The God has created a man in order that he creates that the God fails to do

Monday, 23 May 2011

Non-commutative geometry meets a serious problem

Non-commutative geometry is developed as the differential calculus on modules over a ring A when A is non-commutative. Note that, if A=C(X) is the commutative ring of smooth real functions on a manifold X, we are in the case of familiar differential geometry of vector bundles over X. If A is a graded commutative ring, one comes to supergeometry. However, one meets a difficulty if A is a non-commutative ring. In this case, higher order differential operators are ill defined. A problem lies in the fact that, in contrast with the case of a commutative or graded commutative ring, a multiplication p->ap fail to be a zero order differential operator if an element a of a ring A does not belong to its center.


G. Sardanashvily, Lectures on differentail geometry of modules and rings, arXiv: 0910.1515

1 comment:

  1. I've always been curious, is there some relation between this approach using the commutative ring of smooth real functions on a manifold...and Morse theory?

    [You know, writing "→" will produce an arrow → (which may be used instead of "->").]