Since compact quantum groups naturally coact on interesting algebras like quantum spheres, it would be perhaps more natural to consider Hopf comodule algebras.
Equivariant Cyclic Cohomology of H-Algebras
Then (A, H, M ) is a left Hopf triple, where A = H is the underlying algebra of H and H coacts on H via its comultiplication.
Invariant Cyclic Homology
In particular for V = kσ and H coacting on k = kσ via a grouplike element σ ∈ H, we have a Hopf cotriple (H, H, kσ ).
Invariant Cyclic Homology
We then say that C coacts from the right on M .
Galois corings from the descent theory point of view
Let C be an A-coring, and suppose that C coacts on A.
Galois corings from the descent theory point of view
A Hopf algebra coacting on a algebra in a compatible way.
The universal Hopf cyclic theory
CC] comodule coalgebra A Hopf algebra coacting on a coalgebra in a compatible way.
The universal Hopf cyclic theory
That this is the right noncommutative version of the permutation group can be seen from the fact that adding commutativity of the uij to the above deﬁnition yields the group algebra of the permutation group and that, by a theorem of Wang [Wan], As (n) is the biggest Hopf algebra coacting on a space of n points.
A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
Finally, the Internet Movie Database coacting network is used to illustrate how, for bigger and sparser communities which cannot be considered fully connected, one can still easily approximate the degree distribution.
Structural preferential attachment: Network organization beyond the link
In the next step we deﬁne our coeﬃcients as C to be acted and coacted by B .
Quantum Groupoids and their Hopf Cyclic Cohomology
It should be noted that even for the case K := H is merely a Hopf algebra and T = H coacting on itself via adjoint coaction, the above cocyclic module did not appear in the literature.
Quantum Groupoids and their Hopf Cyclic Cohomology
Quantum groups of function algebra type, in the spirit of, say, [27, 18, 22], appear naturally as Hopf algebras coacting universally on various structures (such as a vector space endowed with an R-matrix in the case of, for example, or a quadratic algebra in ).
Centers, cocenters and simple quantum groups
Raianu, Cosemisimple Hopf algebras coacting on coalgebras, Comm.
Wide Morita contexts, relative injectivity and equivalence results
As for injectivity, this follows from the fact that A with maps as in Proposition 1.1 is a commutative Hopf algebra coacting on Xn, hence corresponds to a group acting on Xn .
Integration over quantum permutation groups
In other words, As (n) is a Hopf algebra coacting on Xn .
Integration over quantum permutation groups
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