### A generalised Hopf algebra for solitons.

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A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.

We show by explicit calculations in the particular case of the 4-dimensional irreducible representation of ${\mathcal{U}}_{q}\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$ that it is not always possible to generalize to the quantum case the notion of symmetric algebra of a Lie algebra representation.

The affine Birman-Wenzl-Murakami algebras can be defined algebraically, via generators and relations, or geometrically as algebras of tangles in the solid torus, modulo Kauffman skein relations. We prove that the two versions are isomorphic, and we show that these algebras are free over any ground ring, with a basis similar to a well known basis of the affine Hecke algebra.

A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group $S{L}_{h}\left(2\right)$.

A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.

In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal...

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM\u25b9\u25c2k\left(G\right)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup ${G}_{\mathbb{O}}$ as transversal in an order 128 group $X$ with subgroup ${\mathbb{Z}}_{2}^{3}$ and hence obtain a Hopf quasigroup $k{G}_{\mathbb{O}}>\u25c2k\left({\mathbb{Z}}_{2}^{3}\right)$ as a particular case of our construction.

We present a generalization of the classical central limit theorem to the case of non-commuting random variables which are bm-independent and indexed by a partially ordered set. As the set of indices I we consider discrete lattices in symmetric positive cones, with the order given by the cones. We show that the limit measures have moments which satisfy recurrences generalizing the recurrence for the Catalan numbers.

We introduce a representation theory of q-Lie algebras defined earlier in [DG1], [DG2], formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules.