Narrow Results

In this paper, we discuss the group consisting of some good automorphisms of representation ring of the non-pointed Hopf algebra $D(n)$, the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by $x,y,z$ with the relations:

The aim of this paper is to present some connections of infinite Cogalois Theory with Clifford extensions and Hopf algebras.

We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter–Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field k is the complex field and G is a finite abelian group, we classify quiver Hopf algebras over G, multiple Taft algebras over G and Nichols algebras in . We show that the quantum enveloping algebra of a complex semisimple Lie algebra is a quotient of a semi-path Hopf algebra.

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

Let A be a Hopf algebra over a field K of characteristic zero such that its coradical H is a finite-dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation ζ on A such that A^ζ ≅ Q#H where A^ζ is the dual quasi-bialgebra obtained from A by twisting its multiplication by ζ, Q is a connected dual quasi-bialgebra in and Q#H is a dual quasi-bialgebra called the bosonization of Q by H.

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

The complete list of the coquasitriangular structures of the graded Hopf algebra over a connected Hopf quiver is given.

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra $A$ as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra. We find connections between the exponent and Frobenius–Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius–Schur indicators of the original Hopf algebra $A$. We study the category of modules of the smash coproduct.

We consider the category of comodules over a smash coproduct coalgebra $C>⊲H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C>⊲H$-colinear, $H$-colinear and rational $C$-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.

We study the pre-Lie algebra of rooted trees $({𝒯},\to )$ and we define a pre-Lie structure on its doubling space $(V,⇝)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted, respectively, by $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and $({{𝒟}}^{\prime },\star ,\chi )$. We prove that $({{𝒟}}^{\prime },\star ,\chi )$ is a module-bialgebra on $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and we find some relations between the two pre-Lie structures.

For $n,m\in \mathbb{N}$, let $H_{n,m}$ be the dual of the Radford algebra of dimension $n^{2}m$. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over $H_{n,m}$. Along the way, we describe the simple objects in ${}_{H_{n,m}}^{H_{n,m}}{𝒴}{𝒟}$ and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case $n=2$. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for $n=2=m$ and $n=2$, $m=3$, which recovers some results of the second and third author in the former case, and of Xiong in the latter.

Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es.

Jorge Luis Borges

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu )$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra $A_{N2n+1}^{\mu \lambda }$ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, Super type $A_2(q;{𝕀}_2)$ and the Nichols algebra $𝔲𝔣𝔬(8)$. And the involved finite-dimensional Nichols algebras of non-diagonal type are $12$, $4m$ and $m^2$-dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over $A_{13}^{+\lambda }$.

In the paper, we discuss the partial actions of Sweedler Hopf algebra on the generalized quaternion algebra, and give the sufficient and necessary conditions to make the actions be the partial actions. As an application, we give a complete description of all partial actions of Sweedler Hopf algebra on the generalized quaternion algebra.