Recovering Quivers from Derived Quiver Representations

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way.     

The Algebras of Semi-Invariants of Euclidean Quivers

We give a new short proof of Skowroński and Weyman's theorem about the structure of the algebras of semi-invariants of Euclidean quivers, in the case of quivers without oriented cycles and in characteristic zero. Our proof is based essentially on Derksen and Weyman's result about the generators of these algebras and properties of Schofield semi-invariants.     

Realizations of Topological Categories

There are functor-preordering-structured categories S(F,P), defined by the Prague School, in which every concrete category over a concretizable basecategory is realizable. Over nice basecategories there are realizations of all topological categories in some topological S(F,L). This gives rise for a new characterization of those concrete categories having a topological hull.     

Derived equivalences from mutations of quivers with potential

We show that Derksen–Weyman–Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi–Yau triangulated categories. Our approach is related to that of Iyama–Reiten and ‘Koszul dual’ to that of Kontsevich–Soibelman. It improves on previous work by Vitória. In Appendix A, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.     

Common properties of bounded linear operators AC and BA: Local spectral theory

Let X, Y   be Banach spaces, A:X→Y$A:X\to Y$ and B,C:Y→X$B,C:Y\to X$ be bounded linear operators satisfying operator equation ABA=ACA$ABA=ACA$. In this paper, we show that AC and BA share the local spectral properties such as Bishop's property (β), subscalarity and Dunford's property (C). Also, the quasi-nilpotent part and the analytic core for AC and BA are studied.     

Convex Realizations of Planar Linear Trees

We prove a necessary condition for the existence of a convex realization of a planar linear tree. In the case of broken lines, it is shown that this condition is sufficient; a continuous algorithm constructing such a realization is found.     

Hereditarily Optimal Realizations of Consistent Metrics

One of the main problems in phylogenetics is to find good approximations of metrics by weighted trees. As an aid to solving this problem, it could be tempting to consider optimal realizations of metrics—the guiding principle being that, the (necessarily unique) optimal realization of a tree metric is the weighted tree that realizes this metric. And, although optimal realizations of arbitrary metrics are, in general, not trees, but rather weighted networks, one could still hope to obtain a phylogenetically informative representation of a given metric, maybe even more informative than the best approximating tree. However, optimal realizations are not only difficult to compute, they may also be non-unique. Here we focus on one possible way out of this dilemma: hereditarily optimal realizations. These are essentially unique, and can be described in a rather explicit way. In this paper, we recall what a hereditarily optimal realization of a metric is and how it is related to the 1-skeleton of the tight span of that metric, and we investigate under what conditions it coincides with this 1-skeleton. As a consequence, we will show that hereditarily optimal realizations for consistent metrics, a large class of phylogentically relevant metrics, can be computed in a straight-forward fashion. Received August 26, 2004     

Finite Dimensional Inductive Algebras Are Trivial#

It is observed that there are no non-trivial finite dimensional inductive algebras for a connected group.     

A CONTINUOUS TIME KRONECKER'S LEMMA AND MARTINGALE CONVERGENCE

A continuous time version of Kronecker's lemma is established. This is used to obtain convergence results for martingales.     

Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.     

Game-theoretic versions of strong law of large numbers for unbounded variables

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: It's Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.     

Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain , which is either the open unit disk or the open right half-plane . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function has a bi-inner dilation mapping into ; here U ₁ and Y ₁ are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions and (namely and ) satisfying and for almost all . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ _r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ _l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.      

Categorical Generalization of a Universal Domain

J. Adámek defined SC categories as a categorical generalization of Scott domains. Namely, an SC category is finitely accessible, has an initial object and is boundedly cocomplete (each diagram with a compatible cocone has a colimit). SC categories are proved to serve well as a basis for the computer language semantics.The purpose of this paper is to generalize the concept of a universal Scott domain to a universal SC category. We axiomatize properties of subcategories of finitely presentable objects of SC categories (generalizing thus semilattices of compact elements of Scott domains). The categories arising are called FCC (finitely consistently cocomplete) categories. It is shown that there exists a universal FCC category, i.e., such that every FCC category may be FCC embedded into it. The result is an application of a general procedure introduced 30 years ago by V. Trnková.     

Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

对偶的Hom型拟Hopf代数的余表示和范畴实现

In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a(braided) dual Hom-quasi-bialgebra is a monoidal category. Finally,we give a categorical realization of dual Hom-quasi-Hopf algebras.     

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Rank 2

We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component B ₀(λ) containing the unit element for arbitrary rank 2 cases. We also present the explicit form of B ₀(λ) containing the highest (or lowest) weight vector by the polyhedral realization method.