by Jacon, Nicolas

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}_q (\widehat{\mathfrak{sl}}_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki–Koike algebras by using Lusztig a-values.

DOI: 10.1007/s10468-007-9081-2
Online Date: 7/28/2007
Print publication date: 12/1/2007
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by Nauwelaerts, Erna; Oystaeyen, Freddy

We develop a generalization of the traditional crossed products and we derive general structural properties. Localization at a particular Ore set is investigated and as a consequence the relation to crossed products is examined. Finally, examples are given.

DOI: 10.1007/s10468-007-9078-x
Online Date: 7/28/2007
Print publication date: 4/1/2008
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by Guédénon, T.

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C).

DOI: 10.1007/s10468-007-9073-2
Online Date: 7/19/2007
Print publication date: 3/1/2008
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by Klep, Igor

We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings.

DOI: 10.1007/s10468-007-9062-5
Online Date: 7/19/2007
Print publication date: 10/1/2007
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by Meltzer, Hagen

Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λ-module can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ.

DOI: 10.1007/s10468-007-9067-0
Online Date: 7/18/2007
Print publication date: 10/1/2007
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by Ene, Viviana; Popescu, Dorin

We study maximal Cohen–Macaulay modules over the hypersurface ring $K[[x, y]]/(x^n),\ n\geq 3,$
K being a field. Infinite families of non-isomorphic indecomposable maximal Cohen–Macaulay modules of arbitrary number of minimal generators or of arbitrary rank are constructed.

DOI: 10.1007/s10468-007-9075-0
Online Date: 7/12/2007
Print publication date: 4/1/2008
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by Rees, Sarah

It is well known that the sets of strings that define all representations of string algebras and many representations of other quotients of path algebras form a regular set, and hence are defined by finite state automata. This short article aims to explain this connection between representation theory and automata theory in elementary terms; no technical background in either representation theory or automata theory is assumed. The article describes the structure of the set of strings of a monomial algebra as a locally testable and hence regular set, and describes explicitly the construction of the automaton, illustrating the construction with an elementary example. Hence it explains how the sets of strings and bands of a monomial algebra correspond to the sets of paths and closed (non-powered) circuits in a finite graph, and how the growth rate of the set of bands is immediately visible from that graph.

DOI: 10.1007/s10468-007-9063-4
Online Date: 7/10/2007
Print publication date: 6/1/2008
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by Bergen, Jeffrey; Grzeszczuk, Piotr; Hryniewicka, Małgorzata

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is

If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then
$R^L \not= 0$.We can then use Theorem 13 to prove

Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R

^H

satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN, where N is the dimension of H.

DOI: 10.1007/s10468-007-9070-5
Online Date: 7/5/2007
Print publication date: 6/1/2008
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