Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector

Let be a -step nilpotent Lie algebra; we say is non-integrable if, for a generic pair of points , the isotropy algebras do not commute: . Theorem: If is a simply-connected -step nilpotent Lie group, is non-integrable, is a cocompact subgroup, and is a left-invariant Riemannian metric, then the geodesic flow of on is neither Liouville nor non-commutatively integrable with first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

Heegner points and Mordell-Weil groups of elliptic curves over large fields

Let be an elliptic curve defined over of conductor and let be the absolute Galois group of an algebraic closure of . For an automorphism , we let be the fixed subfield of under . We prove that for every , the Mordell-Weil group of over the maximal Galois extension of contained in has infinite rank, so the rank of is infinite. Our approach uses the modularity of and a collection of algebraic points on - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer and for a prime prime to , the rank of over all the ring class fields of a conductor of the form is unbounded, as goes to infinity.

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Criteria for large deviations

We give the general variational form of

for any bounded above Borel measurable function on a topological space , where is a net of Borel probability measures on , and a net in converging to . When is normal, we obtain a criterion in order to have a limit in the above expression for all continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.

     

A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders

Let be a complete discrete valuation domain with the unique maximal ideal . We suppose that is an algebra over an algebraically closed field and . Subamalgam -suborders of a tiled -order are studied in the paper by means of the integral Tits quadratic form . A criterion for a subamalgam -order to be of tame lattice type is given in terms of the Tits quadratic form and a forbidden list of minor -suborders of presented in the tables.

     

The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra contains a two-sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form . We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions and by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces and over and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras

Let be a field, a non-zero element of and the Iwahori-Hecke algebra of the symmetric group . If is a block of of -weight and the characteristic of is at least , we prove that the decomposition numbers for are all at most . In particular, the decomposition numbers for a -block of of defect are all at most .

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

Frobenius distributions of Drinfeld modules over finite fields

We express the weighted class number of Drinfeld -modules of rank two with given characteristic polynomial over the finite field , where as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.

     

Uniform Sobolev inequalities and absolute continuity of periodic operators

We establish certain uniform inequalities for a family of second order elliptic operators of the form on the -torus, where and is a symmetric, positive definite matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on with singular potential. The absolute continuity of the elliptic operator div on with a positive periodic scalar function is also studied.

     

A generalization of tight closure and multiplier ideals

We introduce a new variant of tight closure associated to any fixed ideal , which we call -tight closure, and study various properties thereof. In our theory, the annihilator ideal of all -tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal and the multiplier ideal associated to (or, the adjoint of in Lipman's sense) in normal -Gorenstein rings reduced from characteristic zero to characteristic . Also, in fixed prime characteristic, we establish some properties of similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal and the F-rationality of Rees algebras.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

Matrix-weighted Besov spaces

Nazarov, Treil and Volberg defined matrix weights and extended the theory of weighted norm inequalities on to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix weight setting. In particular, we introduce matrix- weighted homogeneous Besov spaces and matrix-weighted sequence Besov spaces , as well as , where the are reducing operators for . Under any of three different conditions on the weight , we prove the norm equivalences , where is the vector-valued sequence of -transform coefficients of . In the process, we note and use an alternate, more explicit characterization of the matrix class. Furthermore, we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on if is doubling. We also obtain the boundedness of almost diagonal operators on under any of the three conditions on . This leads to the boundedness of convolution and non-convolution type Calderón-Zygmund operators (CZOs) on , in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the -transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces and show that results corresponding to those above are true also for the inhomogeneous case.

     

On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces

Let be a simply connected connected real nilpotent Lie group with Lie algebra , a connected closed subgroup of with Lie algebra and satisfying . Let be the unitary character of with differential at the origin. Let be the unitary representation of induced from the character of . We consider the algebra of differential operators invariant under the action of on the bundle with basis associated to these data. We consider the question of the equivalence between the commutativity of and the finite multiplicities of . Corwin and Greenleaf proved that if is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.

     

Analytic contractions, nontangential limits, and the index of invariant subspaces

Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of .

We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0.

It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent:

(1) for some ,

(2) for all , ,

(3) has nonzero Lebesgue measure,

(4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .

     

The autohomeomorphism group of the Cech-Stone compactification of the integers

It is shown to be consistent that there is a nontrivial autohomeomorphism of , yet all such autohomeomorphisms are trivial on a dense -ideal. Furthermore, the cardinality of the autohomeomorphism group of can be any regular cardinal between and . The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.

     

Spectral lifting in Banach algebras and interpolation in several variables

Let be a unital Banach algebra and let be a closed two-sided ideal of . We prove that if any invertible element of has an invertible lifting in , then the quotient homomorphism is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foias, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna-Pick, and Carathéodory type interpolation for , the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra and , the algebra of bounded operators on a finite dimensional Hilbert space . A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for .

In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of , in which one bounds the spectral radius of the interpolant and not the norm.