The Structure And Properties Of Necklace Lie Algebra

Recently.Le Bruyn and Ginzburg have introuced necklace Lie algebra,which is defined on quiver and of infinite dimensional. The necklace Lie algebra plays an inportant role in the; noncommunication geometry . There is few results on the structure of necklace Lie algebra up to now.We study the structure and properties of a necklace Lie algebra.In section 1.1. we define the mapping a on circle in double quiver Q of a quiver Q, and give a strict defition of bracket operation in Lie algebra,then prove it is a Lie bracket.In section 1.2,we introduce the left and right index arrays of an necklace word. Using them, we divided the necklace words into 5 classes,namely:A,B,C,D and E. We also discuss a natural involution of necklace Lie algebra .In section 1.3, taking the advantage of the classification by the index arrays ,pointed out the space spanned by class A,B,C respect!vivety are subalgebras of NQ. We also show that the subspace spanned by the elements of length 2 in C is an abelian subalgebras of NQ,denoted by H.we find H is contained in the center of subalgebra P of the subspace spanned by the elements in class C. In section 2.1, we give a general form of elements in Z(NQ), ,provided that For quiver which has only one circle or a circle with some single arrows,we also decisive some of it's elements in Z(NQ). In section 2.2,we prove that if the set of arrows in a quiver is not empty,then NQ is not semisimple.We construct the nonsovable subalgebra of NQ when there is no less than one circle whose length is bigger than one ,and there is no loop in quiver Q,thus leads to a conclusion that NQ is nonsovable .Using the index arrays to discuss the operation of necklace word .construct a sovable subalgebra of NQ,which is not nilpotent.This shows that NQ is not nilpotent provided that there is a cyclic with length bigger than 1. In section2.3,we give; the decomposition of NQ, for quiver without loop.