Some forms of Kac-Moody algebras (Q1188142)

The author considers a symmetrizable Kac-Moody algebra of rank \(\ell\) over an algebraically closed field \(F\) and gives a way of constructing \(K\)-forms, where \(\overline{K}=F\), by glueing together \(\ell\) copies of quaternion algebras over \(K\). In the case when not all of the copies are isomorphic, he needs to add some compatibility conditions between the copies attached to neighboring vertices. This construction does not exhaust all the possible \(K\)-forms of a Kac-Moody algebra, and an indication is given on how to overcome this problem by considering quaternion algebras over finite extensions of \(K\). For the case \(K=\mathbb{R}\), it is shown how this approach allows the construction of a symmetric invariant bilinear form. Some restrictions to the Kac-Moody algebra were removed in the author's paper [Tôhoku Math. J., II. Ser. 43, 425-445 (1991; Zbl 0723.17022)].