43,666 research outputs found
Krull-Schmidt decompositions for thick subcategories
Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition
theorems for thick subcategories of various triangulated categories including
the derived categories of rings, Noetherian stable homotopy categories, stable
module categories over Hopf algebras, and the stable homotopy category of
spectra. In all these categories, it is shown that the thick ideals of small
objects decompose uniquely into indecomposable thick ideals. We also discuss
some consequences of these decomposition results. In particular, it is shown
that all these decompositions respect K-theory.Comment: Added more references, fixed some typos, to appear in Journal of Pure
and Applied Algebra, 22 pages, 1 figur
Tree modules
After stating several tools which can be used to construct indecomposable
tree modules for quivers without oriented cycles, we use these methods to
construct indecomposable tree modules for every imaginary Schur root. These
methods also give a recipe for the construction of tree modules for every root.
Moreover, we give several examples illustrating the results.Comment: Typos corrected; references added; deleted former Lemma 3.13 which is
not neede
Cluster Complexes via Semi-Invariants
We define and study virtual representation spaces having both positive and
negative dimensions at the vertices of a quiver without oriented cycles. We
consider the natural semi-invariants on these spaces which we call virtual
semi-invariants and prove that they satisfy the three basic theorems: the First
Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition
Theorem. In the special case of Dynkin quivers with n vertices this gives the
fundamental interrelationship between supports of the semi-invariants and the
Tilting Triangulation of the (n-1)-sphere.Comment: 34 page
Generic cluster characters
Let \CC be a Hom-finite triangulated 2-Calabi-Yau category with a
cluster-tilting object . Under a constructibility condition we prove the
existence of a set \mathcal G^T(\CC) of generic values of the cluster
character associated to . If \CC has a cluster structure in the sense of
Buan-Iyama-Reiten-Scott, \mathcal G^T(\CC) contains the set of cluster
monomials of the corresponding cluster algebra. Moreover, these sets coincide
if has finitely many indecomposable objects.
When \CC is the cluster category of an acyclic quiver and is the
canonical cluster-tilting object, this set coincides with the set of generic
variables previously introduced by the author in the context of acyclic cluster
algebras. In particular, it allows to construct -linear bases in acyclic
cluster algebras.Comment: 24 pages. Final Version. In particular, a new section studying an
explicit example was adde
Mixed Tensors of the General Linear Supergroup
We describe the image of the canonical tensor functor from Deligne's
interpolating category to attached to the
standard representation. This implies explicit tensor product decompositions
between any two projective modules and any two Kostant modules of ,
covering the decomposition between any two irreducible
-representations. We also obtain character and dimension formulas. For
we classify the mixed tensors with non-vanishing superdimension. For
we characterize the maximally atypical mixed tensors and show some
applications regarding tensor products.Comment: v3: Improved exposition, corrected minor mistakes v2: shortened and
revised version. Comments welcom
Duality of compact groups and Hilbert C*-systems for C*-algebras with a nontrivial center
In the present paper we prove a duality theory for compact groups in the case
when the C*-algebra A, the fixed point algebra of the corresponding Hilbert
C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies
the minimality condition A.'\cap F = Z as well as a technical condition called
regularity. The abstract characterization of the mentioned Hilbert C*-system is
expressed by means of an inclusion of C*-categories T_\c < T, where T_\c{i}s a
suitable DR-category and T a full subcategory of the category of endomorphisms
of A. Both categories have the same objects and the arrows of T can be
generated from the arrows of T_\c{a}nd the center Z.
A crucial new element that appears in the present analysis is an abelian
group C(G), which we call the chain group of G, and that can be constructed
from certain equivalence relation defined on G^, the dual object of G. The
chain group, which is isomorphic to the character group of the center of G,
determines the action of irreducible endomorphisms of A when restricted to Z.
Moreover, C(G) encodes the possibility of defining a symmetry also
for the larger category T of the previous inclusion.Comment: Final version appeared in Int. J. Math. 15 (2004) 759-812. Minor
changes w.r.t. to the previous versio
- …