Properties And Homomorphisms Of Some Necklace Lie Algebras

Recently,Bodklant,Le Bruyn([8]) and Ginzburg([9, 10]) introduce necklace Lie algebra (?),which is an infinite dimensional lie algebra defined on a quiver. Necklace Lie algebra plays an important role in the noncommunication geometry, quantum groups and other fields.Necklace Lie algebra attract the attention of Mathematicians and several research on its structure arrears recently([22, 23, 24, 25]).Mei gets some results of necklace Lie algebra,she defined the left and right index arrays of a necklace word.Using them,she divided the necklace words into 5 classes and deserves the corresponding subalgebras.She also studied an interesting anti-automorphism of order 2([24]).Yu proved there are some finite-dimensional simple Lie subalgebras in necklace Lie algebra which are isomorphic to sl(n)([25]).He also studies the properties of isomorphism of necklace Lie algebras.In this thesis,we study properties and homomorphisms of the necklace Lie algebras of some special quiver. In charter 1,we review the definitions of quiver and necklace Lie algebra,the left and right index arrays of a necklace word ,and list some basic properties of necklace lie algebra. In section 2.1,we prove that there is no necklace word of type D and E in our case, we prove the. linear subspace N_G and N_M which base on set of type A and right index array has only one element:G = {w|w∈N(?), L_w~0 = kI∪(i_r), R_w~0= (i_r), i_r∈(1, 2,…, n), k∈Z~+} and type A and right index array is empty:M = {w|w∈N(?), L_w~0 = kI, R_w~0= (?), k∈Z~+} are subalgebra of N(?). In section 2.2,using N_G and N_M,we prove that necklace lie algebra is not solvable algebra,nilpotent algebra and semisimple algebra. In section 2.3,we prove the linear K-subspace P spanned by the set NW(?)\H is an idea of N(?) and the quotient algebra N(?)/P is a commutative algebra,so N(?)/P is a nilpotent algebra. In section 3.1,we construct some linear maps of N(?) and give the necessary and sufficient conditions when these linear maps are homomorphisms or anti-homomorphisms. In section 3.2 we study a special subgraph.we prove there is only class A. in necklace word in this case.finally,we construct an exact sequence from idea P and H and prove N(?) is a decomposable algebra.