Coset relation algebras

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to “shift” the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In later articles, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra, and the class of group relation algebras is similarly adequate to the task of representing all measurable relation algebras in which the associated groups are finite and cyclic.     

von Neumann代数中套子代数上的Lie同构

本文给出因子von Neumann代数中的幂等算子在广义Lie积下的一个刻画; 得到因子von Neumann代数中套子代数的幂等算子在Lie积下的一个特征．作为应用, 研究了因子von Neumann代数中套子代数上的Lie同构,并证明因子von Neumann 代数中套子代数之间的Lie同构,要么是同构与广义迹之和,要么是负反同构与广义迹之和．     

On the classification problem of the quasi-isomorphism classes of 1-connected minimal free cochain algebras

The purpose of this paper is to classify the quasi-isomorphism classes of 1-connected minimal free cochain algebras over a commutative ring. Our tool to address this problem is a “certain” long sequence, called the Whitehead exact sequence, which we construct for every such algebra. We introduce the notion of coherent isomorphisms between these exact sequences and we show that two 1-connected minimal free cochain algebras are quasi-isomorphic if and only if their respective Whitehead exact sequences are coherently isomorphic.     

Lie isomorphisms of reflexive algebras

A Lie isomorphism ? between algebras is called trivial if ?=ψ+τ, where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.     

量子代数模的张量积与不可分解模

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.     

Virasoro李代数的子代数间的同构及生成元

证明了无中心Virasoro李代数的有限维子代数同构的充分必要条件,证明了两个元素di,dj作为生成元的充分必要条件,找出了几组互相同构的无限维真子代数,研究了他们的极大性,单性以及其它性质.     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

Classification of solvable Leibniz algebras with null-filiform nilradical

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

量子环面上一类导子李代数的结构和自同构群

本文研究量子环面上的一类导子李代数,它包含了Virasoro-Like代数及其q类似．首先证明了这 类导子李代数之间的同构一定是分次同构,并进一步给出了代数同构的充要条件及同构映射的具体表达 式,最后确定了该类李代数的自同构群．     

李三系与Laurent多项式代数F[t,t~(-1)]

本文通过讨论Laurent多项式代数及其导子代数的对合自同构确定了一类具体的无限维单李三系, 并且提供了一种利用Novikov代数上自然的李代数结构来构造李三系的方法.     

一类无限维李代数的同构与单性

探讨了一类的无限维李代数,并构造了此类李代数的理想,同构,同态,并对其性质作了探讨.     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

Map composition generalized to coherent collections of maps

Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set^?, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements.