The ranks of homotopy groups of Kac-Moody groups

TL;DR

This paper provides a combinatorial method to compute the ranks of homotopy groups of Kac-Moody groups by relating them to the dimensions of graded components of associated Lie algebras, based on Poincaré series.

Contribution

It introduces a new approach to determine the ranks of homotopy groups of Kac-Moody groups using Lie algebra gradings and Poincaré series, extending understanding of their topological structure.

Findings

Constructed Lie algebras from Poincaré series of flag manifolds.

Interpreted even homotopy group ranks as dimensions of graded Lie algebra components.

Provided a combinatorial interpretation for all homotopy group ranks.

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 $\pi_k(G(A))$. For a large class of Kac-Moody groups, we construct Lie algebras with grade from the Poincar\'{e} 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 $k>0$.