Representations of Hardy Algebras: Absolute Continuity,Intertwiners, and Superharmonic Operators

Suppose T₊(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H ^∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T₊(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H ^∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.     

Connected subgroups of SO(2, n) acting irreducibly on ?2,n

We classify all connected subgroups of SO(2, n) that act irreducibly on ℝ^{2, n} . Apart from SO ₀(2, n) itself these are U(1, n/2), SU(1, n/2), if n even, S ¹ · SO(1, n/2) if n even and n ≥ 2, and SO ₀(1, 2) for n = 3. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely SO ₀(2, n), SU(1, n), and SO ₀(1, 2).     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship     

Weak generators of the algebral ∞ and the commutant of a model operator

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H^∞. It is known that ϕ+BH^∞ is a weak^* generator of the algebra H^∞/BH^∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak^* generator of the algebra l^∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B₁·…·B_N, where N ∈ℕ, and B₁, …, B_N are Blaschke products satisfying condition (C), there exists a function ϕ∈H^∞ such that ϕ(Λ) is a weak^* generator of the algebra l^∞, and ϕ+BH^∞ is not a weak^* generator of the algebra H^∞/BH^∞. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85. Translated by M. F. Gamal'.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

A splitting theorem for connected moravaK-theories

LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π_*(k(n)) ≅ ℤ _p [υ _n ] where |v _n | = 2(p ⁿ -1). We prove the following splitting theorem: Thek(n) ^*-torsion ofk(n) ^* (X) is already annihilated byv _n ^e (e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and ^r k(n) (0≤r≤e-1) where ^r k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for ^r k(n)ΛX.     

Asymptotic enumeration ofN-free partial orders

LetN(n) andN ^* (n) denote, respectively, the number of unlabeled and labeledN-free posets withn elements. It is proved thatN(n)=2 ^{n logn+o(n logn)} andN ^*(n)=2^{2n logn+o(n logn)}. This is obtained by considering the class ofN-free interval posets which can be easily counted.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

k-NN METHOD IN PARTIAL LINEAR MODEL UNDER RANDOM CENSORSHIP

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Über die Cohomologie von BSO(2k+1) und Vn,2

If h^* is a multiplicative cohomology theory on the category of CW-pairs, then it is shown that h^*(BSO (2k+1)) is under certain conditions isomorphic to the ring of formal power series over h^*(*) in universal Pontrjagin-classes p₁,...,p_k. In the second part of the paper one finds a calculation of the cohomology of the Stiefel manifolds V_n,2=SO(n)/SO(n–2).     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,