by Hamblin, James

In this paper, finite solvable groups satisfying the “$n$-prime hypothesis” are considered. Specifically, a bound on the number of irreducible character degrees of such a group is obtained when $n=2$. The general situation is also considered, and generalizations of the $n$-prime hypothesis are analyzed.

DOI: 10.1007/s10468-006-9032-3
Online Date: 9/21/2006
Print publication date: 2/1/2007
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by Klep, Igor; Tressl, Marcus

We investigate the prime spectrum of a noncommutative ring and its spectral closure, the extended prime spectrum. We construct a ring for which the prime spectrum is a spectral space different from the extended prime spectrum and we construct a von Neumann regular ring for which the prime spectrum is not a spectral space.

DOI: 10.1007/s10468-006-9009-2
Online Date: 9/20/2006
Print publication date: 6/1/2007
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This is the official blog of Algebras and Representation Theory, moderated by its Editor-in-chief Alain Verschoren, from the University of Antwerp.

The theory of rings, algebras and their representations has evolved into a well-defined subdiscipline of general algebra, combining its proper methodology with that of other disciplines and leading to a wide variety of application fields. Due to this, many papers in these domains got dispersed in the scientific literature, making it extremely difficult for researchers to keep track of recent developments.

Algebras and Representation Theory aims to play a unifying role in this, presenting to its readers both up-to-date information about progress within the field of rings, algebras and their representations, as well as clarifying relationships with other fields. It realizes this by publishing significant and original research papers, as well as expository survey papers written by specialists, wishing to present the ‘state-of-the-art’ of well-defined subjects or subdomains.

Our blog should be viewed as a natural extension of the journal, allowing fellow researchers to comment and expand on published contributions, whose abstracts will be added to the blog as soon as the corresponding paper is published.

Texts submitted to the blog will be reviewed by the moderator, who decides whether to accept it for publication. This decision will be taken rather quickly, in principle within a couple of days.

I hope you will enjoy our blog and make frequent use of it, be it by adding and sharing your own view or by using its contents in your own research.

See you online,
Alain Verschoren

by Generalov, Alexander; Kosmatov, Nikolai

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family $D(3\mathcal R)$ in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.

DOI: 10.1007/s10468-006-9012-7
Online Date: 9/15/2006
Print publication date: 6/1/2007
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by Yanagawa, Kohji

Let $E = K{\left\langle {y_{1} ,…,y_{n} } \right\rangle }$ be the exterior algebra. The (cohomological) distinguished pairs of a graded E-module N describe the growth of a minimal graded injective resolution of N. Römer gave a duality theorem between the distinguished pairs of N and those of its dual N
*. In this paper, we show that under Bernstein–Gel’fand–Gel’fand correspondence, his theorem is translated into a natural corollary of local duality for (complexes of) graded $S=K[x_1, \ldots, x_n]$-modules. Using this idea, we also give a $\mathbb{Z}^{n} $-graded version of Römer’s theorem.

DOI: 10.1007/s10468-006-9037-y
Online Date: 9/14/2006
Print publication date: 12/1/2006
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by Chen, X.-W.; Silvestrov, S. D.; Oystaeyen, F.

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra ${\text{sl}}^{c}_{2} $.

DOI: 10.1007/s10468-006-9027-0
Online Date: 9/13/2006
Print publication date: 12/1/2006
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