by Ágoston, István; Dlab, Vlastimil; Lukács, Erzsébet

The paper is a continuation of the authors’ study of quasi-hereditary algebras whose Yoneda extension algebras (homological duals) are quasi-hereditary. The so-called standard Koszul quasi-hereditary algebras, presented in this paper, have the property that their extension algebras are always quasi-hereditary. In the natural setting of graded Koszul algebras, the converse also holds: if the extension algebra of a graded Koszul quasi-hereditary algebra is quasi-hereditary, then the algebra must be standard Koszul. This implies that the class of graded standard Koszul quasi-hereditary algebras is closed with respect to homological duality. Another immediate consequence is the fact that all algebras corresponding to the blocks of the category O are standard Koszul.

DOI: 10.1023/A:1022373620471
Print publication date: 3/1/2003
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by Moens, M.-A.; Borceux, F.

We study Azumaya multiplicative graphs over a suitable base category, generalizing in this way the theory of Azumaya algebras over a ring, with or without unit, and the theory of enriched Azumaya categories. We exhibit the links with the corresponding notions of centrality, separability, Brauer group and Brauer–Taylor group.

DOI: 10.1023/A:1022395705820
Print publication date: 3/1/2003
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by Schmitt, Alexander H. W.

To every oriented tree we associate vector bundle problems. We define semistability concepts for these vector bundle problems and establish the existence of moduli spaces. As an important application, we obtain an algebraic construction of the moduli space of holomorphic triples.

DOI: 10.1023/A:1022322529046
Print publication date: 3/1/2003
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by Letzter, Edward S.

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F
_∈,p
[G] depends on both a suitable root of unity ∈ and an antisymmetric bicharacter p: Z

ⁿ
×Z

ⁿ
→C
^×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F
_∈,p
[G] depends on a natural ℋ×ℋ-action thereon, where ℋ is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (Prim F
_∈,p
[G])/ℋ×ℋ onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F
_∈,p
[G] is not finitely stratified by the torus action.

DOI: 10.1023/A:1022387504911
Print publication date: 3/1/2003
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by Dubrovin, N. I.; Gräter, J.; Hanke, T.

If R is a ring with subset S then the rational closure Div
_R
(S) of S in R is the smallest subring D of R containing S such that U(D)=D∩U(R) where U(D), resp. U(R), denotes the group of units of D resp. R. In this paper a new approach to the so-called complexity is given in order to describe how the elements of Div
_R
(S) are built from elements of S.

DOI: 10.1023/A:1022320103094
Print publication date: 3/1/2003
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