Actions of pointed Hopf algebras with reduced pi invariants

Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:

1. is semiprime and is almost central in ,
2. is reduced.

If we also assume that is central, then satisfies the standard polynomial identity of degree , where is the greatest integer in .

     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

Transversals for strongly almost disjoint families

For a family of sets , and a set , is said to be a transversal of if and for each . is said to be a Bernstein set for if for each . Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets is said to admit a -transversal if can be written as such that each admits a transversal. We study the question of when an almost disjoint family admits a -transversal and related questions.

     

Decomposing representations of finite groups on Riemann-Roch spaces

If is a finite subgroup of the automorphism group of a projective curve and is a divisor on stabilized by , then we compute a simplified formula for the trace of the natural representation of on the Riemann-Roch space , under the assumption that is ``rational', is nonspecial, and the characteristic is ``good'. We discuss the partial formulas that result if is not rational.

     

On quasifree profinite groups

Recently, it has been shown by Harbater and Stevenson that a profinite group is free profinite of infinite rank if and only if is projective and -quasifree. The latter condition requires the existence of distinct solutions to certain embedding problems for . In this paper we provide several new non-trivial examples of -quasifree groups, projective and non-projective. Our main result is that open subgroups of -quasifree groups are -quasifree.

     

Semiprime smash products and -stable prime radicals for PI-algebras

Assume that is a finite-dimensional Hopf algebra over a field and that is an -module algebra satisfying a polynomial identity (PI). We prove that if is semisimple and is -semiprime, then is semiprime. If is cosemisimple, we show that the prime radical of is -stable.

     

On the degree of Hilbert polynomials associated to the torsion functor

Let be a local, Noetherian ring and an ideal. A question of Kodiyalam asks whether for fixed , the polynomial giving the th Betti number of has degree equal to the analytic spread of minus one. Under mild conditions on , we show that the answer is positive in a number of cases, including when is divisible by or is an integrally closed -primary ideal.

     

Non-reflexivity of the derivation space from Banach algebras of analytic functions

Let be an open connected subset of the plane, and let be a Banach algebra of analytic functions on . We show that the space of bounded derivations from into is not reflexive. We also obtain similar results when for .

     

On local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

Some Hopf Galois structures arising from elementary abelian -groups

Let be an odd prime, , the elementary abelian -group of rank , and let be the group of principal units of the ring . If is a Galois extension with Galois group , then we show that for , the number of Hopf Galois structures on afforded by -Hopf algebras with associated group is greater than , where .

     

A reduction theorem for the topological degree for mappings of class

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class from a real separable reflexive Banach space into the dual space can be reduced into the calculation of degree of mapping from a closed subspace into Since the Leray-Schauder mappings are acting from to and we are dealing with mappings from to the standard `invariant subspace' condition must be replaced by a less obvious one.

     

Perturbations and Weyl's theorem

A Banach space operator is completely hereditarily normaloid, , if either every part, and (also) for every invertible part , of is normaloid or if for every complex number every part of is normaloid. Sufficient conditions for the perturbation of by an algebraic operator to satisfy Weyl's theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator satisfies -Weyl's theorem.

     

Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers is a field generated by an element having a power in , and a cyclotomic extension is an extension generated by a root of unity. We show that a radical extension that is almost Galois over is almost cyclotomic. More precisely, we prove that if is radical with Galois closure , then contains a cyclotomic field such that the degree is bounded above by an almost linear function of . In particular, if is Galois, it contains a cyclotomic field such that .

     

Mapping properties of analytic functions on the disk

There is a universal constant with the following property. Suppose that is an analytic function on the unit disk , and suppose that there exists a constant so that the Euclidean area, counting multiplicity, of the portion of which lies over the disk , centered at and of radius , is strictly less than the area of . Then must send into . This answers a conjecture of Don Marshall.

     

Extensions of orthosymmetric lattice bimorphisms revisited

This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces and with Dedekind-complete and an orthosymmetric lattice bimorphism with lattice bimorphism extension which is not orthosymmetric, where denotes the Dedekind-completion of . On the other hand, there is an associative -multiplication in the same Archimedean Riesz space which extends to a -multiplication in which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.

     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

Extremal extensions for the sum of nonnegative selfadjoint relations

The sum of two nonnegative selfadjoint relations (multi-valued operators) and is a nonnegative relation. The class of all extremal extensions of the sum is characterized as products of relations via an auxiliary Hilbert space associated with and . The so-called form sum extension of is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with and . Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on and .

     

On the topology of the boundary of a basin of attraction

Suppose is a continuous flow on a locally compact metrizable space and is an (asymptotically stable) attractor. Let be the boundary of the basin of attraction of . In the present paper it will be shown how the Conley index of plays an important role in determining the topological nature of and allows one to obtain information about the global dynamics of in .

     

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

find all possible extending to all , with . We also consider the problem of finding metric geodesics joining two given isometries and . It is well known that if there exists a continuous path joining and , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining and .

     

Codimension growth of two-dimensional non-associative algebras

Let be a field of characteristic zero and let be a two- dimensional non-associative algebra over . We prove that the sequence of codimensions of is either bounded by or grows exponentially as . We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is , .