Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology HS ²(G, A). We also give conditions for the map HS ⁿ (G, A) → H ⁿ (G, A) to be injective.     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H ⁰(E) and H ⁰(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π ₁ (G/H) = 0.     

A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

In [5], Navarro defines the set , where Q is a p-subgroup of a p-solvable group G, and shows that if δ is the trivial character of Q, then Irr(G|Q, δ) provides a set of canonical lifts of IBr_p(G), the irreducible Brauer characters with vertex Q. Previously, in [2], Isaacs defined a canonical set of lifts B_π(G) of I_π(G). Both of these results extend the Fong-Swan Theorem to π-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that 2∈π, or if |G| is odd, we have B_π(G) = Irr(G|Q, 1_Q). In this note we give a counterexample to show that this is not the case when . It is known that if and χ∈B_π(G), then the constituents of χ_N are in B_π (N). However, we use the same counterexample to show that if , and χ∈Irr(G|Q, 1_Q) is such that θ ∈Irr(N) and [θ, χ _N] ≠ 0, then it is not necessarily the case that θ ∈Irr(N) inherits this property. Received: 17 October 2005     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.      

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

On the exponent of tensor categories coming from finite groups

We describe the exponent of a group-theoretical fusion category C = C(G, ω, F, α) associated to a finite group G in terms of group cohomology. We show that the exponent of C divides both e(ω)expG and (expG)², where e(ω) is the cohomological order of the 3-cocycle ω. In particular, expC divides (dim C)². This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).     

Geometric minimality conditions for systems of exponentials inL p [−π, π]

A proof is given of the stability theorem for minimal systems of exponentialse(Λ) = {e ^iλx }λ∈Λ inL _p [−π, π], where Λ ⊂ ℂ is a discrete subset. Geometric minimality conditions for such systems are obtained. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 773–777, November, 1995. I wish to express gratitude to A. A. Shkalikov, who posed the problem and paid constant attention to this work.     

Further Results on Finitely Generated Projective Modules

In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K ₀(π) : K ₀(R)→K ₀(R/I) is a group epimorphism with the kernel {[P]−[Q] |P = PI, Q = QI}; there is an isomorphism of ordered groups from K ₀(R) to the gorup of all such functions ƒ _P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers. Received February 2, 1999, Accepted December 9, 1999     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

On derivations of algebras with involution

Summary Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all A∈ A(X) and some B ∈ L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.     

Universal Strict General Actors and Actors in Categories of Interest

For any category of interest ℂ we define a general category of groups with operations \mathbbC_G, \mathbbC\hookrightarrow\mathbbC_G\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of \mathbbC_G\mathbb{C_G}. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product \textUSGA(A)\ltimes A{\text{USGA}}(A)\ltimes A is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

Separable functors for the category of Doi-Hopf <Emphasis Type="Italic">π</Emphasis>-modules

Let π be a discrete group. Given a Doi-Hopf π-datum (H,A,C) and α∈π, we give necessary and sufficient conditions for the functor F ^(α) from the category of Doi-Hopf π-modules to the category of right A _α -modules to be separable. This leads to a generalized notion of integrals. As an application, we prove a Maschke type theorem for Doi-Hopf π-modules and relative Hopf π-modules.