The model of generalized quons is described in a purely algebraic way. Commutation relations and corresponding consistency conditions for our generalized quons system are studied in terms of quantum Weyl algebras. Fock space representation and corresponding scalar product is also given. *
This is the translation to appear in the "SUPERSYMMETRY 2000 -Encyclopaedic Dictionary" of the original Comptes Rendus Note, published in March 1980, in which basic notions of noncommutative geometry were introduced and applied to noncommutative tori. These include connections on finite projective modules, their curvature, and the Chern character. Finite projective modules on the noncommutative two-torus T 2 θ were realized as Schwartz spaces of vector valued functions on R. Explicit constant curvature connections were constructed and a basic integrality phenomenon of the total curvature was displayed. The pseudo-differential calculus and the Atiyah-Singer index theorems were extended to Lie group actions on C * algebras and used to explain the above integrality of the total curvature by an index formula for finite difference-differential operators on the line. Recent interest in the hep-th literature for basic notions of noncommutative geometry in the case of noncommutative tori (cf for instance hep-th/0012145 for an excellent review) prompted us to make the English translation of the original paper available.
The concept of a crossed tensor product of algebras is studied from a few
points of views. Some related constructions are considered. Crossed enveloping
algebras and their representations are discussed. Applications to the
noncommutative geometry and particle systems with generalized statistics are
indicated.Comment: 20 pages, latex2e (latexsym), no figures, accepted for publication in
J. Math. Phys. 41 no10 (2000
A proposal of the concept of n-regular obstructed categories is given. The corresponding regularity conditions for mappings, morphisms and related structures in categories are considered. An nregular TQFT is introduced. It is shown the connection of time reversibility with the regularity.PACS. 03. 65. Fd -Algebraic methods in quantum theory
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