Moduli space and structure of noncommutative 3-spheres

Details Moduli space and structure of noncommutative 3-spheres Moduli space and structure of noncommutative 3-spheres

2003-08-28

arxiv.org/abs/math/0308275v2

Lett.Math.Phys. 66 (2003) 91-121

Mathematics

Quantum Algebra

50 pages. References added

Scientific paper

10.1023/B:MATH.0000017678.10681.

We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the $C^\ast$-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric spaceof unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering $\pi$ by a noncommutative 3-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering $\pi$ by pairing the direct image of the fundamental class of the noncommutative 3--dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function...

Affiliated with

Also associated with

No associations

Rating

Moduli space and structure of noncommutative 3-spheres does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Moduli space and structure of noncommutative 3-spheres, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Moduli space and structure of noncommutative 3-spheres will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-340914