of M,,( A) is.
- In the noncommutative framework, the notion of vector bundles (or better the notion of space of sections of a vector bundle) is replaced by the notion of projective modules of finite type over an algebra A , or equivalently, by projectors in the algebra M~(A).
- Quantum Groups are then obtained by replacing the commutative algebra A by an arbitrary noncommutative algebra.
