### A class of weak Hopf algebras.

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In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category ${}_{H}^{H}$ via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and ${}_{H}^{H}$ is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that $(A{\u266e}_{\diamond}H,\alpha \otimes \beta )$ is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category ${}_{H}^{H}$. Finally,...

Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra and $(A,\beta )$ a right $(H,\alpha )$-Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor $F$ from the category of relative Hom-Hopf modules to the category of right $(A,\beta )$-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H,\alpha )$-coaction to be separable. This leads to a generalized...

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Additive deformations of bialgebras in the sense of J. Wirth [PhD thesis, Université Paris VI, 2002], i.e. deformations of the multiplication map fulfilling a certain compatibility condition with respect to the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.

We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of Doi Hom-Hopf...

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring...

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products ${H}_{4}\bowtie {H}_{4}$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $({H}_{4},{H}_{4},\u25b9,\u25c3)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for...

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A{\u266e}_{R}B,\alpha \otimes \beta )$. Moreover, necessary and sufficient conditions for $(A{\u266e}_{R}B,\alpha \otimes \beta )$ to be a cobraided Hom-Hopf algebra are given.

We construct quantum commutators on comodule algebras over coquasitriangular Hopf algebras, so that they are quantum group coinvariant and have the generalized antisymmetry and Leibniz properties. If the coquasitriangular Hopf algebra is additionally cotriangular, then the quantum commutators satisfy a generalized Jacobi identity, and turn the comodule algebra into a quantum Lie algebra. Moreover, we investigate the projective and injective dimensions of some Doi-Hopf modules over a quantum commutative...

Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution $(F,G)$ of the first equation of the Kashiwara-Vergne conjecture$$x+y-log\left({\mathrm{e}}^{y}{\mathrm{e}}^{x}\right)=(1-{\mathrm{e}}^{-\mathrm{ad}\phantom{\rule{0.166667em}{0ex}}x})F(x,y)+({\mathrm{e}}^{\mathrm{ad}\phantom{\rule{0.166667em}{0ex}}y}-1)G(x,y).$$Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates $x$ and $y$ thanks to the kernel of the Dynkin idempotent.

We define polynomial $H$-identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$-ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E\left(n\right)$, we exhibit a finite set of polynomial $H$-identities which distinguish the Galois objects over $H$ up to isomorphism.

In this communication, I recall the main results [BDK1] in the classification of finite Lie pseudoalgebras, which generalize several previously known algebraic structures, and announce some new results [BDK2] concerning their representation theory.

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical $S{U}_{q}\left(2\right)$ studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $\mathbb{S}$-modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.Let $\mathcal{O}$ denote the forgetful functor from $\mathbb{S}$-modules to graded vector spaces. Left modules over an operad $\mathcal{P}$ are treated as $\mathcal{P}$-algebras in the category of $\mathbb{S}$-modules. We generalize the results obtained...

In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a...

Let $kG$ be a group algebra, and $D\left(kG\right)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D\left(kG\right)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$. As a special case, we then give an application to the group algebra $k{D}_{n}$, where $k$ is a field of characteristic $2$ and ${D}_{n}$ is a dihedral group of order $2n$.