by Bessenrodt, Christine; Olsson, JØrn B.

Generalizing results on spin character degrees, we determine for a given conjugacy class of odd type in the double cover of Sn spin characters of Sn which have the minimal 2-power on this class in their character value. Surprisingly, the Glaisher map plays an important rôle here.

DOI: 10.1007/s10468-004-0191-9
Print publication date: 3/1/2005
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by Andruskiewitsch, Nicolás; Dăscălescu, Sorin

This is a contribution to the classification of finite-dimensional pointed Hopf algebras. We are concerned with the case when the group of group-like elements is Abelian of exponent 2. We attach to such a pointed Hopf algebra a generalized simply-laced Cartan matrix; we conjecture that the Hopf algebra is finite-dimensional if and only if the Cartan matrix is of finite type. We prove the conjecture for the types An and An(1). We obtain the classification of all possible Hopf algebras with Cartan matrix An. We use the lifting method developed by Hans-Jürgen Schneider and the first-named author.

DOI: 10.1007/s10468-004-6008-z
Print publication date: 3/1/2005
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by Futorny, Vyacheslav M.; Grishkov, Alexander N.; Melville, Duncan J.

Let
$\mathfrak{g}$
be an untwisted affine Kac–Moody algebra and MJ(λ) a Verma-type module for
$\mathfrak{g}$
with J-highest weight λ∈P. We construct quantum Verma-type modules MJq(λ) over the quantum group
$U_{q}(\mathfrak{g})$
, investigate their properties and show that MJq(λ) is a true quantum deformation of MJ(λ) in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules.

DOI: 10.1007/s10468-004-5765-z
Print publication date: 3/1/2005
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by Xie, Guangming; Kobayashi, Shigeru; Marubayashi, Hidetoshi; Popescu, Nicolea; Vraciu, Constantin

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,σ] be the skew polynomial ring over Q in an indeterminate X, where σ is an automorphism of Q. Consider the natural map φ from Q[X,σ]XQ[X,σ] to Q, where Q[X,σ]XQ[X,σ] is the localization of Q[X,σ] at the maximal ideal XQ[X,σ] and set
$\widetilde{R}=\varphi^{-1}(R)$
, the complete inverse image of R by φ. It is shown that
$\widetilde{R}$
is a Dubrovin valuation ring of Q(X,σ) (the quotient ring of Q[X,σ]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism σ is classified into five types, in order to study the structure of
$\Gamma_{\widetilde{R}}$
(the value group of
$\widetilde{R}$
). It is shown that there is a commutative valuation ring R with automorphism σ which belongs to each type and which makes
$\Gamma_{\widetilde{R}}$
Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional.

DOI: 10.1007/s10468-004-5766-y
Print publication date: 3/1/2005
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by Ara, Pere; Pedersen, Gert K.; Perera, Francesc

We prove a new extension result for QB-rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of QB-rings. More concretely, we show that a surjective pullback of two QB-rings is usually again a QB-ring. Specializing to the case of an extension of a semi-prime ideal I of a unital ring R, the pullback setting leads naturally to the study of rings whose multiplier rings are QB-rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be QB-rings. Our analysis is based on the study of extensions and the use of nonstable K-theoretical techniques.

DOI: 10.1007/s10468-004-5767-x
Print publication date: 3/1/2005
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by Zelikson, Shmuel

Let Q be a quiver of type ADE. We construct the corresponding Auslander–Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. In An case, we recover special wiring diagrams.

DOI: 10.1007/s10468-004-6117-8
Print publication date: 3/1/2005
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by Smoktunowicz, Agata

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.

DOI: 10.1007/s10468-004-6118-7
Print publication date: 3/1/2005
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by Bocklandt, Raf

In this paper, we classify all the symmetric quivers and corresponding dimension vectors whose quotient space, classifying the semisimple representation classes, is a complete intersection. The result we obtain is that such quivers can be reduced to a few number of basic quivers, using some elementary types of reduction.

DOI: 10.1007/s10468-004-8324-8
Print publication date: 3/1/2005
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