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A note on coalgebra gauge theory

Tomasz Brzeziński (1997)

Banach Center Publications

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

A U q s l 2 -representation with no quantum symmetric algebra

Olivia Rossi-Doria (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show by explicit calculations in the particular case of the 4-dimensional irreducible representation of U q s l 2 that it is not always possible to generalize to the quantum case the notion of symmetric algebra of a Lie algebra representation.

Affine Birman-Wenzl-Murakami algebras and tangles in the solid torus

Frederick M. Goodman, Holly Hauschild (2006)

Fundamenta Mathematicae

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.

An infinite torus braid yields a categorified Jones-Wenzl projector

Lev Rozansky (2014)

Fundamenta Mathematicae

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.

Bicovariant differential calculi and cross products on braided Hopf algebras

Yuri Bespalov, Bernhard Drabant (1997)

Banach Center Publications

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...

Bicrossproduct Hopf quasigroups

Jennifer Klim, Shahn Majid (2010)

Commentationes Mathematicae Universitatis Carolinae

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup k M k ( G ) from every group X with a finite subgroup G X and IP quasigroup transversal M X subject to certain conditions. We identify the octonions quasigroup G 𝕆 as transversal in an order 128 group X with subgroup 2 3 and hence obtain a Hopf quasigroup k G 𝕆 > k ( 2 3 ) as a particular case of our construction.

bm-independence and central limit theorems associated with symmetric cones

Janusz Wysoczański (2007)

Banach Center Publications

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.

Braided modules and reflection equations

Dimitri Gurevich (1997)

Banach Center Publications

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.

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