Standard Envelopes of Commutative Triple Systems

Under certain constraints on the characteristic of a field , the commutative standard enveloping q-algebra >B of a commutative triple system A over is defined. It is proved that(1) if the algebra B is simple, then the system A is simple;(2) if the system A is simple, then B either is simple or decomposes into the direct sum of two isomorphic simple subalgebras (as of ideals).     

Structure of Degenerate Block Algebras

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map : A A F, we define a Lie algebra = (A, ) over F with basis {e_x | x A/{0}} and Lie product [e_x,e_y] = (x,y)e_x+y. We show that is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der of is a complete Lie algebra. We describe the double extension D( , T) of by T, where T is spanned by the locally finite derivations of , and determine the second cohomology group H²(D( , T),F) using anti-derivations related to the form on D( , T). Finally, we compute the second Leibniz cohomology groups HL²( , F) and HL²(D( , T), F).2000 Mathematics Subject Classification: 17B05, 17B30This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee.     

A note on the spectrum of an upper triangular operator matrix

Let be a upper triangular operator matrix acting on the Banach space . We investigate the set of the operators for which , where denotes the spectrum.

     

Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let and be two nest algebras on X. Suppose that φ is an additive bijective mapping from onto such that φ(A²)=φ(A)² for every . Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A)=TAT⁻¹ for every or φ(A)=TA∗T−1 for every .     

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if lies in the class and is a torus of positive dimension, then ``most' vector bundles over admit no complete nonnegatively curved metrics.

     

Linear maps preserving ideals of -algebras

We show that unital self-adjoint linear bijections of matrix algebras, type factors and abelian -algebras preserving maximal left ideals are isomorphisms and we show that a unital continuous linear map of a -algebra that maps the minimal left ideal into itself is the identity map.     

On an asymptotic behavior of elements of order in irreducible representations of the classical algebraic groups with large enough highest weights

The behavior of the images of a fixed element of order in irreducible representations of a classical algebraic group in characteristic with highest weights large enough with respect to and this element is investigated. More precisely, let be a classical algebraic group of rank over an algebraically closed field of characteristic 2$">. Assume that an element of order is conjugate to that of an algebraic group of the same type and rank naturally embedded into . Next, an integer function on the set of dominant weights of and a constant that depend only upon , and a polynomial of degree one are defined. It is proved that the image of in the irreducible representation of with highest weight contains more than Jordan blocks of size if and are not too small and .

     

On the ε-entropy of classes of holomorphic functions

Suppose thatB _R ^d is a ball of radiusR in ℂ ^d and σ is the standard measure on the unit sphere in ℂ ^d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH _R ⁰ (l, p, d) we denote the class of functionsf holomorphic inB _R ^d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH ^p (B _R ^d ). In this paper an asymptotic formula for the ε-entropy of the classH _R ⁰ (l, p, d) in the spacesL ^p (σ), 1≤p<∞, and is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 286–293, August, 2000.     

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ), (R, K, )) of congruence classes of extensions of (R, K, ) by (U, Q, ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.2000 Mathematics Subject Classification: 17B56, 17B99, 18G99     

The atomic and molecular reaction statics

This work presents a new science called atomic and molecular reaction statics (AMRS). There are four parts for AMRS, i.e. the group theoretical derivation of molecular electronic states, the principle of microscopic reversibility, the principle of microscopic transitivity and the optimum energy process rule. AMRS has been developed for about twenty years.