Maschke Type Theorems for Nonassociative Hopf Algebras

In this article, we give necessary and sufficient conditions for a possibly nonassociative comodule algebra over a nonassociative Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the associative setting. Also, from this result we deduce a version of Maschke's Theorems and the consequent characterization of projectives for (H, B)-Hopf triples associated with a nonassociative Hopf algebra H and a nonassociative right H-comodule algebra B.     

关于四元数代数同构的条件

引入广义四元数代数的 K上表示矩阵的概念 ,探讨复线性表示与 K上表示矩阵的关系 .在此基础上 ,给出两个广义四元数代数同构的一个新的判定条件     

Strongly Anisotropic Involutions on Central Simple Algebras

The classical theorem of Bröcker and Prestel on quadratic forms over formally real fields determines a valuation theoretic condition under which all totally indefinite forms are weakly isotropic. In this article, we look for analogues of such a result in a more general setting of algebras with involutions. We prove that for involutions of first kind over central simple algebras of index two, one indeed has a Bröcker–Prestel like statement. The connection between two conditions, namely, total indefiniteness and weak isotropy is made via so called gauge functions on central simple algebras.     

L 2-Cohomology for Von Neumann Algebras

We study L ²-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L ²-cohomology and show how the study of the first L ²-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. In particular, we show that the first L ²-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest. Received: March 2006 Revision: October 2006 Accepted: October 2006     

A Few Properties of Just Infinite Algebras

In this article we prove a few interesting properties of just infinite algebras. Bartholdi (2006 Bartholdi , L. ( 2006 ). Branch rings, thinned rings, tree enveloping rings . Israel J. Math. 154 : 93 – 139 .[Crossref], [Web of Science ®] , [Google Scholar]), defines a particular class of just infinite algebras and demonstrates various properties of these examples. One such property, which is tedious to prove for his specific examples, is primality. We prove here that, in fact, all just infinite algebras are prime. We then consider two corollaries of this theorem; one suggests a weaker definition of just infinite for finitely generated algebras and the other examines the specific case of just infinite algebras which also satisfy a polynomial identity.     

The Category of Firm Modules for Nonunital Monomial Algebras

Let k be a field and X a set and P be a set of words over X. Consider the free nonunital k-algebra over X generated by the nonempty words over X and let R be the quotient of this algebra modulo the ideal generated by the words in P. R is called a “nonunital monomial algebra”. A right R-module M is said to be “firm” if M? _R R → M given by m ? r? mr is an isomorphism. In this article we prove that if R is a nonunital monomial algebra, the category of firm modules is Grothendieck.     

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coar...     

A Construction Method for Albert Algebras over Algebraic Varieties

We construct cubic Jordan algebras over an integral proper scheme X such that 2, 3 ∈ H ⁰(X, 𝒪 _X ), generalizing a construction by B. N. Allison and J. R. Faulkner. In the process, we obtain admissible cubic algebras and pseudocomposition algebras over X. Results on the structure of these algebras are obtained, as well as examples over elliptic curves.     

An Improved Multiplicity Conjecture for Codimension 3 Gorenstein Algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen–Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case.     

A New Cyclic Module for Hopf Algebras

We define a new cyclic module, dual to the Connes–Moscovici cocyclic module, for Hopf algebras, and give a characteristic map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.     

A Decomposition Theorem for CK1 of Central Simple Algebras

Let A be a central simple algebra over its center F. Define CK₁ A = Coker(K₁ F → K₁ A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK₁(A? _F B) = CK₁ A × CK₁ B.     

A Trace-Like Invariant for Representations of Hopf Algebras

We introduce an invariant for the irreducible representations of finite dimensional Hopf algebras, defined as the trace of a map induced by the antipode on the endomorphisms of each corresponding simple module. We also compute the value of this invariant for the representations of two families of non-semisimple Hopf algebras.     

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.     

Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory

We study modules over the ring of complex generalized numbers from a topological point of view, introducing the notions of -linear topology and locally convex -linear topology. In this context particular attention is given to completeness, continuity of -linear maps and elements of duality theory for topological -modules. As main examples we consider various Colombeau algebras of generalized functions.Mathematics Subject Classifications (2000) 46F30, 13J99, 46A20.Claudia Garetto: Current address: Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, 6020 Insbruck, Austria. e-mail: claudia@mat1.uibk.ac.at     

四元数分析中无界域上正则函数的带位移非线性边值问题

讨论了四元数分析中无界域上正则函数的一类带位移非线性边值问题.利用积分方程方法和Schauder不动点理论证明了问题解的存在性,并给出了解的积分表达式.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

Higher Algebraic <Emphasis Type="BoldItalic">K</Emphasis>-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F-algebra Σ, α an R-automorphism of Λ. Denote the extension of α to Σ also by α. Let Λ _α [T] (resp. Σ _α [T] be the α-twisted Laurent series ring over Λ (resp. Σ). In this paper we prove that (i) There exist isomorphisms ) for all n ≥ 1. (ii) is an l-complete profinite Abelian group for all n≥2. (iii)for all n≥2. (iv)is injective with uniquely l-divisible cokernel (for all n≥2). (v) K _–1(Λ), K _–1(Λ _α [T]) are finitely generated Abelian groups. Presented by Alain Verschoren.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝ^d be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫_Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.