Regular Multiplier Research Articles

QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of Snon the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

A Note on the Antipode for Algebraic Quantum Groups

AbstractRecently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Constructing New Braided T-Categories Over Regular Multiplier Hopf Algebras

Let A be a regular multiplier Hopf algebra, and let Aut(A) denote the set of all isomorphisms α from A to itself that are algebra maps satisfying (Δ ○ α)(a) = (α ⊗ α) ○ Δ(a) for all a ∈ A. Let G be a certain crossed product group Aut(A) × Aut(A). The main purpose of this article is to provide a class of new braided T-categories in the sense of Turaev [\\citealp9]. For this, we introduce a class of new categories A 𝒴𝒟 A (α, β) of (α, β)-Yetter–Drinfel'd modules with α, β ∈Aut(A), and we show that the category ℳ𝒴𝒟(A) = { A 𝒴𝒟 A (α, β)}(α, β)∈G becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic [6].

Bicrossproducts of multiplier Hopf algebras

In this paper, we generalize Majidʼs bicrossproduct construction. We start with a pair ( A , B ) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. The right action of A on B gives rise to the smash product A # B . The left coaction of B on A gives a possible coproduct Δ # on A # B . We discuss in detail the necessary compatibility conditions between the action and the coaction for Δ # to be a proper coproduct on A # B . The result is again a regular multiplier Hopf algebra. Majidʼs construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair ( C , D ) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D yields a duality between A # B and the smash product C # D . We show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The ⁎-algebra case is also considered. Some special cases are treated and they are related with other constructions available in the literature. The basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = K H and H ∩ K = { e } (where e is the identity of G) is used to illustrate our theory. More examples will be considered in forthcoming papers on the subject.

A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

A new method of constructing quasitriangular group-cograded multiplier Hopf algebras is provided. For a multiplier Hopf dual pairing σ between regular multiplier Hopf algebras A and B, we introduce the concept of a σ-compatible pairing (Φ, Ψ, σ ), where Φ and Ψ are actions of the twisted semi-direct group Open image in new window of a group G on A and B, respectively. We construct a twisted double group-cograded multiplier Hopf algbera D(A, B; σ, Φ, Ψ). Furthermore, if there is a canonical multiplier in M(B ⊗ A) we show existence of quasitriangular structure on D(A, B; σ, Φ, Ψ). As an application, special cases and examples are given.

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman. We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.

Multiplier Hopf Algebras Imbedded in Locally Compact Quantum Groups

Let (A,∆) be a locally compact quantum group and (A0,∆0) a regular multiplier Hopf algebra. We show that if (A0,∆0) can be imbedded in (A,∆), then A0 will inherit some of the analytic structure of A. Under certain conditions on the imbedding, we will be able to conclude that (A0,∆0) is actually an algebraic quantum group with a full analytic structure. The techniques used to show this can also be applied to obtain the analytic structure of a ∗-algebraic quantum group in a purely algebraic fashion. Moreover, the reason that this analytic structure exists at all, is that the associated one-parameter groups, such as the modular group and the scaling group, are diagonizable. As an immediate corollary, we will show that the scaling constant μ of a ∗-algebraic quantum group equals 1. This solves an open problem posed in [13].

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux’s main theorem in the case of smash products.

Boundary optimal flow control with state constraints

AbstractThe numerical solution of the Dirichlet boundary optimal control problem of the Navier‐Stokes equations in presence of pointwise state constraints is investigated. A Moreau‐Yosida regularization of the problem is proposed to obtain regular multipliers. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is presented. The paper ends with a numerical experiment. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct.

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

Let ⟨ A, B⟩ be a pairing of two regular multiplier Hopf algebras A and B. One method of constructing the Drinfel'd double 𝒟 = A⋈ B cop is by the use of an invertible twist map R: B⊗ A→ A⊗ B defining an associative product on A⊗ B. In Delvaux (2003) and Drabant and Van Daele (2001), the authors construct R by , where R 2 and is only related to the module actions between A and B. Another way is given in Delvaux and Van Daele (2004a) in which the authors also just consider the module actions and then construct the Drinfel'd double 𝒟 as an algebra of operators on the vector space B⊗ A. In this article we will give two different points of view of constructing the Drinfel'd double 𝒟 for multiplier Hopf algebras. The first is that the Drinfel'd double 𝒟 associated to the pairing ⟨ A, B⟩ is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfel'd double 𝒟 is given in the framework of a special twisted tensor product algebra structure A ⊡ A op ⊗ A B. The second is as follows: If P is a multiplier Hopf algebra and a reduced (A, B)-bicomodule algebra (an A-Long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P, and we show that the Drinfel'd double 𝒟 is isomorphic as a multiplier Hopf algebra to the opposite twisting of (A op, cop ⊗ B op, cop ). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras.

Larson–Sweedler Theorem and Some Properties of Discrete Type in (G-Cograded) Multiplier Hopf Algebras

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000). Finally, we study regular multiplier Hopf algebras of η -discrete type.

Minimizing Sequences in Class-Qualified Deposit Problems

We study multidimensional control problems involving first-order partial differential equations. To ensure the existence of sufficiently regular multipliers (from the space \({C^{\ast}}\)) in the first-order necessary optimality conditions, some restrictions of the feasible domain have to be added. In particular, we investigate ‘class-qualified’ problems where the weak derivatives of \(x\) can be represented within a Baire function class. In the present paper, we prove conditions under which the original and the modified problems possess the same minimal values.

Existence of Regular Lagrange Multipliers for a Nonlinear Elliptic Optimal Control Problem with Pointwise Control-State Constraints

A class of optimal control problems for semilinear elliptic equations with mixed control-state constraints is considered. The existence of bounded and measurable Lagrange multipliers is proven. As a particular application, the Lavrentiev-type regularization of pointwise state constraints is discussed. Here, the existence of associated regular multipliers is shown, too.

Piecewise Continuous Controls in Dieudonné-Rashevsky Type Problems

This paper considers multidimensional control problems governed by a first-order PDE system. It is known that, if the structure of the problem is linear-convex, then the so-called e-maximum principle, a set of necessary optimality conditions involving a perturbation parameter e > 0, holds. Assuming that the optimal controls are piecewise continuous, we are able to drop the perturbation parameter within the conditions, proving the Pontryagin maximum principle with piecewise regular multipliers (measures). The Lebesgue and Hahn decompositions of the multipliers lead to refined maximum conditions. Our proof is based on the Baire classification of the admissible controls.

Regular Lagrange Multipliers for Control Problems with Mixed Pointwise Control-State Constraints

A class of quadratic optimization problems in Hilbert spaces is considered, where pointwise box-constraints and constraints of bottleneck type are given. The main focus is on proving the existence of regular Lagrange multipliers in $L^2$-spaces. This question is solved by investigating the solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control problems for elliptic and parabolic partial differential equations with mixed pointwise control-state constraints.

Maternal and infant factors associated with excess kindergarten costs.

To estimate the excess educational costs at kindergarten from infant and maternal factors that are reported routinely at birth. Birth and school records were analyzed for all children who were born in Florida between September 1, 1990, and August 31, 1991, and entered kindergarten from 1996 through 1999 (N = 120,554). Outcome measure was cost to state, derived from base allocation for students in regular classrooms plus multiplier weights for those who were assigned to 8 mutually exclusive special education categories or who repeated kindergarten. More than one quarter of the study cohort was found to be assigned to special education classes at kindergarten. Regression model estimates indicated that children who were born at <1000 g (n = 380) generated 71% higher costs in kindergarten than children who were born at >or=2500 g. Children who were born at 1000 to 1499 g (n = 839) generated 49% higher costs. Other birth conditions, independent of birth weight, were associated with higher kindergarten costs: family poverty (31%), congenital anomalies (29%), maternal education less than high school (20%), and no prenatal care (14%). Because of their prevalence, family poverty and low maternal education accounted for >75% of excess kindergarten costs. If 9% of infants who weighed between 1500 and 2499 g (n = 1027) could be delivered at 2500 g, then the state of Florida potentially could save 1 million dollars in kindergarten costs. Savings of a similar magnitude might be achieved if 3% of mothers who left school without a diploma (n = 1528) were to graduate. Any policy recommendation aimed at reducing education costs in kindergarten must take into consideration 3 factors: the prevalence of risk conditions whose amelioration is desired, the potential cost savings associated with reducing those conditions, and the costs of amelioration. Projecting these costs from information that is available at birth can assist school districts and state agencies in allocating resources.

The Drinfel'd double versus the Heisenberg double for an algebraic quantum group

Let A be a regular multiplier Hopf algebra with integrals. The dual of A, denoted by Â, is a multiplier Hopf algebra so that 〈 A ̂ ,A〉 is a pairing of multiplier Hopf algebras. We consider the Drinfel'd double, D= A ̂ ⋈A cop , associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair 〈 A ̂ ,A〉 has a “canonical multiplier” W∈M( A ̂ ⊗A) . The image of W in M( D⊗ D) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra D̂ via the right action of D on D̂ which defines the pair 〈 D ̂ ,D〉 . As expected from the finite-dimensional case, we find that the deformation of the product in D̂ is related to the Heisenberg double A# A ̂ .

The Drinfel'd double of multiplier Hopf algebras

Let 〈 A, B〉 be a pairing of two regular multiplier Hopf algebras A and B. The Drinfel'd double associated to this pairing is constructed by using appropriate representations of A and B on the same vector space B⊗ A. We realize the Drinfel'd double, denoted by D, as an algebra of operators on the vector space B⊗ A. In the case that 〈 A, B〉 is a multiplier Hopf ∗ -algebra pairing, we prove that D is again a multiplier Hopf ∗ -algebra. If A and B carry positive integrals, we prove that D also has a positive integral. This proof is not given before.