Root Systems Of A Family Of Indefinite Kac-Moody Algebras

In 1960's,the theory of Kac-Moody algebra was built by V.G.Kac and R.V.Moody.Root systems of Kac-Moody algebra are studied in depth and scope.In this paper,we discuss a family of indefinite root systems of Kac-Moody algebra whose Cartan martices(A_n) are generalization of the Cartan matrices of the hyperbolic Kac-Moody algebra H₅⁶.First,the root system of the hyperbolic Kac-Moody algebra H₅⁶, which is the simplest case,is studied.The explicit description of the set of all real roots is obtained by action of Weyl Gruop on the set of all real roots of Kac-Moody algebra D₄¹.At the same time,we describe the imaginary roots of H₅⁶,including detailed description of K and the multiplicities of some low level roots.Second,the description of the set of all real roots of g(A₁) is obtained.Third,the way of describing the set of all real roots of g(A₁) is applied to the case of g(A_r),r = 2,3,4.