by Kuhn, G.; Vershik, A.

Let G=A
ut(T) be the group of automorphisms of a homogeneous tree and let d(v,g⋅v) denote the natural tree distance. Fix a base vertex e in T. The function φ^μ(g)=exp(−μd(e,g⋅e)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator dφ^μ/dμ|_μ=0=log(φ) is conditionally positive definite but not positive definite. Hence, log(φ) corresponds to a nontrivial cocycle β(g): G→H
_π in some representation space H
_π. In contrast with the case of PGL(2,ℝ), the representation π is not irreducible.Let ψ
_o
(g) be the derivative of the spherical function corresponding to the complementary series of A
ut(T). We show that −d(e,g⋅e) and ψ
_o
(g) come from cohomologous cocycles. Moreover, ψ
_o
is associated to one of the two (irreducible) special representations of A
ut(T).

DOI: 10.1023/A:1025163802191
Print publication date: 8/1/2003
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by Bovdi, Victor

We give an explicit list of all p-groups G with a cyclic subgroup of index p
², such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also prove that such a K-basis does not exist for the group algebra KG, in the case when G is either a non-Abelian powerful p-group or a two generated p-group (p≠2) with a central cyclic commutator subgroup. This paper is a continuation of the paper which appeared in Arch. Math. (Basel)
(2000), 217–285.

DOI: 10.1023/A:1025168211902
Print publication date: 8/1/2003
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by Puninski, Gena

Using geometrical invariants we classify those pure injective modules over a commutative valuation domain which are envelopes of one element.

DOI: 10.1023/A:1025186829954
Print publication date: 8/1/2003
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by Kleiner, Mark; Perez, Efren

Let Λ be an Artin algebra, let modΛ be the category of finitely generated Λ-modules, and let A⊂modΛ be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module C∈A, we compute the almost split sequence 0→A→B→C→0 in A from the almost split sequence 0→D TrC→E→C→0 in modΛ. Since the computation is particularly simple if the minimal right A-approximation of D TrC is indecomposable for all indecomposable and not Ext-projective C∈A, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.

DOI: 10.1023/A:1025182712207
Print publication date: 8/1/2003
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by Farinati, Marco A.

We study the derived invariance of the cohomology theories Hoch
^*, H
^* and HC
^* associated with coalgebras over a field. We prove a theorem characterizing derived equivalences. As particular cases, it describes the two following situations: (1) f: C→D a quasi-isomorphism of differential graded coalgebras, (2) the existence of a ‘cotilting’ bicomodule
_C

T

_D
. In these two cases we construct a derived-Morita equivalence context, and consequently we obtain isomorphisms Hoch
^*(C)≅Hoch
^*(D) and H
^*(C)≅H
^*(D). Moreover, when we have a coassociative map inducing an isomorphism H
^*(C)≅H
^*(D) (for example, when there is a quasi-isomorphism f: C→D), we prove that HC
^*(C)≅HC
^*(D).

DOI: 10.1023/A:1025114311299
Print publication date: 8/1/2003
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by Bekkert, Viktor; Merklen, Héctor A.

We give a description of the indecomposable objects in the derived category of a finite-dimensional gentle algebra.

DOI: 10.1023/A:1025142023594
Print publication date: 8/1/2003
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