Representation-tame locally hereditary algebras

Zbigniew Leszczyński

doi:10.4064/cm99-2-3

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

Algebra Colloquium ◽

10.1142/s1005386713000114 ◽

2013 ◽

Vol 20 (01) ◽

pp. 123-140

Author(s):

Teng Zou ◽

Bin Zhu

Keyword(s):

Simplicial Complex ◽

Root System ◽

Cluster Complex ◽

Cluster Complexes ◽

Endomorphism Algebras ◽

Tilted Algebras ◽

Cluster Categories ◽

Hereditary Algebras ◽

Cluster Tilting ◽

Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

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Hilbert $C^{*}$-modules in which all relatively strictly closed submodules are complemented

Glasnik Matematicki ◽

10.3336/gm.56.2.08 ◽

2021 ◽

Vol 56 (2) ◽

pp. 343-374

Author(s):

Boris Guljaš ◽

Keyword(s):

Hilbert Space ◽

Hilbert Modules ◽

Strict Topology ◽

Direct Sums ◽

Essential Ideal ◽

Closed Submodule ◽

Closed Submodules ◽

Hereditary Algebras ◽

Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in $\ell_\infty$ and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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