On bimeasurings

We introduce and study bimeasurings from pairs of bialgebras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler's universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant functor on the category of bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in the category of Hopf modules is considered.     

Quantum group actions, twisting elements, and deformations of algebras

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for catagories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.     

Representations and Cocycle Twists of Color Lie Algebras

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra .Presented by A. Verschoren.     

Enveloping ringoids

We present notions of module over a universal algebra, and linear representation of a universal algebra, which have gained currency with categorical algebraists, we give several intriguing examples of these objects, showing that important aspects of universal algebras can be studied in this context, and we describe the theory ofenveloping ringoids, which have categories of left modules equivalent to corresponding categories of modules and linear representations. Algebras in many of the familiar varieties of algebras, which have underlying groups, turn out to have enveloping ringoids that are equivalent to familiar rings. Nonempty algebras in an abelian varietyV have an enveloping ringoid equivalent toR(V), the so-calledring of the variety V.Dedicated to the memory of Alan DayPresented by J. Sichler.     

Elliptic units and sign functions

In the first part of this paper we give a new definition of the elliptic analogue of Sinnott’s group of circular units. In this we essentially use the ideas discussed in Oukhaba (in Ann Inst Fourier, 55(33):753–772, 2005). In the second part of the paper we are interested in computing the index of this group of elliptic units. This question is closely related to the behaviour of the universal signed ordinary distributions introduced in loc. cit. Such distributions have a natural resolution discovered by Anderson. Consequently, we can apply Ouyang’s general index formula and the powerful Anderson’s theory of double complex to make the computations     

When is a Cleft Extension H-Azumaya?

We show which H ^op -cleft extensions of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).Presented by A. Verschoren     

Rota-Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.     

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category for which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category, with respect to the long forgetful functor, is self-dual as a Weak Hopf Algebra.     

RADICAL THEORY FOR SEMIRINGS

Abstract

In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ?_∞(S) and ?^ε(S) and two similar ideals β_∞ (S) and β^ε(S) is widely solved. We prove ?_∞(S) ? ?^ε(S) = β^∞(S) = β^ε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M_ε(U) is always semisimple, a result which is not true for the special class M_∞(U).     

Universal deformation rings and cyclic blocks

In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory. Received July 19, 1999 / Revised May 13, 2000 / Published online October 30, 2000     

Lie metabelian restricted universal enveloping algebras

We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian.Received: 1 October 2004     

Application of Koszul complex to Wronski relations for $U(\frak{gl}_n)$

Explicit relations between two families of central elements in the universal enveloping algebra $U(\frak{gl}_n)$ of the general linear Lie algebra $\frak{gl}_n$ are presented. The two families of central elements in question are the ones expressed respectively by the determinants and the permanents: the former are known as the Capelli elements, and the latter are the central elements obtained by Nazarov. The proofs given are based on the exactness of the Koszul complex and the Euler-Poincaré principle.     

The Cohomology of the Universal Steenrod Algebra

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.     

Real Taylor shifts and simultaneous approximation

We are interested in the Taylor shift operator acting on the space of infinitely differentiable functions. In particular if we choose a countable family of centers of Taylor expansion, we prove that the associated family of real Taylor shifts fulfills the approximation of any given family of infinitely differentiable functions with a common subsequence of iterates applied on a common vector. We obtain similar conclusions in the context of universal series improving recent statements. Finally we introduce the notion of doubly universal Taylor shift. All these results give new and natural examples of disjoint universality.     

Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X. As an application, we reduce the study of invariant integrally closed ideals of V×X to that of some smaller variety in the case of X=M_n,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237-262] for two special infinite series.     

Twisted descent algebras and the Solomon-Tits algebra

The notion of descent algebra of a bialgebra is lifted to the Barratt-Joyal setting of twisted bialgebras. The new twisted descent algebras share many properties with their classical counterparts. For example, there are twisted analogs of classical Lie idempotents and of the peak algebra. Moreover, the universal twisted descent algebra is equipped with two products and a coproduct, and there is a fundamental rule linking all three. This algebra is shown to be naturally related to the geometry of the Coxeter complex of type A.     

Hyperbolicity of algebras with involution over a given extension

Let F be a field and (A,σ) a central simple F-algebra with involution. Let π(t) be a separable polynomial over F. Let F(π)=F[t]/(π(t)). Tignol considered the question whether the algebra A⊗F(π) is hyperbolic. He introduced the algebra H_π, which is universal for this question. Haile and Tignol determined the structure of the algebra H_π and introduced a certain homomorphic image C_π of H_π. In this paper, we give a new characterization of C_π and introduce a new algebra A_π that classifies the commutative algebras with involution that become hyperbolic over F(π). We determine the structure of A_π and use it to examine some examples.     

Universal homogeneous derivations of graded ϵ-commutative algebras

Let K be a commutative ring, let ? be an abelian group, and let ?:?x?→K be a commutation factor over ?.A ? graded K-algebra is said to be ?-commutative if its ?-bracket is identically zero, (K,?) derivations from a given ?-commutative ?-graded K-algebra A into bimodules are studied. It is proved that for each λ?? there exists a universal initial (k,?)-derivation of degree λ of A. For each λ?? a natural module of (K, ?, λ)-differentials of A along with a differential map is constructed. It is proved that each derivation of A canonically equipps this module with a structure of differential module. Applications and examples are given. It is shown that the first order exterior differentials which are known from the theory of smooth graded manifolds are universal initial homogeneous derivations of the sort considered hereby.     

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of finitely generated algebras U over a field containing a finitely generated commutative Harish-Chandra subalgebra Γ. This is an important class of associative algebras, which includes all finite W-algebras of type A over an algebraically closed field of characteristic zero, in particular, the universal enveloping algebra of (or ) for all n. We show that any Γ-torsion theory defined by the coheight of the prime ideals of Γ is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in have the same coheight. Hence, the coheight of these associated prime ideals is an invariant of a given simple U-module. This implies the stratification of the category of U-modules controlled by the coheight of the associated prime ideals of Γ. Our approach can be viewed as a generalization of the classical paper by Block (1981) [4]; it allows, in particular, to study representations of beyond the classical category of weight or generalized weight modules.     

Quantum homogeneous spaces,duality and quantum 2-spheres

For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra ofG. These methods are applied to the quantum group SU(2). As two-sided coideals we take the subspaces spanned by twisted primitive elements in the sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podle is established.