Two Dimensional Nilpotent Euclidean Nearrings

We find all multiplications on the two dimensional Euclidean group (IR²,+) such that (IR², +, ·) is a nilpotent topological nearring of rank three. We determine the left, right, and two sided ideals of these nearrings and we determine when two of these nearrings are isomorphic. As a consequence, we are able to conclude that there are infinitely many isomorphism classes of such nearrings. This is in considerable contrast to the case for rings. There are only eight isomorphism classes of topological rings with additive group (IR²,+).AMS Subject Classification (2000) 16Y30     

The numerical range of a nilpotent operator on a Hilbert space

We prove that the numerical range of an arbitrary nilpotent operator on a complex Hilbert space is a circle (open or closed) with center at and radius not exceeding where is the power of nilpotency of

     

Subcubic Growth of the Averaged Dehn Function for a Class 2 Nilpotent Group

We show that the averaged Dehn function with respect to each finite presentation of an arbitrary finitely generated class 2 nilpotent group is subcubic. For the finite rank 2 free class 2 nilpotent group this implies the subasymptoticity of the averaged Dehn function in the sense of M. Gromov, confirming his conjecture.Original Russian Text Copyright © 2005 Romankov V. A.The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00489) and the Scientific Program Universities of Russia of the Ministry for Education of the Russian Federation (Grant 362-05).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 663–672, May–June, 2005.     

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ?^? where G is a compact, simple group and ? = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if ? is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKähler metric with G-invariant Kähler potential.     

A Generalization of <Emphasis Type="Italic">T</Emphasis>-Groups

This paper identifies a certain class of locally supersoluble groups (called soluble hall-T groups) which contains the soluble T-groups as well as the nilpotent groups. The main result states that the product of a normal soluble hall-T subgroup and a subnormal locally supersoluble subgroup is always locally supersoluble.AMS Subject Classification (1991): 20E25, 20F16, 20F19     

On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold where is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and a discrete subgroup of acting on by left translations. For this purpose, we shall first show that if is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric then the restriction of to the center of is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group . If is one such group, we prove that no timelike geodesic in can be translated by an element of By the way, we rediscover that the Heisenberg Lie group admits a flat left-invariant Lorentz metric if and only if

     

Solving systems of linear equations over lie nilpotent rings

We show how the so called right and left determinants can be used in the solution of certain systems of linear equations over Lie niipotent rings, A close analogue of Cramer's rule is formulated for right linear equations over the Grassmann algebra.     

The domains of partial attraction of probabilities on groups and on vector spaces

It is shown that for exponential Lie groupsG the limit behavior of i.i.d. triangular arrays on the groupG and on the tangent spaceG coincide. This result is used to obtain a characterization of domains of partial attraction (resp. semistable attraction) on exponential (resp. simply connected nilpotent) Lie groups via the corresponding domains on the tangent space.     

Groups with a Polynomial Dimension Growth

We show that finitely generated groups with a polynomial dimension growth have Yu’s property A and give an example of such groups.