1,828 research outputs found
Distinguished bases of exceptional modules
Exceptional modules are tree modules. A tree module usually has many tree
bases and the corresponding coefficient quivers may look quite differently. The
aim of this note is to introduce a class of exceptional modules which have a
distinguished tree basis, we call them radiation modules (generalizing an
inductive construction considered already by Kinser). For a Dynkin quiver,
nearly all indecomposable representations turn out to be radiation modules, the
only exception is the maximal indecomposable module in case E_8. Also, the
exceptional representation of the generalized Kronecker quivers are given by
radiation modules. Consequently, with the help of Schofield induction one can
display all the exceptional modules of an arbitrary quiver in a nice way.Comment: This is a revised and slightly expanded version. Propositions 1 and 2
have been corrected, some examples have been inserte
Invariant Subspaces of Nilpotent Linear Operators. I
Let be a field. We consider triples , where is a finite
dimensional -space, a subspace of and a linear
operator with for some , and such that . Thus,
is a nilpotent operator on , and is an invariant subspace with
respect to .
We will discuss the question whether it is possible to classify these
triples. These triples are the objects of a category with the
Krull-Remak-Schmidt property, thus it will be sufficient to deal with
indecomposable triples. Obviously, the classification problem depends on ,
and it will turn out that the decisive case is For , there are
only finitely many isomorphism classes of indecomposables triples, whereas for
we deal with what is called ``wild'' representation type, so no
complete classification can be expected.
For , we will exhibit a complete description of all the indecomposable
triples.Comment: 55 pages, minor modification in (0.1.3), to appear in: Journal fuer
die reine und angewandte Mathemati
A characterization of admissible algebras with formal two-ray modules
In the paper we characterize, in terms of quivers and relations, the
admissible algebras with formal two-ray modules introduced by G. Bobi\'nski and
A. Skowro\'nski [Cent. Eur. J.Math.1 (2003), 457--476].Comment: Mainly correcting typos. Also a new abstract and minor changes in the
introduction and subsection 3.
Which canonical algebras are derived equivalent to incidence algebras of posets?
We give a full description of all the canonical algebras over an
algebraically closed field that are derived equivalent to incidence algebras of
finite posets. These are the canonical algebras whose number of weights is
either 2 or 3.Comment: 8 pages; slight revision; to appear in Comm. Algebr
Determining topological order from a local ground state correlation function
Topological insulators are physically distinguishable from normal insulators
only near edges and defects, while in the bulk there is no clear signature to
their topological order. In this work we show that the Z index of topological
insulators and the Z index of the integer quantum Hall effect manifest
themselves locally. We do so by providing an algorithm for determining these
indices from a local equal time ground-state correlation function at any
convenient boundary conditions. Our procedure is unaffected by the presence of
disorder and can be naturally generalized to include weak interactions. The
locality of these topological indices implies bulk-edge correspondence theorem.Comment: 7 pages, 3 figures. Major changes: the paper was divided into
sections, the locality of the order in 3D topological insulators is also
discusse
The Ingalls-Thomas Bijections
Given a finite acyclic quiver Q with path algebra kQ, Ingalls and Thomas have
exhibited a bijection between the set of Morita equivalence classes of
support-tilting modules and the set of thick subcategories of mod kQ and they
have collected a large number of further bijections with these sets. We add
some additional bijections and show that all these bijections hold for
arbitrary hereditary artin algebras. The proofs presented here seem to be of
interest also in the special case of the path algebra of a quiver.Comment: This is a modified version of an appendix which was written for the
paper "The numbers of support-tilting modules for a Dynkin algebra" (see
arXiv:1403.5827v1
Optimal Renormalization Group Transformation from Information Theory
Recently a novel real-space RG algorithm was introduced, identifying the
relevant degrees of freedom of a system by maximizing an information-theoretic
quantity, the real-space mutual information (RSMI), with machine learning
methods. Motivated by this, we investigate the information theoretic properties
of coarse-graining procedures, for both translationally invariant and
disordered systems. We prove that a perfect RSMI coarse-graining does not
increase the range of interactions in the renormalized Hamiltonian, and, for
disordered systems, suppresses generation of correlations in the renormalized
disorder distribution, being in this sense optimal. We empirically verify decay
of those measures of complexity, as a function of information retained by the
RG, on the examples of arbitrary coarse-grainings of the clean and random Ising
chain. The results establish a direct and quantifiable connection between
properties of RG viewed as a compression scheme, and those of physical objects
i.e. Hamiltonians and disorder distributions. We also study the effect of
constraints on the number and type of coarse-grained degrees of freedom on a
generic RG procedure.Comment: Updated manuscript with new results on disordered system
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