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In this paper we introduce a kind of 'noncommutative neighbourhood' of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a 'distortion' of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow. [ABSTRACT FROM AUTHOR]

Given the ring of integers O of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL( n, O) such that O G, the O-span of G, coincides with M( n, O), the ring of ( n × n)-matrices over O? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2 m there is also a finite group G ⊂ GL(2 m, O) such that O G = M(2 m, O). [ABSTRACT FROM AUTHOR]

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product. [ABSTRACT FROM AUTHOR]

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H *-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products. [ABSTRACT FROM AUTHOR]

We apply to Hopf algebras a construction from graded rings, called “the group ring of a graded ring”. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra. [ABSTRACT FROM AUTHOR]