Hopf Ore Extensions Research Articles

Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field \Bbbk of positive characteristic p are studied. We show that every IHOE over \Bbbk satisfies a polynomial identity (PI), with PI-degree a power of p , and that it is a filtered deformation of a commutative polynomial ring. We classify all 2 -step IHOEs over \Bbbk , thus generalising the classification of 2 -dimensional connected unipotent algebraic groups over \Bbbk . Further properties of 2 -step IHOEs are described: for example their simple modules are classified, and every 2 -step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie \Bbbk -algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over \Bbbk .

The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

Representations of Hopf-Ore Extensions of Group Algebras

In this paper, we study the representations of the Hopf-Ore extensions kG(χ− 1,a,0) of group algebra kG, where k is an algebraically closed field. We classify all finite dimensional simple kG(χ− 1,a,0)-modules under the assumption $|\chi |=\infty $ and $|\chi |=|\chi (a)|<\infty $ respectively, and all finite dimensional indecomposable kG(χ− 1,a,0)-modules under the assumption that kG is finite dimensional and semisimple, and |χ| = |χ(a)|. Moreover, we investigate the decomposition rules for the tensor product modules over kG(χ− 1,a,0) when char(k) = 0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra kDn, where n = 2m, m > 1 odd, and char(k) = 0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.

Generalized Hopf–Ore Extensions of Hopf Group-Coalgebras

In this paper, we introduce the concept of a generalized Hopf–Ore extension of a Hopf group-coalgebra and give the necessary and sufficient conditions for the Ore extension of a Hopf group-coalgebra to be a Hopf group-coalgebra. Moreover, an isomorphism theorem on generalized Hopf group-coalgebra Ore extensions is given and specific cases in a simple type called special Hopf group-coalgebra Ore extensions are also considered. Our results are generalizations of Hopf–Ore extensions on Hopf algebras and also are very useful to construct new Hopf group-coalgebras.

The structure of connected (graded) Hopf algebras

In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.

Hopf algebras and skew PBW extensions

In this article we relate some Hopf algebra structures over Ore extensions and over skew PBW extensions of a Hopf algebra. These relations are illustrated with examples. We also show that Hopf Ore extensions and generalized Hopf Ore extensions are Hopf skew PBW extensions.

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

In this paper, we continue our study of the tensor product structure of category of weight modules over a Hopf–Ore extensions of a group algebra kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and are different. Then we describe the Green ring of the tensor category . It is shown that is isomorphic to the polynomial algebra over the group ring in one variable when , and that is isomorphic to the quotient ring of the polynomial algebra over the group ring in two variables modulo a principle ideal when . When , is isomorphic to the quotient ring of a skew group ring modulo some ideal, where is a polynomial algebra over in infinitely many variables.

D-groups and the Dixmier–Moeglin equivalence

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions

Generalized Hopf–Ore extensions

We investigate some Hopf algebra structures over an Ore extension of a Hopf algebra. Necessary and sufficient conditions for an Ore extension of a Hopf algebra to have a Hopf algebra structure of a certain type are given. This construction, called a generalized Hopf–Ore extension, generalizes Hopf–Ore extensions. We describe the generalized Hopf–Ore extensions of the enveloping algebras of Lie algebras. For some Lie algebras g, the generalized Hopf–Ore extensions of U(g) are classified.

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

ABSTRACTIn this paper, we investigate the tensor structure of the category of finite- dimensional weight modules over the Hopf–Ore extensions kG(χ−1,a,0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.

Almost involutive Hopf-Ore extensions of low dimension

The purpose of this paper is to complete the classification of almost involutive Hopf algebras up to dimension 23. For this, we first present a brief survey of these algebras, showing that most of them can be obtained by iterated Hopf-Ore extensions. Then the classification follows using a technique we developed for determining when a Hopf-Ore extension is almost involutive.

Representations of Hopf-Ore Extensions of Group Algebras and Pointed Hopf Algebras of Rank One

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let $H=kG(\chi, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a central element of $G$ and $\chi$ is a $k$-valued character for $G$ with $\chi(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H'$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H'$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H'$, and classify them. Finally, when $\chi(a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H'$.

ORE EXTENSIONS OF HOPF GROUP COALGEBRAS

The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.

Yetter-Drinfeld modules over the Hopf-Ore extension of the group algebra of dihedral group

Let k be an algebraically closed field of characteristic zero, and Dn be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A(n, 0) of kDn. We describe the structures and properties of simple Yetter-Drinfeld modules over A(n, 0), and classify all simple Yetter-Drinfeld modules over A(n, 0).