The Green Ring of Drinfeld Double D(H 4)

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double D(H ₄) of Sweedler’s four-dimensional Hopf algebra H ₄. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over D(H ₄). Then we describe the structure of the Green ring r(D(H ₄)) of D(H ₄) and show that r(D(H ₄)) is generated, as a ring, by infinitely many elements subject to a family of relations.     

Harmonic spinors on semisimple symmetric spaces

Let G/H be a semisimple symmetric space. We consider a Dirac operator D on G/H twisted by a finite dimensional H-representation. We give an explicit integral formula for certain solutions of the equation D=0. In particular, some quotients of standard principal series representations are seen to occur in the kernel of D.     

RELATIONS AMONG SOME PARAMETERS OF HYPERGRAPHS

The relations among the dominating number, independence number and covering number of hypergraphs are investigated. Main results are as follows：Dv（H）≤min{α≤（H）, p（H）, p（H）, T（H）}; De（H）≤min{v（H）, T（H）, p（H）}; DT（H） ≤αT（H）; S（H）≤ Dv （H） ＋ α（H）≤n; 2≤ Dv （H） ＋ T（H） ≤n; 2 〈 Dv （H） ＋ v（H）≤n/2 ＋ [n/r]; Dv （H） ＋ p（H） 〈_n;2≤De（H） ＋ Dv（H）≤n/2 ＋ [n/r];α（H） ＋ De（H）≤n;2 ≤ De（H） ＋ v（H）≤2[n/r]; 2 De（H） ＋ p（H）≤n-r ＋ 2.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

Decomposition of Completely Positive Maps

The paper is concerned with completely positive maps on the algebra of unbounded operatore L+(D) and on its completion L(D, D⁺). A decomposition theorem for continuous positive functionals is proved in [Tim. Loef.), and [Scholz 91] contains a generalization to maps into operator algebra on finite dimensional Hilbert spaces H₀. The aim of the present paper is to construct an analogous decomposition without the assumption that H₀ is finite dimensional. Moreover, the Kraus - theorem [Kraus] is proved for normal completely positive mappings on L(D, D⁺). The paper is organized as follows. Section 1 contains the necessary definitions and notations. In Section 2 we prove the decomposition theorem. Section 3 deal with the structure of the normal completely positive mappings.     

Domains with growth conditions for the quasihyperbolic metric

For a given growth functionH, we say that a domainD ?R ⁿ is anH-domain if δ_D x≤δ_D(x ₀)H(k _D(x,x ₀)),x ∈D, where δ_D(x)=d(x?D) andk _D denotes the quasihyperbolic distance. We show that ifH satisfiesH(0)=1, |H'|≤H, andH"≤H, then there exists an extremalH-domain. Using this fact, we investigate some fundamental properties ofH-domains.     

Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics

We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: (1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; (2) the unconditional partial regularity is necessarily limited.     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Shilov boundary and p-adic injective analytic functions

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f _* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f _* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality $\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} $ .     

Minimum cost homomorphisms to semicomplete multipartite digraphs

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism ofDtoH if uv∈A(D) implies f(u)f(v)∈A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced recently by the authors and M. Tso.Suppose we are given a pair of digraphs D,H and a cost c_i(u) for each u∈V(D) and i∈V(H). The cost of a homomorphism f of D to H is ∑_u∈V(D)c_f(u)(u). Let H be a fixed digraph. The minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For input digraph D and costs c_i(u) for each u∈V(D) and i∈V(H), verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of when H is a semicomplete digraph. In this paper we extend the classification to semicomplete k-partite digraphs, k≥3, and obtain such a classification for bipartite tournaments.     

The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism of D to H if uv∈A(D) implies f(u)f(v)∈A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H).An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u∈V(D) is associated with costs c_i(u),i∈V(H), then the cost of the homomorphism f is ∑_u∈V(D)c_f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem forH and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs c_i(u),u∈V(D),i∈V(H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost.Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12] and [11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].     

On Semi-Finite Hexagons of Order (2, <Emphasis Type="Italic">t</Emphasis>) Containing a Subhexagon

The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi–finite thick generalized polygons. We show here that no semi–finite generalized hexagon of order (2, t) can have a subhexagon H of order 2. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon H(2) or its point–line dual H ^D (2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon \({\mathcal{S}}\) of order (2, t) which contains a generalized hexagon H of order 2 as an isometrically embedded subgeometry must be finite. Moreover, if \({H \cong H^{D}}\)(2) then \({\mathcal{S}}\) must also be a generalized hexagon, and consequently isomorphic to either H ^D (2) or the dual twisted triality hexagon T(2, 8).     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

De Branges-Rovnyak spaces and Dirichlet spaces

Sarason has shown that the local Dirichlet spaces Dλ may be considered as manifestations of de Branges-Rovnyak spaces H(b), and has used this identification to give a new proof that the spaces Dλ are star-shaped. We investigate which other Dirichlet spaces D(μ) arise as de Branges-Rovnyak spaces, and which other de Branges-Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces inside Dλ.     

Minimum cost and list homomorphisms to semicomplete digraphs

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism of D to H if uv∈A(D) implies f(u)f(v)∈A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected input graph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x∈V(D) is assigned a set L_x of possible colors (vertices of H).The following optimization version of these decision problems generalizes the list homomorphism problem and was introduced in Gutin et al. [Level of repair analysis and minimum cost homomorphisms of graphs, Discrete Appl. Math., to appear], where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs D,H and a positive integral cost c_i(u) for each u∈V(D) and i∈V(H). The cost of a homomorphism f of D to H is . For a fixed digraph H, the minimum cost homomorphism problem for H is stated as follows: for an input digraph D and costs c_i(u) for each u∈V(D) and i∈V(H), verify whether there is a homomorphism of D to H and, if one exists, find such a homomorphism of minimum cost.We obtain dichotomy classifications of the computational complexity of the list homomorphism and minimum cost homomorphism problems, when H is a semicomplete digraph (digraph in which there is at least one arc between any two vertices). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when H is a semicomplete digraph: both problems are polynomial solvable if H has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for the minimum cost homomorphism problem is different: the problem is polynomial time solvable if H is acyclic or H is a cycle of length 2 or 3; otherwise, the problem is NP-hard.     

On the set of hypercyclic vectors for the differentiation operator

Let D be the differentiation operator Df = f′ acting on the Fréchet space H of all entire functions in one variable with the standard (compact-open) topology. It is known since the 1950’s that the set H(D) of hypercyclic vectors for the operator D is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepúlveda whether the set H(D) contains (up to the zero function) a non-trivial subalgebra of H or an infinite-dimensional closed linear subspace of H. In the present article both questions are answered affirmatively.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.