A Representation of the Generators of the Quotients Group of By Matrices with Special Properties

Nader Taffach¹

¹Faculty of Science, Department of Mathematics, Idlib University, Syria.

Abstract

The problem of the existence and construction of a resolution of singularities is one of the central questions of algebraic geometry. In this paper, we study this problem in connecting with the quotients for . It is known that the action of on its Lie algebra is corresponding to the action of on . As a result of this action, it will be an invariant ring, which determines the quotients for . This paper is devoted to studying the singularity of these quotients. We write this singularity as a matrix with interesting features such as, for example, its quadratic is a zero matrix and its rank is less than or equal to 1. Therefore, in this paper, we reduce the studying of the singularity of the quotients of , which is a hard problem, to the studying of a matrix of invariants which is an easy problem.

Keywords: Singularities; Lie algebra; Lie groups; symplectic doubling; quotients