by Dugas, Alex S.

Over an Artin algebra Λ many standard concepts from homological algebra can be relativized with respect to a contravariantly finite subcategory $\mathcal{C}$ of mod-Λ, which contains the projective modules. The main aim of this article is to prove that the resulting relative homological dimensions of modules are preserved by stable equivalences between Artin algebras. As a corollary, we see that Auslander’s notion of representation dimension is invariant under stable equivalence (a result recently obtained independently by Guo). We then apply these results to the syzygy functor for self-injective algebras of representation dimension three, where we bound the number of simple modules in terms of the number of indecomposable nonprojective summands of an Auslander generator.

DOI: 10.1007/s10468-006-9015-4
Online Date: 11/25/2006
Print publication date: 6/1/2007
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by Ara, P.; Moreno, M. A.; Pardo, E.

We compute the monoid V(L

_K
(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L

_K
(E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L

_K
(E) and the lattice of order-ideals of V(L

_K
(E)). When K is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a dense subalgebra of the graph C
^*-algebra C
^*(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

DOI: 10.1007/s10468-006-9044-z
Online Date: 11/25/2006
Print publication date: 4/1/2007
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by Abd El-Hafez, A. T.; Delvaux, L.; Daele, A. Van

Let G be a group and assume that (A

_p
)
_p∈G
is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ

_p,q
: A

_pq
→ A

_p
⊗ A

_q
satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ

_p,q
can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.

DOI: 10.1007/s10468-006-9043-0
Online Date: 11/25/2006
Print publication date: 2/1/2007
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