8 research outputs found
Kleiner's theorem for unitary representations of posets
A subspace representation of a poset is given by
a system consisting of a vector space and its subspaces
such that if . For each real-valued
vector with positive components, we define a unitary
-representation of as a system that
consists of a unitary space and its subspaces such that if and satisfies , in
which is the orthogonal projection onto . We prove that
has a finite number of unitarily nonequivalent indecomposable
-representations for each weight if and only if has a
finite number of nonequivalent indecomposable subspace representations; that
is, if and only if contains any of Kleiner's critical posets.Comment: 12 pages, paper reorganized and rewritten. some statements were adde
Systems of subspaces of a unitary space
For a given poset, we consider its representations by systems of subspaces of
a unitary space ordered by inclusion. We classify such systems for all posets
for which an explicit classification is possible.Comment: 20 page
Unitarizable representations of quivers
We investigate representations of *-algebras associated with posets.
Unitarizable representations of the corresponding (bound) quivers (which are
polystable representations for some appropriately chosen slope function) give
rise to representations of these algebras. Considering posets which correspond
to unbound quivers this leads to an ADE-classification which describes the
unitarization behaviour of their representations. Considering posets which
correspond to bound quivers, it is possible to construct unitarizable
representations starting with polystable representations of related unbound
quivers which can be glued together with a suitable direct sum of simple
representations. Finally, we estimate the number of complex parameters
parametrizing irreducible unitary non-equivalent representations of the
corresponding algebras.Comment: 27 pages, Section 2.3 reorganized, final version, to appear in
Algebras and Representation Theor
Unitarization of linear representations of non-primitive posets
We prove that partially ordered set has finite number of finite-dimensional
indecomposable nonequivalent Hilbert representations with orthoscalarity
condition if and anly if it has finite number of indecomposable linear
representations. We show that each indecomposable representation of the poset
of finite type could be unitarized with some weight.Comment: 32 pages, Several Appendix are added. Proofs completed. Several typos
were fixe