Cohomological Obstructions to Cleft Extensions over Cocommutative Hopf Algebras

We extend the cohomological treatment of cleft extensions over cocommutative Hopf algebras by giving an interpretation of degree three Sweedler cohomology classes as obstructions to extensions. We show that every twisted action of a cocommutative Hopf algebra on an algebra R gives rise to an obstruction, a degree three Sweedler cohomology class of H with values in the center of R. The obstruction vanishes if and only if the twisted action belongs to a crossed product extension. We also show that every Sweedler three-cocycle can be realized as an obstruction.     

Galois theory for Hopf algebroids

An extensionB⊂A of algebras over a commutative ringk is anH-extension for anL-bialgebroidH ifA is anH-comodule algebra andB is the subalgebra of its coinvariants. It isH-Galois if the canonical mapA ⊗_B A →A ⊗_L H is an isomorphism or, equivalently, if the canonical coringA ⊗_L H:A is a Galois coring. In the case of Hopf algebroid anyH _R-extension is shown to be also anH _L-extension. If the antipode is bijective then also the notions ofH _R-Galois extensions and ofH _L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroidH with bijective antipode. It states that anyH-Galois extensionB⊂A is projective, and ifA isk-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

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Sunto Un'estensioneB(A di algebre su un anello commutativok è unaH-estensione per unL-bialgebroideH seA è unaH-comodulo algebra eB è la sottoalgebra dei suoi coinvarianti. Essa èH-Galois se l'applicazione canonicaA ⊗_A B→A ⊗_L H è un isomorfismo o, equivalentemente, se il coanello canonicoA ⊗_L H:A è un coanello di Galois. Nel caso di un algebroide di Hopf si dimostra che ogniH _R-estensione è unaH _L-estensione. Se l'antipode è biiettivo allora si dimostra che anche le nozioni di estensioniH _R-Galois eH _L-Galois coincidono. I risultati per le strutture biiettive entwining sono estesi alle strutture entwining su algebre non commutative, al fine di dimostrare un teorema simile al Teorema dii Kreimer-Takeuchi per un Hopf algebroideH proiettivo finitamento generato con antipode biiettivo. Il teorema afferma che ogni estensioneH-GaloisB ⊂A è proiettiva e seA èk-piatto allora la suriettività dell'applicazione canonica è sufficiente a garantire la proprietà di Galois. La teoria di Morita, sviluppata per i coanelli da Caenepeel, Vercruysse e Wang, viene applicata per ottenere criteri equivalenti per la proprietà di Galois per estensioni di algebroidi di Hopf. Questo conduce a risultati analoghi, per algebroidi di Hopf, a quelli ottenuti da Doi per estensioni di algebre di Hopf e da Cohen Fishman e Montgomery nel caso degli algebroidi di Hopf Frobenius.

     

On Cleft Extensions of the Tensor Product of Two Hopf Algebras

In this article, we first decompose a cleft extension for X ? Y into two cleft extensions for X and Y respectively, where X and Y are Hopf algebras on a commutative ring R. Conversely, we introduce the concept of consistent cleft Hopf extensions and prove that one can construct a cleft extension for X ? Y by two cleft extensions for X and Y if and only if these two cleft extensions are consistent. An example is also given to show an application of our main results.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Morita theory for coring extensions and cleft bicomodules

A Morita context is constructed for any comodule of a coring and, more generally, for an L-C bicomodule Σ for a coring extension (D:L) of (C:A). It is related to a 2-object subcategory of the category of k-linear functors MC→MD. Strictness of the Morita context is shown to imply the Galois property of Σ as a C-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold.Cleft property of an L-C bicomodule Σ—implying strictness of the associated Morita context—is introduced. It is shown to be equivalent to being a GaloisC-comodule and isomorphic to EndC(Σ)L_⊗D, in the category of left modules for the ring EndC(Σ) and right comodules for the coring D, i.e. satisfying the normal basis property.Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules.     

A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

To any finite symmetric subsetR ⊂ O₃ corresponds a Hecke operatorT _R on L²(S ²) which leaves the eigenspaces ℋ_n (n ≥ 0) of the Laplacian invariant. We compute the trace ofT _R | ℋ_n and prove that the sum of the positive eigenvalues ofT _R on ⊕_k=0 ^n-1ℋ_k prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator , and renormalizing the traces ofT _{R m} | ℋ _n slightly, we obtain sequences of Fourier coefficients of modular forms on Γ₀(4). Dedicated to R. Remmert on the occasion of his seventieth birthday     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

The spectral radius of matrix continuous refinement operators

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space L₂^m(\mathbb R^s)L_2^m({{\mathbb R}}^s), m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces L _p for p ∈ [1, ∞ ), p ≠ 2. However, additional restrictions on the kernel c are imposed in this case.     

Shades of orthodoxy in Rees-Sushkevich varieties

It is shown that, within the class of Rees-Sushkevich varieties that are generated by completely (0-) simple semigroups over groups of exponent dividing n, there is a hierarchy of varieties determined by the lengths of the products of idempotents that will, if they fall into a group ℋ-class, be idempotent. Moreover, the lattice of varieties generated by completely (0-) simple semigroups over groups of exponent dividing n, with the property that all products of idempotents that fall into group ℋ-classes are idempotent, is shown to be isomorphic to the direct product of the lattice of varieties of groups with exponent dividing n and the lattice of exact subvarieties of a variety generated by a certain five element completely 0-simple semigroup.     

On the isomorphic classes of hopf-galois extensions with normal bases over a commutative ring

In this paper we consider the isomorphic classes of H-Galois extensions with the normal basis property over a fixed commutative ring B for a finite Hopf algebra H. The result is that when H is cocommutative, then the above isomorphic classes and the 2-nd cohomology group H ²(L*, U) are isomorphic as groups under the product defined in this paper, where L=B∝H, L*=Hom _B (L, B). Supported by the NNSF of China     

Irreducible Width 2 Posets of Linear Discrepancy 3

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |h _L (x) − h _L (y)| ≤ k, where h _L (x) is the height of x in L. Tannenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem for characterizing the posets of linear discrepancy 2. Howard et al. (Order 24:139–153, 2007) showed that this problem is equivalent to finding all posets of linear discrepancy 3 such that the removal of any point reduces the linear discrepancy. In this paper we determine all of these minimal posets of linear discrepancy 3 that have width 2. We do so by showing that, when removing a specific maximal point in a minimal linear discrepancy 3 poset, there is a unique linear extension that witnesses linear discrepancy 2. The first author was supported during this research by National Science foundation VIGRE grant DMS-0135290.     

WeightedL p closure theorems for spaces of entire functions

For a fixed weight Δ(dx) onR ¹ and a linear space ℋ ⊆L ^p(Δ) of entire functions that is closed under difference quotientsh(·)→(z−·) ⁻¹[h(z)−h(·)], theL ^p(Δ) closure of ℋ is studied and characterized in terms of the normsL(z), (z∈C ¹ of the evaluation functionalsh→h(z),h∈ℋ. Partially supported by DA-ARO-31-124-71-6182 and NSF GP-43011.     

A Characterization and the Intersection of the Maximal Compatible Extensions of a Partial Order

We present a characterization of the maximal compatible extensions of a given compatible partial order ≤  _r on a unary algebra (A,f ). These extensions can be constructed by using the compatible linear extensions of ≤  _r*, where (A*,f*) is the so called contracted quotient algebra of (A,f) and the compatible partial order ≤  _r* on (A*,f*) is naturally induced by ≤  _r . Using this characterization, we determine the intersection of the maximal compatible extensions of ≤  _r .      

Invariant properties of representations under cleft extensions

The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.     

Weighted Polynomials on Discrete Sets

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ , deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete H?lder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities.     

Estimates for the norms of fractional derivatives in terms of integral moduli of continuity and their applications

For functions defined on the entire real axis or a semiaxis, we obtain Kolmogorov-type inequalities that estimate the L _p -norms (1 ≤ p < ∞) of fractional derivatives in terms of the L _p -norms of functions (or the L _p -norms of their truncated derivatives) and their L _p -moduli of continuity and establish their sharpness for p = 1: Applications of the obtained inequalities are given.