- overview-regular-functions
Replace X with algebra of functions on X (called "regular functions")
[can recover space as maximal ideals]
Esp:
(geometrically)
- regular functions on algebraic variety
- smooth functions on manifold
- continuous functions on locally compact Hausdorff space
(analog of algebraic:)
- linear algebra: dual
- representation theory: characters
[EG of duality / Galois connection]
point is that you have algebraic objects that capture geometry / topology
(you can now apply tools of algebra;
some/many things much clearer)
[analytic versions:
Laplacian, heat kernel]
...and now replace algebra with operators on the algebra ?
(or not? C^* algebras are functions on a space, or operators on a Hilbert space...)
(C^*-algebras:
recover X as: the space of characters equipped with the weak* topology;
C^* ~= C_0(X))
Leads to:
-> quantum groups (Hopf algebras; deformations of nice Hopf algebras, like funcs on group, or UEA)