On the Topology of Kac–Moody groups

Abstract

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C$, let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$. In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F$ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$. We illustrate our results with the example of rank two Kac–Moody groups.

Appendix

In this section we establish some basic facts about the topology of the Kac–Moody groups $\text {K}$, and their classifying spaces $\text {BK}$.

Recall that a subset $J \subseteq I$ is called spherical if the subgroup $\text {K}_J \subseteq \text {K}$ is a compact Lie group. The poset of spherical subsets of $I$ is denoted by $\mathcal S (A)$. In [9] (Theorem A) it is shown that as an abstract group, $\text {K}$ is an amalgamated product of subgroups of the form $\text {K}_J$, where $J \in \mathcal S (A)$ has cardinality at most two. In other words, the following canonical map is an isomorphism:

$$\begin{aligned} \text {colim}_{J \in \mathcal S (A), \, |J| \le 2} \text {K}_J \longrightarrow \text {K}, \end{aligned}$$

where the colimit is taken in the category of groups. Now given $J \in \mathcal S (A)$ it is easy to see that $\text {K}_J$ is generated by the groups $\text {K}_j$ for $j \in J$. Hence, the map above factors through a sequence of two surjective maps:

$$\begin{aligned} \text {colim}_{J \in \mathcal S (A), \, |J| \le 2} \text {K}_J \longrightarrow \text {colim}_{J \in \mathcal S (A)} \text {K}_J \stackrel{\varphi }{\longrightarrow }\text {K}. \end{aligned}$$

As a consequence we see that the map $\varphi$ above is an isomorphism of (abstract) groups.

The next step is to understand the topology on $\text {K}$. Let us begin by recalling some constructions from Sect. 8. The reader is referred to [8, 15] for details regarding the Bruhat decomposition of $\text {K}$ that is used in the arguments that follow. Let $\mathrm{N(T)}\subset \text {K}$ denote the normalizer of ${{\mathrm{T}}}$. Given $w \in \text {W}$, let $\tilde{w} \in \mathrm{N(T)}$ denote any lift of $w$ in $\mathrm{N(T)}$. We will denote the space ${{\mathrm{B}}}\tilde{w} {{\mathrm{B}}}\cap \text {K}$ by $\tilde{\text {Y}}_w$. This is a well defined subspace of $\text {K}$ homeomorphic, as a right ${{\mathrm{T}}}$-space, to $\mathbb C ^{l(w)} \times {{\mathrm{T}}}$. Now for a generating reflection $r_i$, let $\text {Y}_i \subset \tilde{\text {Y}}_{r_i}$ be the subspace $\mathbb C \times \{ 1 \} \subset \mathbb C \times {{\mathrm{T}}}$ under the above identification. Then the group product in $\text {K}$ induces a homeomorphism:

$$\begin{aligned} \tilde{\text {Y}}_w = \text {Y}_{i_1} \times \cdots \times \text {Y}_{i_s} \times {{\mathrm{T}}}, \end{aligned}$$

where $w = r_{i_1} \ldots r_{i_s}$ is a reduced expression. Furthermore, the closure of $\tilde{\text {Y}}_w$ is given by:

$$\begin{aligned} \text {Z}_w := \bigcup _{v \le w} \tilde{\text {Y}}_v. \end{aligned}$$

With this structure, $\text {K}$ becomes a ${{\mathrm{T}}}$ -CW complex, constructed by successively attaching ${{\mathrm{T}}}$-cells. The topology is generated by the closed subspaces $\text {Z}_w$. Hence a subspace $\text {Z}\subseteq \text {K}$ is closed if and only if $\text {Z}\cap \text {Z}_w$ is closed for all $w \in \text {W}$. Now given $J \in \mathcal S (A)$, let $w_0 \in \text {W}_J$ denote the longest element. It follows from the closure relation that $\text {Z}_{w_0} = \text {K}_J$ as compact subspaces of $\text {K}$.

Assume now that $H$ is any topological group and that we are given a homomorphism $\phi : \text {K}\rightarrow H$ that restricts to a continuous map on each $\text {K}_J$ for $J \in \mathcal S (A)$. Given an element $w \in \text {W}$, let $w = r_{i_1} \ldots r_{i_s}$ be a reduced expression. Notice that $\phi$ extends to a canonical continuous map $\tilde{\phi }$ from the product $\text {K}_{i_1} \times \cdots \times \text {K}_{i_s}$ to $H$ given by the product of the individual restriction maps. Moreover, $\tilde{\phi }$ factors through the projection map from $\text {K}_{i_1} \times \cdots \times \text {K}_{i_s}$ onto the subspace $\text {Z}_w$. It follows that $\phi$ restricts to a continuous map on $\text {Z}_w$. By the definition of the topology on $\text {K}$, we see and that $\phi$ is in fact a continuous homomorphism. The upshot of the argument given above is that $\text {K}$ is in fact the colimit of the groups $\text {K}_J$ indexed over the poset $\mathcal S (A)$ in the category of Topological Groups. We conclude:

Theorem 12.1

The topological group $\text {K}$ has the following properties:

• (a) $\text {K}$ is a free ${{\mathrm{T}}}$-CW complex of finite type under the right action of ${{\mathrm{T}}}$. This structure is compatible with the CW structure on the homogeneous space $\text {K}/{{\mathrm{T}}}$.

• (b) $\text {K}$ is equivalent to the colimit, in the category of topological groups, of the compact Lie groups $\text {K}_J$ indexed over the poset $\mathcal S (A)$.

Remark 12.2

Since $\text {K}$ is a ${{\mathrm{T}}}$-CW complex, it is built by successively attaching ${{\mathrm{T}}}$-cells. Decomposing ${{\mathrm{T}}}$ as a CW-complex, we see that $\text {K}$ may be constructed by successively attaching (standard) cells. However, it fails to be a CW complex by virtue of the fact that the boundary of cells being attached may glue to cells of higher dimension. We will call a space built by attaching cells in a possibly non-sequential order a Cell Complex (there is some conflict in the literature on the terminology for such an object). Working inductively with the stages, we see that a cell complex is homotopy equivalent to a CW complex.

Remark 12.3

Principal $\text {K}$-bundles that appear in physical applications tend to be defined over base spaces (like solutions of differential equations) that may not have an obvious homotopy type of a CW complex. It is therefore desirable to study the homotopy type of the space ${\mathcal{B }\text {K}}$ that classifies numerable $\text {K}$-bundles. This is Milnor’s model [17] of the classifying space of $\text {K}$, which is defined as a colimit of certain spaces $\mathcal{B }_n\text {K}$ under inclusions $\mathcal{B }_n\text {K}\subset \mathcal{B }_{n+1}\text {K}$. The space $\mathcal{B }_n\text {K}$ is defined as the quotient $\mathcal{E }_n\text {K}/\text {K}$, where $\mathcal{E }_n\text {K}$ is the $n$-fold join of $\text {K}$ with itself given the quotient (weak) topology, and the diagonal action of $\text {K}$. Hence $\mathcal{E }_n\text {K}$ can be seen as a quotient of $\Delta ^{n-1} \times \text {K}^{\times n}$. Using the fact that $\text {K}$ is a cell complex of finite type, we see that $\mathcal{B }_n\text {K}$ has the structure of a cell complex of finite type. It is clear that the inclusions $\mathcal{B }_n\text {K}\subset \mathcal{B }_{n+1}\text {K}$ are cellular, and therefore, by the previous remark, $\mathcal{B }\text {K}$ has the homotopy type of a CW complex.