The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties

We study the following question: when is the right adjoint of the forgetful functor from the category of -Doi-Hopf modules to the category of -modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that and the smash product are isomorphic as -bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case , and this leads to the notion of -Frobenius -module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if is finite dimensional and unimodular.

     

A note on a paper by Molai and Khorram

The aim of this note is to show that the optimization algorithm proposed in [6] may not lead to the optimal solution in some cases. In fact, the optimization problem remains open if we do not apply the branch-and-bound method to solve it or compute all the objective values of minimal solutions of the feasible domain.     

A note on a paper of Sperb

Summary Two lower bounds of Sperb for the first eigenvalue of an elastically attached membrane are proved in an elementary way.

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Zusammenfassung Zwei untere Schranken von Sperb für den tiefsten Eigenwert einer elastisch gestützten Membran werden in einfacher Weise nachgewiesen.

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This work supported by the Department of the Navy, Naval Ordnance Systems Command under Contract N00017-72-C-4401.     

Hopf modules,Frobenius functors and (one-sided) Hopf algebras

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted m_H, and prove that the value of this invariant is connected to the order of S. In the case where char k?=?0, it is shown that if S has finite order then it is either the identity or has order 2?m_H. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if m_H is finite. We also consider the case where char k?=?p?>?0, generalizing the results of [7] to the infinite-dimensional setting.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

A note on a paper by G.Pólya

This note is concerned with the computer implementation of Pólya's paper [8] for determining the zeros of a special class of entire functions. Grau's modification to the Graeffe process [6] is utilised to bound the iterated coefficients.     

A note on the Auslander-Reiten conjecture

It is proved that the Auslander-Reiten conjecture is true for local Artin algebras with radical cube zero.     

A note on uniform approximation by the indestructibles

We modify an idea in an earlier paper to enlarge the collection of inner functions that are known to be in the uniform closure of the indestructible Blaschke products.     

A note on a paper by G. Mastroianni and G. Monegato

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral

where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

Algebras, hyperalgebras, nonassociative bialgebras and loops

Sabinin algebras are a broad generalization of Lie algebras that include Lie, Malcev and Bol algebras as very particular examples. We present a construction of a universal enveloping algebra for Sabinin algebras, and the corresponding Poincaré-Birkhoff-Witt Theorem. A nonassociative counterpart of Hopf algebras is also introduced and a version of the Milnor-Moore Theorem is proved. Loop algebras and universal enveloping algebras of Sabinin algebras are natural examples of these nonassociative Hopf algebras. Identities of loops move to identities of nonassociative Hopf algebras by a linearizing process. In this way, nonassociative algebras and Hopf algebras interlace smoothly.     

A note on Tang and He’s paper

In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72-85] when 1 < p_i < 2 for i = 1, 2, … , n.