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Moduli space and structure of noncommutative 3-spheres. (English) Zbl 1052.58012

The paper deals with so-called noncommutative three-dimensional spheres which are defined via duality by some deformations of the spaces of polynomial functions on the standard sphere \(S^3 \subset {\mathbb R}^4\). These deformations have to satisfy some equations of the \(K\)-theoretic origin, This paper is a sequel to a paper by the same authors [Commun. Math. Phys. 230, No. 3, 539–579 (2002; Zbl 1026.58005)] where a three-parameter family of such deformations was found and, in particular, it was shown that the corresponding deformations of the polynomial algebra on \({\mathbb R}^4\) are isomorphic to the Sklyanin algebras introduced in the framework of the Yang–Baxter equation.
The results of the present paper are summarized in the authors’ abstract as follows: “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 space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified with 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”.

MSC:

58B34 Noncommutative geometry (à la Connes)
53C35 Differential geometry of symmetric spaces
46L87 Noncommutative differential geometry
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory

Citations:

Zbl 1026.58005
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