The ranks of homotopy groups of Kac-Moody groups

2015/01/19 by Xuan Zhao, Zhao Xu-an, Xu-an, Zhao +2
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #math.AG #math.AT

paper · pdf · doi:10.48550/arxiv.1501.04380

arxiv created 2015/01/19 · openalex publication_date 2015/01/19 · arxiv updated 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a Cartan matrix and G(A) be the Kac-Moody group associated to Cartan matrix A. In this paper, we discuss the computation of the rank i_k of homotopy group π_k(G(A)). For a large class of Kac-Moody groups, we construct Lie algebras with grade from the Poincaré series of their flag manifolds. And we interpret i_2k as the dimension of the degree 2k homogeneous component of the Lie algebra we constructed. Since the computation of i_2k-1 is trivial, this gives a combinatorics interpretation of i_k for all k>0.

Citations

• Poincaré series and rational cohomology rings of Kac-Moody groups and their flag manifolds

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