The Radicals of Crossed Products

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra, and $(A,\beta)$ a Hom-algebra. In this paper we will introduce the crossed product $(A\#_{\sigma}H,\beta\otimes\alpha)$, which is a Hom-algebra. Then we will introduce the notions of cleft extensions and Galois extensions respectively, and prove that a crossed product is equivalent to a cleft extension and a cleft extension is equivalent to a Galois extension with normal bases property.

We provide a differential structure on arbitrary cleft extensions $B:=A^{\mathrm{co}H}\subseteq A$ for an $H$-comodule algebra $A$. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra $B\#_\sigma H$ from the data of a bicovariant calculus on the structure Hopf algebra $H$ and a calculus on the base algebra $B$, which is compatible with the $2$-cocycle and measure of the crossed product. The result is a quantum principal bundle wi...

In this paper we explore the concept of depth of a ring extension when the overall algebra factorises as a product of two subalgebras, in particular the case of finite dimensional Hopf algebras. As a result we generalise the results by Kadison and Young \cite{HKY} on depth of a Hopf algebra $H$ in its smash product with a finite dimensional left $H$-module algebra $A$, $A#H$ to the context of generalised smash products $Q^{*op}#_\psi H$ \cite{Bz1} where $Q$ is the quotient mo...

Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for $A\ot B$ to be a bialgebra, and sufficient conditions for $A\ot B$ to be a Hopf algebra. We discuss when such a cross product Hopf algebra is a double cross (co)product, a biproduct, or, more generally, a smash (co)product Hopf algebra. In each...

Hopf crossed products, or in other words, cleft comodule algebras form a special but important class in Hopf-Galois extensions. To discuss this interesting subject, we will start with the more familiar group crossed products, and then see that they are naturally generalized by Hopf crossed products; these Hopf crossed products are characterized as Hopf-Galois extensions with normal basis. After showing this characterization due to Doi and Takeuchi, we will proceed to two appl...

Let $H$ be a finite-dimensional Hopf algebra. We study the behaviou r of primitive and maximal ideals in certain types of ring extensions determined by $H$. The main focus is on the class of faithfully flat Galois extensions, which includes includes smash and crossed products. It is shown how analogous results can be obtained for the larger class of extensions possessing a total integral, which includes extensions $A^H\subseteq A $ when $H$ is semisimple. We use Passman's "pr...

The paper presents a construction of the crossed product of a C*-algebra by a semigroup of endomorphisms generated by partial isometries.

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by means of extensions, or from Radford's biproduct $H\cong R#kG$, where $kG$ is the group algebra of $G$ of order 2, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^3$.

The present article represents a step forward in the study of the following problem: If $\mathbb{A}=(A_{1},A_{2})$ and $\mathbb{H}=(H_{1},H_{2})$ are Hopf braces in a symmetric monoidal category C such that $(A_{1},H_{1})$ and $(A_{2},H_{2})$ are matched pairs of Hopf algebras, then we want to know under what conditions the pair $(A_{1}\bowtie H_{1},A_{2}\bowtie H_{2})$ constitutes a new Hopf brace. We find such conditions for the pairs $(A_{1}\otimes H_{1},A_{2}\bowtie H_{2}...

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional quasi-Hopf algebras started in math.QA/0310253 (p=2) and math.QA/0402159 (p>2). More specifically, we completely describe the class RG(p) for p>2. Namely, we show that if H\in RG(p) has a nontrivial associator, then the rank of H[1] over H[0] is \...