On radical square zero rings

Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modules which have a non-projective module M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n

The strong side of weak topological insulators

Three-dimensional topological insulators are classified into "strong" (STI) and "weak" (WTI) according to the nature of their surface states. While the surface states of the STI are topologically protected from localization, this does not hold for the WTI. In this work we show that the surface states of the WTI are actually protected from any random perturbation that does not break time-reversal symmetry, and does not close the bulk energy gap. Consequently, the conductivity of metallic surfaces in the clean system remains finite even in the presence of strong disorder of this type. In the weak disorder limit the surfaces are found to be perfect metals, and strong surface disorder only acts to push the metallic surfaces inwards. We find that the WTI differs from the STI primarily in its anisotropy, and that the anisotropy is not a sign of its weakness but rather of its richness.Comment: 12 pages, 7 figure

Deformed Algebras from Inverse Schwinger Method

We consider a problem which may be viewed as an inverse one to the Schwinger realization of Lie algebra, and suggest a procedure of deforming the so-obtained algebra. We illustrate the method through a few simple examples extending Schwinger's $su(1,1)$ construction. As results, various q-deformed algebras are (re-)produced as well as their undeformed counterparts. Some extensions of the method are pointed out briefly.Comment: 14 pages, Jeonju University Report, Late

LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio

Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Comment: 33 page

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat