31 research outputs found
Bialgebra cohomology, pointed Hopf algebras, and deformations
We give explicit formulas for maps in a long exact sequence connecting
bialgebra cohomology to Hochschild cohomology. We give a sufficient condition
for the connecting homomorphism to be surjective. We apply these results to
compute all bialgebra two-cocycles of certain Radford biproducts
(bosonizations). These two-cocycles are precisely those associated to the
finite dimensional pointed Hopf algebras in the recent classification of
Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as
graded bialgebra deformations of Radford biproducts.Comment: Cohomological results in the paper were significantly improved and
generalized. See new abstract for detail
The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables
We identify a collection of primitive elements generating the Hopf algebra
NCSym of symmetric functions in noncommuting variables and give a combinatorial
formula for the antipode.Comment: 8 pages; footnote added; references added; further remarks adde
Double-ended queues and joint moments of left-right canonical operators on full Fock space
We follow the guiding line offered by canonical operators on the full Fock
space, in order to identify what kind of cumulant functionals should be
considered for the concept of bi-free independence introduced in the recent
work of Voiculescu. By following this guiding line we arrive to consider, for a
general noncommutative probability space (A, phi), a family of "(l,r)-cumulant
functionals" which enlarges the family of free cumulant functionals of the
space. In the motivating case of canonical operators on the full Fock space we
find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple
(A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the
left and B_1, ... , B_d canonical operators on the right. This extends a known
one-sided formula for free cumulants of A_1, ..., A_d, which establishes a
basic operator model for the R-transform of free probability.Comment: In this (final) version, the introduction was re-written to better
show the motivation for the question considered in the pape
Deformation by cocycles of pointed Hopf algebras over non-abelian groups
We introduce a method to construct explicitly multiplicative 2-cocycles for
bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles
arise as liftings of H-invariant linear functionals on V tensor V and give a
close formula to deform braided commutator-type relations.
Using this construction, we show that all known finite dimensional pointed
Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric
group S_3 and some families over S_4 are cocycle deformations of bosonizations
of Nichols algebras.Comment: 20 pages. This version: extended version following the referee's
suggestions. Intended for non-expert
On Hopf algebras whose coradical is a cocentral abelian cleft extension
This paper is a first step toward the full description of a family of Hopf
algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}
obtained by a cocentral abelian cleft extension. We describe the simple
Yetter-Drinfeld modules, compute the fusion rules and determine the
finite-dimensional Nichols algebras for some of them. In particular, the
well-known Fomin-Kirillov algebras appear as Nichols algebras over K_{3}. As a
byproduct we obtain new Hopf algebras of dimension 216.Comment: 24 pages. Comments are welcome
On rigidity of Nichols algebras
We study deformations of graded braided bialgebras using cohomological
methods. In particular, we show that many examples of Nichols algebras,
including the finite-dimensional ones arising in the Andruskiewitsch-Schneider
program of classification of pointed Hopf algebras, are rigid. This result can
be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket.Comment: 22 page