Cohomological construction of quantized universal enveloping algebras

Given an associative algebra and the category of its finite dimensional modules, additional structures on the algebra induce corresponding ones on the category . Thus, the structure of a rigid quasi-tensor (braided monoidal) category on is induced by an algebra homomorphism (comultiplication), coassociative up to conjugation by (associativity constraint) and cocommutative up to conjugation by (commutativity constraint), together with an antiautomorphism (antipode) of satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element with suitable induced actions on , and . Drinfeld defined such a structure on for any semisimple Lie algebra with the usual comultiplication and antipode but nontrivial and , and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), .

In the paper we give a direct cohomological construction of the which reduces to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of , in particular, gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements , and can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to with depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.

     

Approximation by harmonic functions

For a compact set we construct a restoring covering for the space of real-valued functions on which can be uniformly approximated by harmonic functions. Functions from restricted to an element of this covering possess some analytic properties. In particular, every nonnegative function , equal to 0 on an open non-void set, is equal to 0 on . Moreover, when , the algebra of complex-valued functions on which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set has a nontrivial Jensen measure, then contains a nontrivial compact set with analytic algebra .

     

Automorphism Groups and Invariant Subspace Lattices

Let be a - dynamical system and let be the analytic subalgebra of . We extend the work of Loebl and the first author that relates the invariant subspace structure of for a -representation on a Hilbert space , to the possibility of implementing on We show that if is irreducible and if lat is trivial, then is ultraweakly dense in We show, too, that if satisfies what we call the strong Dirichlet condition, then the ultraweak closure of is a nest algebra for each irreducible representation Our methods give a new proof of a ``density' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid -algebras.

     

Differential operators on Stanley-Reisner rings

Let be an algebraically closed field of characteristic zero, and let be a polynomial ring. Suppose that is an ideal in that may be generated by monomials. We investigate the ring of differential operators on the ring , and , the idealiser of in . We show that and are always right Noetherian rings. If is a square-free monomial ideal then we also identify all the two-sided ideals of . To each simplicial complex on there is a corresponding square-free monomial ideal , and the Stanley-Reisner ring associated to is defined to be . We find necessary and sufficient conditions on for to be left Noetherian.

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Group actions on arrangements of linear subspaces and applications to configuration spaces

For an arrangement of linear subspaces in that is invariant under a finite subgroup of the general linear group we develop a formula for the -module structure of the cohomology of the complement . Our formula specializes to the well known Goresky-MacPherson theorem in case , but for the formula shows that the -module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space . Our motivating examples are arrangements in that are invariant under the action of by permuting coordinates. A particular case is the ``-equal' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases and are spaces of ordered and unordered point configurations in many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in and provide explicit results for the ``-equal' and the ``-divisible' cases.

     

Linear isometries between subspaces of continuous functions

We say that a linear subspace of is strongly separating if given any pair of distinct points of the locally compact space , then there exists such that . In this paper we prove that a linear isometry of onto such a subspace of induces a homeomorphism between two certain singular subspaces of the Shilov boundaries of and , sending the Choquet boundary of onto the Choquet boundary of . We also provide an example which shows that the above result is no longer true if we do not assume to be strongly separating. Furthermore we obtain the following multiplicative representation of : for all and all , where is a unimodular scalar-valued continuous function on . These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

     

Twin trees and -gons

We define a natural generalization of generalized -gons to the case of -graphs (where is a totally ordered abelian group and ). We term these objects -gons. We then show that twin trees as defined by Ronan and Tits can be viewed as -gons, where is ordered lexicographically. This allows us to then generalize twin trees to the case of -trees. Finally, we give a free construction of -gons in the cases where is discrete and has a subgroup of index that does not contain the minimal element of .

     

On the rational homotopy type of function spaces

The main result of this paper is the construction of a minimal model for the function space of continuous functions from a finite type, finite dimensional space to a finite type, nilpotent space in terms of minimal models for and . For the component containing the constant map, in positive dimensions. When is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for and the coproduct of . We also give a version of the main result for the space of cross sections of a fibration.

     

Signed Quasi-Measures

Let be a compact Hausdorff space and let denote the subsets of which are either open or closed. A quasi-linear functional is a map which is linear on singly generated subalgebras and such that for some . There is a one-to-one correspondence between the quasi-linear functional on and the set functions such that i) , ii) If with and disjoint, then , iii) There is an such that whenever are disjoint open sets, , and iv) if is open and , there is a compact such that whenever is open, then . The space of quasi-linear functionals is investigated and quasi-linear maps between two spaces are studied.

     

The Group of Galois Extensions Over Orders in

In this paper we characterize all Galois extensions over where is an arbitrary -Hopf order in . We conclude that the abelian group of -Galois extensions is isomorphic to a certain quotient of units groups in . This result generalizes the classification of -Galois extensions, where , due to Roberts, and also to Hurley and Greither.

     

Extensions of modules over Weyl algebras

In this paper we calculate some groups of singular modules over the complex Weyl algebra . In particular we determine conditions under which is an infinite dimensional vector space when or .

     

The image of the Thom map for Eilenberg-MacLane spaces

Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Necessary and sufficient conditions are given for the fractional integral operator to be bounded from weighted strong and weak spaces within the range into suitable weighted and Lipschitz spaces. We also characterize the weights for which can be extended to a bounded operator from weighted into a weighted Lipschitz space of order . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of between weighted Lipschitz spaces.

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Expansive Subdynamics

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let be a continuous action of on an infinite compact metric space. For each subspace of we introduce a notion of expansiveness for along , and show that there are nonexpansive subspaces in every dimension . For each the set of expansive -dimensional subspaces is open in the Grassmann manifold of all -dimensional subspaces of . Various dynamical properties of are constant, or vary nicely, within a connected component of , but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an . The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.

     

Strict definiteness of integrals via complete monotonicity of derivatives

Let be a nonnegative integer and let be a function with completely monotone and not constant. If is a signed measure on any euclidean space , with vanishing moments up to order , then the integral is strictly positive whenever it exists. For general no larger class of continuous functions seems to admit the same conclusion. Examples and applications are indicated. A section on 'bilinear integrability' might be of independent interest.

     

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions of subalgebras of a Hopf algebra are -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra , we show that when are Hopf algebras in the Yetter-Drinfeld category for , the extension is -Frobenius provided is finite over and the extension of biproducts is cleft.

More generally we give conditions for an extension to be -Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.