633 research outputs found
Half-liberated real spheres and their subspaces
We describe the quantum subspaces of Banica-Goswami's half-liberated
real-spheres, showing in particular that there is a bijection between the
symmetric ones and the conjugation stable closed subspaces of the complex
projective spaces.Comment: 10 page
Hopf-Galois Systems
We introduce the concept of Hopf-Galois system, a reformulation of the notion
of Galois extension of the base field for a Hopf algebra. The main feature of
our definition is a generalization of the antipode of an ordinary Hopf algebra.
The main application of Hopf-Galois systems is the construction of monoidal
equivalences between comodule categories over Hopf algebras.Comment: 15 page
Algebraic quantum permutation groups
We discuss some algebraic aspects of quantum permutation groups, working over
arbitrary fields. If is any characteristic zero field, we show that there
exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra
: this is a refinement of Wang's universality theorem for the (compact)
quantum permutation group. We also prove a structural result for Hopf algebras
having a non-ergodic coaction on the diagonal algebra , on which we
determine the possible group gradings when is algebraically closed and has
characteristic zero.Comment: 11 page
Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit
We show that if and are Hopf algebras that have equivalent tensor
categories of comodules, then one can transport what we call a free
Yetter-Drinfeld resolution of the counit of to the same kind of resolution
for the counit of , exhibiting in this way strong links between the
Hochschild homologies of and . This enables us to get a finite free
resolution of the counit of , the Hopf algebra of the bilinear
form associated to an invertible matrix , generalizing an ealier
construction of Collins, Hartel and Thom in the orthogonal case . It
follows that \B(E) is smooth of dimension 3 and satisfies Poincar\'e duality.
Combining this with results of Vergnioux, it also follows that when is an
antisymetric matrix, the -Betti numbers of the associated discrete quantum
group all vanish. We also use our resolution to compute the bialgebra
cohomology of \B(E) in the cosemisimple case.Comment: 17 page
Galois reconstruction of finite quantum groups
We describe a universal factorization for a functor with values in
finite-dimensional measured algebras. More precisely we contruct the quantum
automorphism group of this functor. This general recontruction result allows us
to recapture a finite-dimensional Hopf algebra from the category of
finite-dimensional measured comodule algebras.Comment: Latex, 10 page
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