Related papers: Contractions, Hopf algebra extensions and cov. dif…
A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a…
The $\hat\kappa$-deformed extended Galilei Hopf group algebra, ${\rm Fun}_{\hat\kappa}(\tilde G_{(m)})$, is introduced. It provides an example of a cocycle bicrossproduct structure, and is shown to be the contraction limit of a…
In this report we give an intrinsic treatment of the results we developed in a previous work connecting the differential calculi on Hopf algebras to the Drinfeld double. In the first place we recover that bicovariant bimodules are in one to…
We discuss quantum deformation of the affine transformation group and its Lie algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. The deformation of…
We study the relations between $q$-deformations and $q$-coherent states of the single oscillator representations for $su_q(1,1)$ and $su_q(2)$ algebras; Dyson and Holstein-Primakoff type in terms of Biedenharn, Macfarlane and anyonic…
We describe and study a four parameters deformation of the two products and the coproduct of the Hopf algebra of plane posets. We obtain a family of braided Hopf algebras, generally self-dual. We also prove that in a particular case (when…
We present an approach to classical definitions and results on cumulant--moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand…
We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is…
An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation…
We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such…
A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the…
The representation and the cohomology theory of associative 2-algebras are developed. We study the deformations and abelian extensions of associative 2-algebras in details.
Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times…
The most common nonlinear deformations of the su(2) Lie algebra, introduced by Polychronakos and Ro\v cek, involve a single arbitrary function of J_0 and include the quantum algebra su_q(2) as a special case. In the present contribution,…
Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided…
In this paper, we define and develop a cohomology and deformation theories of Jacobi-Jordan algebras. We construct a cohomology based on two operators, called zigzag cohomology, and detail the low degree cohomology spaces. We describe the…
We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally , we also consider the isomorphism problem for…
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform…
We introduce a notion of extraction-contraction coproduct on twisted bialgebras, that is to say bialgebras in the category of linear species. If $P$ is a twisted bialgebra, a contraction-extraction coproduct sends $P[X]$ to…
A q-analogue of four dimensional conformally invariant field theory based on the quantum algebra U_{q}(so(4,2)) is proposed. The two- and three-point correlation functions are calculated. The construction is elaborated in order to fit the…