The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven't seen before. Let me outline the construction.
Let $A$ be a symmetrizable generalized Cartan matrix. Define $\frak{g}$ by the usual, Serre-type relations. Denote by $\frak{g}'$ the derived algebra of $\frak{g}$. Let $F$ denote a field of characteristic 0.
The following is a quote from the above-mentioned paper, section 1C.
Let $G^*$ be the free product of the additive groups $\frak{g}_\alpha$, $\alpha \in \Delta^{\mathrm{re}}$ [the real roots], with canonical inclusions $i_\alpha: \frak{g}_\alpha$$ \to G^*$. For any integrable $\frak{g}'$-module $(V, \pi)$, define a homomorphism $\pi^*: G^* \to \operatorname{Aut}_F(V)$ by $\pi^*(i_\alpha(e)) = \operatorname{exp} \pi(e)$. Let $N^*$ be the intersection of all $\operatorname{Ker}\pi^*$, put $G = G^* / N^*$[…].
There are two examples given below the definition:
a) Let $A$ be the Cartan matrix of a split simple finite-dimensional Lie algebra $\frak{g}$ over $F$. Then the group $G$ associated to $\frak{g} := \frak{g}'$$(A)$ is the group $\underline{G}(F)$ of $F$-valued points of the connected simple-connected algebraic group $\underline{G}$ associated to $\frak{g}$ […]
b) Let $\frak{g}$ be as in a), and let $\tilde{A}$ denote the extended Cartan matrix of $\frak{g}$. Then the group $G$ associated to $\frak{g}'$$(\tilde{A})$ is a central extension by $F^*$ of $\underline{G}(F[z, z^{-1}])$ […]
I have managed to verify a) over the complex numbers for certain classical groups using basic Lie theory, though ideally I would like to see a proof using purely algebraic methods. I have no clue how to tackle the second one. I am thankful for references hinting to any of the following:
- Example calculations using this definition
- Other papers which make use of this definition (I am aware of the predecessor paper "Infinite flag varieties and conjugacy theorems" by the same authors.)
as well as for any advice on how to retrace the examples, especially in cases where one cannot read "integrable" as "comes from a representation of $G$".
Best Answer
Note first that the derived Kac-Moody algebra $\mathfrak g'$ is the Kac-Moody algebra $\mathfrak g_{\mathcal D}$ associated to the Kac-Moody root datum $\mathcal D$ of simply connected type (see [1, Example 7.11]), in the sense of [1, Definition 7.13]. Let $\mathfrak G_{\mathcal D}$ denote the constructive Tits functor of type $\mathcal D$ (see [1, Definition 7.47]), restricted to the category of fields of characteristic zero.
Let $F$ be a field of characteristic zero, and let $G$ be the group constructed by Kac-Peterson over $F$. In other words, if $\mathrm{exp}_V: G^*\to \mathrm{GL}(V)$ is the representation of $G^*$ over the integrable $\mathfrak g_{\mathcal D}$-module $V$, and if $\exp=\bigoplus_V\exp_V$ is the direct sum of all these representations of $G^*$, then $G=\exp(G^*)$.
The group $\mathfrak G_{\mathcal D}(F)$ also acts on the direct sum of all the integrable $\mathfrak g_{\mathcal D}$-modules (with similar formulas), and this action factors through $G$ (see [1, Section 7.4.3]). In other words, there is a surjective morphism $\pi:\mathfrak G_{\mathcal D}(F)\to G$. Its kernel lies in the standard torus of $\mathfrak G_{\mathcal D}(F)$ by the Recognition Theorem [1, Theorem 7.71], and since it is easy to check that $\pi$ is injective on this torus (see e.g. [1, Definition 7.31]), $\pi$ is actually an isomorphism. In other words, $G$ is just $\mathfrak G_{\mathcal D}(F)$.
The fact that $\mathfrak G_{\mathcal D}(F)$ satisfies the properties (a) and (b) in the question is then well-known: see for instance [1, Exercice 7.50] for (a) and [1, Section 7.6] for (b).
Note that, modulo some technical details that need some ironing out (whence the reference to the book 1 rather than Tits' papers), everything above follows from the work of Jacques Tits (see in particular 2).
1 T. Marquis, An introduction to Kac-Moody groups over fields, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2018
2 J. Tits, Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra 105 (1987), no. 2, 542–573.