1. Introduction. This is a first of our papers devoted to " noncom-mutative topology and graph theory ". Its origin is the paper [GRW] where a new class of noncommutative algebras Q n was introduced. As explained in [GRW], the algebra Q n is closely related to decompositions of a generic polynomial P (t) of degree n over a division algebra into linear factors. The structure of the algebra Q n s...

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard q-deformation U ′ q(son) (which does not coincide with the Drinfel’d-Jimbo quantum algebra Uq(son)) of the universal enveloping algebra U(son(C)) of the Lie algebra son(C) when q is not a root of unity. These representations are exhausted by irreducible representations...

Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James’ q-Schur algebra, and prove a graded analogue of the Leclerc and Thibon’s conjecture on the decomposition numbers of the q-Schur algebra.

Using q-operator product expansions between U (1) current fields and also the corresponding energy-momentum tensors, we furnish the q-analoques of the generalized Heisenberg algebra and the Krichever-Novikov algebra.

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove analogue de Finetti’s theorem for this type In addition, since vacuum state on q-deformed $$C^*$$ -algebra is only when $$|q|<1$$ , we draw our attention its algebra, which turns out obey a zero-one law.

We consider the conditions under which the q-oscillator algebra becomes a Hopf *-algebra. In particular, we show that there are at least two real forms associated with the algebra. Furthermore, through the representations, it is shown that they are related to su q 1/2 (2)

We show that the differential complex Ω B over the braided matrix algebra BM q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra Ω A which is a differential extension of GL q (N). On the other hand, the algebra Ω A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra Ω B. Geometrical aspects of these results are discussed.

Let S(Λ) be the cyclotomic q-Schur algebra associated to the ArikiKoike algebra H . We construct a certain subalgebra S(Λ) of S(Λ), and show that it is a standardly based algebra in the sense of Du and Rui. S(Λ) has a natural quotient S(Λ), which turns out to be a cellular algebra. In the case where the modified Ariki-Koike algebra H ♭ is defined, S(Λ) coincides with the cyclotomic q-Schur alge...