Strictly imaginary roots of Kac-Moody algebras
The theory of Kac-Moody algebras has been generalized to a wider class of Lie algebras called affine Kac-Moody algebras. A Kac-Moody algebra \({\mathfrak g}\) has the root space decomposition \({\mathfrak g}= \bigoplus_{\alpha\in {\mathfrak h}^*} {\mathfrak g}_ \alpha\), where \({\mathfrak h}\) is the Cartan subalgebra of \({\mathfrak g}\) and \({\mathfrak g}_ \alpha= \{x\in {\mathfrak g}\mid [h,x]= \alpha(h)x\) for all \(h\in {\mathfrak h}\}\). We call \(\alpha\in {\mathfrak h}^*\) a root if \(\alpha\neq 0\) and \({\mathfrak g}_ \alpha\neq 0\). For the finite dimensional simple Lie algebras, every root is real. That is, every root is conjugate to a certain simple root under the Weyl group action. Contrary to the finite dimensional case, infinite dimensional Kac-Moody algebras have imaginary roots that are not Weyl group conjugate to simple roots. Moreover, their multiplicities may be \(>1\). In [\textit{V. G. Kac}, Adv. Math. 30, 85-136 (1978; Zbl 0391.17010)], the notion of strictly imaginary roots was introduced. A root \(\gamma\) is called strictly imaginary if for every real root \(\alpha\) either \(\gamma+ \alpha\) or \(\gamma- \alpha\) is a root. In this paper, the author provides a simple test to determine whether a given imaginary root is strictly imaginary or not. Moreover, he gives a characterization of those Kac-Moody algebras for which all the imaginary roots are strictly imaginary. A complete classification of the strictly imaginary roots of \(2\times 2\) Kac-Moody algebras is also given.
scientific article; zbMATH DE number 645072