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Christian Lomp

Publications and source records attributed to Christian Lomp.

At least 19 recordsLinked to original sources
arXiv2026

Affine cellular algebras and asymptotic algebras

The theory of affine cellular algebras $A$ is extended to incorporate their asymptotic algebras $\hat{A}$, clarifying unexpected differences between classical and affine situations and comparing with Lusztig's asymptotic Hecke algebras. The main new results are about a double centraliser property between $A$ and $\hat{A}$, about constructing $\hat{A}$ from cell modules of $A$, about existence of an embedding $A \rightarrow \hat{A}$ and about a faithful functor from torsionless $\hat{A}$-modules to $A$-modules as well as about the embedding being weakly spectrum preserving (in the sense of Baum and Nistor) and about non-zero endomorphisms of cell modules being injective, while there are no non-zero homomorphisms between non-isomorphic cell modules.

math.RT↗
arXiv2026

Iterated Hopf Ore Extensions over Group Rings

We introduce and study a class of Hopf algebras $H(G, \chi, \eta, b, c, \beta)$ which are two-step Ore extensions of a group algebra $\mathbb{K}[G]$. This construction unifies and generalizes some known families of Hopf algebras such as generalized Taft algebras and Hopf algebras related to $\mathfrak{sl}_2$ constructed by Wang, Wu, and Tan. We analyze the ring theoretical properties of these algebras and classify all finite dimensional simple modules over them. We also consider the tensor products of simple modules in the zero derivation case.

math.RA↗
arXiv2025

Generalized Kac-Paljutkin algebras

In this note, we construct a family of semisimple Hopf algebras $H_{n,m}$ of dimension $n^m m!$ over a field of characteristic zero containing a primitive $n$th root of unity, where $n, m \geq 2$ are integers. The well-known eight-dimensional Kac--Paljutkin algebra arises as the special case $H_{2,2}$, while the Hopf algebras previously constructed by Pansera correspond to the instances $H_{n,2}$. Each algebra $H_{n,m}$ is defined as an extension of the group algebra $\mathbb{K} \Sigma_m$ of the symmetric group by the $m$-fold tensor product $R = \mathbb{K} \mathbb{Z}_n^{\otimes m}$, where $\mathbb{Z}_n$ denotes the cyclic group of order $n$. This extension admits a realization as a crossed product: $H_{n,m} = \mathbb{K} \mathbb{Z}_n^{\otimes m} \#_\gamma \Sigma_m$. In the final section, we construct a family of irreducible $m$-dimensional representations of $H_{n,m}$ that are inner faithful as $R$-modules and exhibit a nontrivial inner-faithful action of a subalgebra of $H_{n,m}$ on a quantum polynomial algebra.

math.QA↗
arXiv2023

Hypersimple Rings and Modules

In this paper a simple right R-module S over a ring R is called hypersimple if its injective hull E(S) is cyclic, and a ring R is called right hypersimple if every simple right R-module is hypersimple. We initiate a study of these new notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose cyclic right modules have cyclic injective hulls.

math.RA↗
arXiv2022

Dual Kasch Rings

It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field $k$, a finite dimensional $k$-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.

math.RA↗
arXiv2020

Locally Finite Representations Over Noetherian Hopf algebras

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.

math.RT↗
arXiv2019

Stable range one for rings with central units

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring $R$ with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: $R$ is semiprime or $R$ is one-sided Noetherian or $R$ has unit-stable range $1$ or $R$ has classical Krull dimension $0$ or $R$ is an algebra over a field $K$ such that $K$ is uncountable and $R$ has only countably many primitive ideals or $R$ is affine and either $K$ has characteristic $0$ or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.

math.RA↗
arXiv2017

A note on simple modules over quasi-local rings

Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. Many non-commutative Noetherian rings whose injective hulls of simple modules are locally Artinian have been extensively studied recently. This property had been denoted by property $(\diamond)$. In this paper we investigate, which non-Noetherian semiprimary commutative quasi-local rings $(R, m)$ satisfy property $(\diamond)$. For quasi-local rings $(R,m)$ with $m^3=0$, we prove a characterisation of this property in terms of the dual space of $Soc(R)$. Furthermore, we show that $(R,m)$ satisfies $(\diamond)$ if and only if its associated graded ring $gr(R)$ does. Given a field $F$ and vector spaces $V$ and $W$ and a symmetric bilinear map $\beta:V\times V\rightarrow W$ we consider commutative quasi-local rings of the form $F\times V \times W$, whose product is given by $(\lambda_1, v_1,w_1)(\lambda_2,v_2,w_2) = (\lambda_1\lambda_2, \lambda_1v_2+\lambda_2v_1, \lambda_1w_2+\lambda_2w_1+\beta(v_1,v_2))$ in order to build new examples and to illustrate our theory. In particular we prove that any quasi-local commutative ring with radical cube-zero does not satisfy $(\diamond)$ if and only if it has a factor, whose associated graded ring is of the form $F\times V \times F$ with $V$ infinite dimensional and $\beta$ non-degenerated.

math.RA↗
arXiv2017

Panov's theorem for weak Hopf algebras

Panov proved necessary and sufficient conditions to extend the Hopf algebra structure of an algebra $R$ to an Ore extension $R[x;\sigma,\delta]$ with $x$ being a skew-primitive element. In this paper we extend Panov's result to Ore extensions over weak Hopf algebras. As an application we study Ore extensions of connected groupoid algebras.

math.RA↗
arXiv2016

Differential smoothness of skew polynomial rings

It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.

math.RA↗
arXiv2016

Ring theoretical properties of affine cellular algebras

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, KLR algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra $A$ satisfies a polynomial identity. Furthermore, we show that $A$ can be embedded into its asymptotic algebra if the occurring commutative affine algebra $B_j$ are reduced and the determinants of the swich matrices are non-zero divisors. As a consequence, we show that the Gelfand-Kirillov dimension of $A$ is less than or equal to the largest Krull dimension of the algebras $B_j$ and that equality hold, in case all affine cell ideals are idempotent or if the Krull dimension of the algebras $B_j$ is less than or equal to $1$. Special emphasis is given to the question when an affine cell ideal is idempotent, generated by an idempotent or finitely generated.

math.RT↗
arXiv2016

A note on semicentral Idempotents

In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.

math.RA↗
arXiv2015

A note on a paper by Cuadra, Etingof and Walton

We analyse the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra $H$ on the $n$th Weyl algebra $A=A_n(K)$ over a field $K$ of characteristic $0$ factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra $H$ on an iterated Ore extension of derivation type $A=K[x_1;d_1][x_2;d_2][\cdots][x_n;d_n]$ in characteristic zero factors through a group algebra.

math.RA↗
arXiv2014

On the semiprime smash product question

This is a survey article on a question, posed in 1986 by M.Cohen and D.Fishman, whether the smash product $A\#H$ of a semisimple Hopf algebra and a semiprime left $H$-module algebra $A$ is itself semiprime.

math.RA↗
arXiv2013

On Topological Lattices and an Application to First Submodules

We introduce the notion of a (strongly) topological lattice $\mathcal{L}=(L,\wedge ,\vee)$ with respect to a subset $X\subsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $\mathrm{Spec}^{\mathrm{f}}(M)$ of \textit{first submodules} of $M.$ We topologize $\mathrm{Spec}^{\mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the topological properties of the associated space.

math.RA↗