Relationship between Reflections Determined by Imaginary Roots and the Weyl Group for a Special GKM Algebra

Abstract:

      It's well known that a reflectin r_α associated to every root α belongs to the Weyi group of a Lie algebra g(A) of finite type. When g(A) is a symmetrizable Kac-Moody algebra of indefinite type, one of can define a reflection r_α for every imzginary root α satisfying (α, α) < 0. From [3] we know r_α ∈-W or r_α is an element of-W mutiplied by a diagram automorphism . How about the relationship between reflections associated to imaginary root and the Weyl group of a symmetrized Kac-Moody algebra (GKM algebra for short)? We shall discuss it for a special GKM algebra in present paper (see 3). In sections 1 and 2 we introduce some basic concepts and give the set of imaginary roots of a class of rank 3 GKM algebras.