Infinitesimal extensions of ℙ1 and their Hilbert schemes

 In order to calculate the multiplicity of an isolated rational curve C on a local complete intersection variety X, i.e. the length of the local ring of the Hilbert Scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of ℙ¹ by locally free sheaves. In this paper we study infinitesimal extensions of ℙ¹, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [ℙ¹]. Received: 11 June 2001 / Revised version: 28 January 2002     

Odd central extensions of Lie superalgebras

Institute of Mathematics, National Academy of Sciences of the Republic of Kazakhstan. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 69–71, July–September, 1995.     

First order deformations of schemes with normal crossing singularities

We study the sheaf T ¹(X) of first order deformations of a reduced scheme with normal crossing singularities. In particular, we obtain a formula for T ¹(X) in a suitable log resolution of X.     

Central extensions and deformations of Hom-Lie triple systems

In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems and prove that it is governed by the cohomology group.     

Selfadjoint extensions of Jacobi matrices of limit-circle type

A boundary condition at ∞ is constructed which characterizes all selfadjoint extensions of the minimal closed operator associated with a Jacobi matrix of limit-circle type.     

Renormalization and central extensions

The stochastic limit of quantum theory [1] motivated a new approach to the renormalization program. Subsequent investigations brought to light unexpected connections with conformal field theory and some subtle relationships between renormalization and central extensions. In the present paper we review the path that has lead to these connections at the light of some recent results.     

On the Krein and Friedrichs extensions of a positive Jacobi operator

We show that for a positive linear operator acting in ℓ² and defined from

a_nx_n+1+b_nx_n+a_n-1x_n-1