Free Kac-Moody groups and their Lie algebras
Let \(I\) be a nonempty set. Suppose that \(K\) is a commutative ring, and for each \(i\in I\) consider a copy \(E_i(K)=\{E_i(\lambda)|\lambda\in K\}\) of the additive group of \(K\). Thus, \(E_i(\lambda)E_i(\mu)=E_i(\lambda+\mu)\) for all \(\lambda,\mu\in K\). The authors consider the free product \({\mathcal F}(K)=*_{i\in I}E_i(K)\) and call this object the free Kac-Moody group. The goal is to attach a Lie algebra to this group as follows. Using the fact that elements of \({\mathcal F}={\mathcal F}(K)\) are ``locally parametrized'' by finitely many elements of \(K\) the authors naturally define the algebra of polynomial functions \(\text{ Pol}({\mathcal F})\) on the group \({\mathcal F}\). The product in this algebra is described in terms of a shuffle algebra, a version of a Magnus-type group is also used. The group \({\mathcal F}\) naturally acts on \(\text{ Pol}({\mathcal F})\) by translations. Next, \(L\) is the Lie algebra of derivations of \(\text{ Pol}({\mathcal F})\) which are right invariant under the group action. Finally, the authors describe the structure of \(L\) as a certain completion of a free Lie algebra.
scientific article; zbMATH DE number 1783444