Actions of Boolean rings on sets

LetB be a Boolean ring (with 1),S a sheaf of sets on the Stone space Spec(B), andS the set of global sections of S. For everya B ands, t S, leta(s, t) denote the element ofS which agrees withs on the support ofa, and witht elsewhere.We set down identities satisfied by this ternary operationB×S×SS (involving also the Boolean operations ofB). For a fixed Boolean ringB, we call a setS given with a ternary operation satisfying these identities aBset. The above construction is shown to give a functorial equivalence between sheaves of setsS on Spec(B) with nonempty sets of global sections, and nonemptyB-setsS. For any setA, the bounded Boolean powerA[B]^* is the freeB-set onA. The varieties ofB-sets, asB ranges over all Boolean rings, constitute (together with one trivial variety) the least nontrivial hypervariety of algebras, in the sense of W. Taylor.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by R. S. Pierce.     

Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 –245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3–ary bracket. Further, we derive the so-called ρ–compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.     

Grobner-Shirshov Basis for Degenerate Ringel-Hall Algebras of Type F4

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr\"{o}bner-Shirshov basis for degenerate Ringel-Hall algebras of type $F_4.$     

Strong skew commutativity preserving maps on rings with involution

Let R be a unital *-ring with the unit I.Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I-P) = 0 implies A = 0.In this paper,it is shown that a surjective map Φ:R→R is strong skew commutativity preserving(that is,satisfiesΦ(A)Φ(B)-Φ(B)Φ(A)~w= AB-BA~w for all A,B∈R) if and only if there exist a map f:R→Z_s(R)and an element Z∈Z_s(R) with Z~2=I such that Φ(A)=ZA +f(A) for all A∈R,where Z_s(R) is the symmetric center of R.As applications,the strong skew commutativity preserving maps on unital prime *-rings and von Neumann algebras with no central summands of type I_1 are characterized.     

An equivariant higher index theory and nonpositively curved manifolds

In this paper, we define an equivariant higher index map from to K_∗(C^∗Γ(X)) if a torsion-free discrete group Γ acts on a manifold X properly, where C^∗Γ(X) is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum-Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1-20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C^∗-algebras, in: C^∗-Algebras: 1943-1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291]. When Γ is trivial, this equivariant higher index map is the coarse Baum-Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective.     

Metric geometry of partial isometries in a finite von Neumann algebra

We study the geometry of the set

     

General form of the deformation of the Poisson superbracket on a (2, <Emphasis Type="Italic">n</Emphasis>)-dimensional superspace

Up to an equivalence transformation, we describe continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on ℝ² taking values in a Grassmann algebra . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 265–286, May, 2008.     

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

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.