一类特殊广义Kac-Moody代数虚根决定的反射与Weyl群间的关系(英)

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

英文关键词:generalized Kac-Moody algebra  imaginary root system  the Weyl group  special imaginary root

中文摘要:

      对有限型李代数g（A），相应于每个根α的反射r_α均在g（A）的Weyl群W中．当g（A）为可对称化的不定型Kac－Moody代数时，若α为一虚根且（α，α）＜0，则亦可定义反射r_α并有r_α∈－W或r_α是－W中元与一个图自同构之积（见[3]）．本文给出了一类秩为3的广义Kac－Moody代数的虚根系，然后讨论了一类特殊的广义Kac－Moody代数的虚根决定的反射与Weyl群之间的关系．

英文摘要:

      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.