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 $k$ be a field. We consider triples $(V,U,T)$, where $V$ is a finite dimensional $k$-space, $U$ a subspace of $V$ and $T \:V \to V$ a linear operator with $T^n = 0$ for some $n$, and such that $T(U) \subseteq U$. Thus, $T$ is a nilpotent operator on $V$, and $U$ is an invariant subspace with respect to $T$. We will discuss the question whether it is possible to classify these triples. These triples $(V,U,T)$ 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 $n$, and it will turn out that the decisive case is $n=6.$ For $n < 6$, there are only finitely many isomorphism classes of indecomposables triples, whereas for $n > 6$ we deal with what is called ``wild'' representation type, so no complete classification can be expected. For $n=6$, 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

Delayed currents and interaction effects in mesoscopic capacitors

We propose an alternative derivation for the dynamic admittance of a gated quantum dot connected by a single-channel lead to an electron reservoir. Our derivation, which reproduces the result of Pr\^{e}tre, Thomas, and B\"{u}ttiker for the universal charge-relaxation resistance, shows that at low frequencies, the current leaving the dot lags after the entering one by the Wigner-Smith delay time. We compute the capacitance when interactions are taken into account only on the dot within the Hartree-Fock approximation and study the Coulomb-blockade oscillations as a function of the Fermi energy in the reservoir. In particular we find that those oscillations disappear when the dot is fully `open', thus we reconcile apparently conflicting previous results.Comment: 9 pages, 8 figure

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category $\mathscr{H}^{(\rho)}$ of semi-stable coherent sheaves of a fixed slope $\rho$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(\rho)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.Comment: 29 page

Stability conditions and Stokes factors

Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The residues of these connections are given by the holomorphic generating function for counting invariants in A constructed by D. Joyce.Comment: Very minor changes. Final version. To appear in Inventione

Quantum groups and double quiver algebras

For a finite dimensional semisimple Lie algebra ${\frak{g}}$ and a root $q$ of unity in a field $k,$ we associate to these data a double quiver $\bar{\cal{Q}}.$ It is shown that a restricted version of the quantized enveloping algebras $U_q(\frak g)$ is a quotient of the double quiver algebra $k\bar{\cal Q}.$Comment: 15 page

Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time

Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1-plane re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016). This is an extended abstract. For a full version of this paper, see Hong S-H, Nagamochi H.: Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time, Technical Report TR 2016-002, Department of Applied Mathematics and Physics, Kyoto University (2016

Direct Measurement of Quantum Confinement Effects at Metal to Quantum-Well Nanocontacts

Model metal-semiconductor nanostructure Schottky nanocontacts were made on cleaved heterostructures containing GaAs quantum wells (QWs) of varying width and were locally probed by ballistic electron emission microscopy. The local Schottky barrier was found to increase by ∼0.140 eV as the QW width was systematically decreased from 15 to 1 nm, due mostly to a large (∼0.200 eV) quantum-confinement increase to the QW conduction band. The measured barrier increase over the full 1 to 15 nm QW range was quantitatively explained when local "interface pinning" and image force lowering effects are also considered