Pseudosymmetric braidings, twines and twisted algebras
Faculty of Sciences. Mathematics and Computer Science

article

2010
Amsterdam
, 2010

Mathematics

Journal of pure and applied algebra. - Amsterdam

214(2010)
:6
, p. 867-884

0022-4049

000274928100014

E

English (eng)

University of Antwerp

A laycle is the categorical analogue of a lazy cocycle. Twines (introduced by Bruguières) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If c is a braiding, the double braiding c2 is always a twine; we prove that it is a strong twine if and only if c satisfies a sort of modified braid relation (we call such c pseudosymmetric, as any symmetric braiding satisfies this relation). It is known that the category of YetterDrinfeld modules over a Hopf algebra H is symmetric if and only if H is trivial; we prove that the YetterDrinfeld category View the MathML source over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the 2n+1-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by the properties of laycles and twines.

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