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Some complex quantum manifolds and their geometry. (English) Zbl 0938.58005

’t Hooft, Gerard (ed.) et al., Quantum fields and quantum space time. Summer school, Cargèse, Corsica, France, July 22 - August 3, 1996. New York, NY: Plenum Publishing Corporation. NATO ASI Ser., Ser. B, Phys. 364, 281-322 (1997).
From the introduction: “Quantum spheres can be defined in any number of dimensions by normalizing a vector of quantum Euclidean space. The differential calculus on quantum Euclidean space induces a calculus on the quantum sphere. The case of two-spheres in three-space is special in that there are many more possibilities. These have been studied by P. Podleś who defined quantum spheres as quantum spaces on which quantum \(SU_q(2)\) coacts. He also developed a noncommutative differential calculus on them.
In these lectures we consider, following [C. Chu, P. Ho and B. Zumino, Z. Phys. C 70, No. 2, 339-344 (1996; Zbl 0915.58010)], a special case of Podleś spheres which is one of those special to three space dimensions. In this case, the quantum sphere \(S^2_q\) is the analogue of the classical sphere defined as \(SU(2)/U(1)\) or as isomorphic to \(\mathbb{C} P(1)\). We also define a stereographic projection and describe the coaction of \(SU_q(2)\) on the sphere by fractional transformations on the complex variable in the plane analogous to the classical ones. The quantum sphere appears then as the quantum deformation of the classical two-sphere described as a complex Kähler manifold. We discuss the differential and integral calculus on \(S^2_q\) and the action of \(SU_q(2)\) vector fields on it. Finally, following [C. Chu, P. Ho and B. Zumino, Mod. Phys. Lett. A 11, No. 4, 307-316 (1996)], we show that one can define on braided copies of \(S^2_q\) invariant anharmonic cross ratios analogous to the classical ones. All this is done in Section 3, after recalling briefly in Section 2 the basic properties of \(GL_q(2)\) and \(SU_q(2)\).
The above results are generalized in Section 4 and 5 to quantum \(\mathbb{C} P_q(N)\) and in Section 6 to quantum Grassmannians. These quantum spaces appear as complex Kähler quantum manifolds which can be described in terms of homogeneous or inhomogeneous coordinates. Differential and integral calculus can be defined on them as well as the quantum analogues of projective invariants. For the general case of Grassmannians, we do not give explicit formulas for the integral and for the projective invariants. They should not be hard to derive by analogy with the \(\mathbb{C} P_q(N)\) case. The type of quantization described here has the property that there exists a special quantum (connection) one-form which generates the differential calculus by taking commutators or anticommutators of it with functions or forms. This one-form is closely related to the Kähler form which can be obtained from it by differentiation. In the Poisson limit, our quantization gives Poisson brackets not only between functions but also between functions and forms and between forms. The special one-form generates the calculus by taking Poisson brackets with functions or forms and the Kähler form can still be obtained from it by differentiation. Our Poisson structure on the manifold is singular and is not the standard one which is obtained by taking the Kähler form as symplectic form. Nevertheless, our Poisson structure is intimately related to the Kähler form, as just explained.
All formulas and derivations of Section 3 can be easily modified, with a few changes of signs, to describe the quantum unit disk and the coaction of quantum \(SU_q(1,1)\) on it, as well as the corresponding invariant anharmonic ratios. This provides a quantum deformation of the Bolyai-Lobachevskij non-Euclidean plane and of the differential and integral calculus on it. The modified equations can be guessed very easily and will not be given here. It should be mentioned that the commutation relations between \(z\) and \(\overline z\) for the unit disk are consistent with a representation of \(z\) and \(\overline z\) as bounded operators in a Hilbert space. This is to be contrasted with the case of the quantum sphere where \(z\) and \(\overline z\) must be unbounded operators. The developments of Sections 4 and 6 can similarly be modified, again with some changes of signs, to describe a quantum deformation of various higher dimensional non-Euclidean geometries.
Finally, in Appendix A, we try to re-formulate the differential and integral calculus on the quantum sphere in a way as close as possible to Connes’ formulation of quantum Riemannian geometry”.
For the entire collection see [Zbl 0894.00053].

MSC:

58B32 Geometry of quantum groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Citations:

Zbl 0915.58010
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