Characteristic Polynomials Of Finitely Generated Groups And Finite Dimensional Lie Algebras

The notion of projective spectrum was defmed by Professor Rongwei Yang.For a tuple A=(A1,…,An)of elements in a unit algebra B,its projective spectrum is defined as P(A)={z=(z1,…,zn)?Cn:A(z)=z1A1+…+znAn is not invertible}.Inspired by this,Professor Zhiguang Hu et al.studied the characteristic poly-nomials of finitely generated groups and finite dimensional Lie algebras,ob-tained a series of profound results.The characteristic polynomial of Lie algebra L with respect to its basis S=(e1,…,es)and its adjoint representation is defined as pL(z)=det(z0I+z1adLe1 +…zsadLes).This paper continues to study the characteristic polynomials of Lie algebras with respect to its adjoint representation.Firstly,this paper proves that if the dimension of the center of Lie algebra L be k,then PL(z)must have a factor z0k+1.Next,proves that when L is a semisimple Lie algebra,if L can be decomposed into the direct sum of k simple Lie algebras,then pL(z)must have a factor Finally,this paper shows that for a semisimple Lie algebra L,the rank of the spectral matrix of its Borel subalgebra is equal to the dimension of t he Cart an subalgebra of L.