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.     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

Querying Historical Data Over Multiple Time-Lines

In a temporal database, time-varying relations may be defined over multiple time-lines with varying rates of sampling and/or progress, for instance, multiple time series. There is a need to represent such data in a temporal model and provide query languages. In this paper, we propose a clocked temporal relational algebra, called , which supports temporal relations based on multiple time-lines. In the underlying model, temporal relations (historical data) are defined over clocks which are subsequences of an assumed global time-line. The algebra is uniform, it is a consistent extension of the relational algebra, and it includes a number of temporal operators to combine data based on different time-lines, as well as extensions of the operators of the relational algebra. The semantics of an operation of depends on the clocks of the relations involved in the operation as well as the relations. We outline a formal interpretation of expressions of , which also represent clocked relations, and sketch a na?ve expression evaluation method.      

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.