Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

Gravity and the Noncommutative Residue for Manifolds with Boundary

We prove a Kastler–Kalau–Walze type theorem for the Dirac operator and the signature operator for 3,4-dimensional manifolds with boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.      

An application of Bell's inequalities to a quantum state extension problem

Using techniques from the study of quantum violations of Bell's inequalities, we give examples of three C ^*-algebras A, B, C, and states ₁₂ on A B, and ₂₃ on B C, which agree on B, but do not have a common extension to A B C. This situation cannot occur in classical probability, i.e. for commutative algebras.     

Remarks on a quantum state extension problem

The problem is considered of finding, for a given pair of states on C ^*-algebras A ₁ A ₂ and A ₂ A ₃, a joint extension to A ₁ A ₂ A ₃. The fact that, in contrast to classical probability, such an extension may fail to exist, is related to the fact that different convex decompositions of the same quantum state need not have a common refinement. Improved necessary criteria for extensibility in terms of Bell's inequalities are derived, and are compared to the necessary and sufficient criteria, as well as to entropic bounds in the simplest case.     

Homogeneous Finsler spaces of negative curvature

We prove that a homogeneous Finsler space with non-positive flag curvature and strictly negative Ricci scalar is a simply connected manifold.     

Poisson Lie groups and pentagonal transformations

Symplectic pentagonal transformations are intimately related to global versions of Poisson Lie groups (Manin groups, S ^*-groups, or symplectic pseudogroups). Symplectic pentagonal transformations of cotangent bundles, preserving the natural polarization, are shown to be in one to one correspondence with pentagonal transformations of the base manifold with a cocycle (if the base is connected and simply connected). By the results of Baaj and Skandalis, this allows to quantize (at the C ^*-algebra level!) those Poisson Lie groups, whose associated symplectic pentagonal transformation admits an invariant polarization. The (2n)²-parameter family of Poisson deformations of the (2n+1)-dimensional Heisenberg group described by Szymczak and Zakrzewski is shown to fall into this case.Supported by Alexander von Humboldt Foundation. On leave from Department of Mathematical methods in Physics, Warsaw University, Poland.     

Bicovariant differential calculi on Sμ U(2)

Bicovariant, differential calculi on S U(2) are investigated and the nonexistence of three-dimensional calculus and the uniqueness of 4D_± calculi are proved. Formulae for the external product, external derivative, and classical limit are presented.     

An introduction to the theory of generalized conics and their applications

A generalized conic is a set of points with the same average distance from the pointset Γ$\Gamma$ in the Euclidean coordinate space. The measuring of the average distance is realized via integration over Γ$\Gamma$ as the set of foci. Using generalized conics we give a process for constructing convex bodies which are invariant under a fixed subgroup G$G$ of the orthogonal group in Rⁿ${\mathbb{R}}^n$. The motivation is to present the existence of non-Euclidean Minkowski functionals with G⊂O(n)$G\subset O\left(n\right)$ in the linear isometry group provided that the closure of G$G$ is not transitive on the unit sphere. As an application, consider Rⁿ${\mathbb{R}}^n$ as the tangent space at a point of a connected Riemannian manifold M$M$ and G$G$ as the holonomy group. If the holonomy group is not transitive on the unit sphere in the tangent space, then the Lévi-Civita connection is (re)metrizable in the sense that there is a smooth collection of non-Euclidean Minkowski functionals on the tangent spaces such that it is invariant under parallel transport with respect to the Lévi-Civita connection (according to Berger’s list of possible Riemannian holonomy groups, all of them are transitive on the unit sphere in the tangent space except in the case where the manifold is a symmetric space of rank≥2). We present the (re)metrizability theorem in a more general context of metrical linear connections with a torsion tensor that is not necessarily vanishing. This allows us to declare eight classes of manifolds equipped with an invariant smooth collection of Minkowski functionals on the tangent spaces. They are called Berwald manifolds in a general sense.     

Differential calculus on quantum Euclidean spheres

We study covariant differential calculus on the quantum Euclidean spheres S _q ^N−1 which are quantum homogeneous spaces with coactions of the quantum groups O _q (N). First order differential calculi on the quantum Euclidean spheres satisfying a dimension constraint are found and classified: ForN≥6, there exist exactly two such calculi one of which is closely related to the classical differential calculus in the commutative case. Higher order differential forms and symmetry are discussed. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Remarks on bicovariant differential calculi and exterior Hopf algebras

We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.     

Invariant fourth root Finsler metrics on the Grassmannian manifolds

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds SO(p+q)/SO(p)×SO(q)$SO(p+q)/SO\left(p\right)\times SO\left(q\right)$. By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2m$2m$-th root Finsler metrics are also given.     

Braided groups and algebraic quantum field theories

We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics ±1 replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.     

Matrix pseudogroups associated with anti-commutative plane

A C ^*-formulation of quantum 2×2 matrix groups for the deformation parameter q=–1 is given.     

A quantum-group-like structure on noncommutative 2-tori

We show that on noncommutative 2-tori, there are natural structures which have analogous formal properties as Hopf algebra structures, but where the comultiplication has values in a deformation of the tensor product.Supported by Project P 7724 PHY of Fonds zur Förderung der wissenschaftlichen Forschung.     

Conformal gauge fixing in Minkowski space

Conformal gauge fixing on simply connected parts of world sheets in Minkowski space is examined in detail. It is proved that conformal gauge fixing can be imposed, including the usual boundary conditions.