by Masuoka, Akira; Oka, Tatsuya

We characterize hereditary (as coalgebras) Hopf algebras by the property of ‘equivariant smoothness’, and apply the result to generalize to the super-context, the category equivalence, due to Hochschild, between the unipotent algebraic affine groups and the finite-dimensional nilpotent Lie algebras, in characteristic zero. The global dimension of commutative Hopf algebras, regarded as coalgebras, is also discussed.

DOI: 10.1007/s10468-005-8204-x
Print publication date: 8/1/2005
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by Fayers, Matthew

In Fayers and Martin [J. Algebra 240 (2001), 859–873], the author and Martin constructed embeddings of Schur algebras S(2,r)↪S(2,R). Here, we generalise to the q-Schur algebras Sq(2,r).

DOI: 10.1007/s10468-005-8146-3
Print publication date: 8/1/2005
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by An, Jianbei; Eaton, Charles W.

We show that Uno's refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Robinson's conjecture, Alperin's weight conjecture, the Isaacs–Navarro conjecture, the Alperin–McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks.

DOI: 10.1007/s10468-005-8144-5
Print publication date: 8/1/2005
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by Jara, P.; Merino, L.; Llena, D.; Ştefan, D.

We define formally smooth coalgebras and we study their relation with hereditary coalgebras. The main result of the paper establishes that a coalgebra with separable coradical is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra
$T_{C_{0}}(N)$
, where C0 is the coradical of C and N is a certain (C0,C0)–bicomodule.

DOI: 10.1007/s00000-005-8110-3
Print publication date: 8/1/2005
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by Rump, Wolfgang

We study a one-dimensional analogue of representation-finite rings. For a left Noetherian semilocal ring R, we define an R-lattice to be a finitely generated R-module with zero socle. We call R lattice-finite if the number of isomorphism classes of indecomposable R-lattices is finite. Under this assumption, we give several equivalent criteria for the existence of Auslander–Reiten sequences in the category of R-lattices. A necessary condition is that the maximal left quotient ring of R is semisimple, and the main sufficient criterion states that R admits a semiperfect semiprime Asano left overorder.

DOI: 10.1007/s10468-005-3656-6
Print publication date: 8/1/2005
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by Rump, Wolfgang

It is well-known that for a lattice-finite order Λ over a complete discrete valuation domain, the radical of Λ-lat (the category of Λ-lattices) is nilpotent modulo projectives. Iyama has shown that this property merely depends on the combinatorial data given by the Auslander–Reiten quiver of Λ. Moreover, he established a criterion for a finite (symmetrizable) translation quiver Q to be the Auslander–Reiten quivers of an order Λ. We improve his characterization by showing that the remaining conditions on Q can be replaced by the existence of an additive function on the vertices of Q (Theorem 4). Our proof rests on a functorial theory of ladders, expressing the Auslander–Reiten structure of Λ-lat by means of an adjoint pair of functors L⊣L− in the homotopy category of two-termed complexes over Λ-lat.

DOI: 10.1007/s10468-005-3655-7
Print publication date: 8/1/2005
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by Iyama, Osamu

In this series of papers, we introduce τ-categories, which are additive categories with some kind of Auslander–Reiten sequences. We apply them to study the category of lattices over orders. In this first paper, we study minimal projective resolutions in functor categories over τ-categories. Then we give a structure theorem of completely graded τ-categories using mesh categories.

DOI: 10.1007/s10468-005-0968-5
Print publication date: 8/1/2005
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by Didt, Daniel

We show that a large class of finite-dimensional pointed Hopf algebras is quasi-isomorphic to their associated graded version coming from the coradical filtration, i.e. they are 2-cocycle deformations of the latter. This supports a slightly specialized form of a conjecture of Masuoka.

DOI: 10.1007/s10468-004-6343-0
Print publication date: 8/1/2005
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