Admissible positive systems of affine Kac-Moody Lie algebras: the twisted cases (Q330171)

The paper deals with admissible positive systems and Hermitian real forms of affine twisted Kac-Moody Lie algebras. These notions have been already dealt with by the authors for non-twisted types in [J. Algebra 453, 561--577 (2016; Zbl 1397.17010)].NEWLINENEWLINE Let \({\mathfrak g}\) be a complex affine Kac-Moody Lie algebra. Let \({\mathfrak g}_{\mathbb{R}}\) be a real form of \({\mathfrak g}\) with Cartan involution \(\theta\) and let \({\mathfrak g}_{\mathbb{R}}={\mathfrak k}_{\mathbb{R}}+{\mathfrak p}_{\mathbb{R}}\) be its Cartan decomposition. Drop the subscript \(\mathbb{R}\) for complexification, so for example \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\). Let \({\mathfrak h}_{\mathbb{R}}\) be a \(\theta\)-stable Cartan subalgebra of \({\mathfrak g}_{\mathbb{R}}\), with root space decomposition \({\mathfrak g}={\mathfrak h}+\sum_\Delta{\mathfrak g}_\alpha\). Let \(\dot{\mathfrak g}\subseteq{\mathfrak g}\) be the underlying finite-dimensional simple Lie algebra, and let \(\dot\Delta\subseteq\Delta\) be its root system. Let \(\delta\) be a generator of the imaginary roots.NEWLINENEWLINE Suppose that \(\theta\) acts as the identity map on \({\mathfrak h}\). Then \(\theta\) acts as 1 or \(-1\) on each root space. We let \(\theta_\alpha\in\{\pm 1\}\) denote its eigenvalue. Also, \(\theta\) is an extension of an involution \(\dot\theta\) on \(\dot{\mathfrak g}\). Then by definition, an admissible positive system for \(\theta\) (or \({\mathfrak g}_{\mathbb{R}}\)) is a positive system \(\Delta^+\) of a \(\theta\)-stable Cartan subalgebra such that NEWLINE\[NEWLINE{\mathfrak p}={\mathfrak p}^+{\mathfrak p}^-,\qquad [{\mathfrak k},{\mathfrak p}^{\pm}]\subseteq{\mathfrak p}^{\pm},\qquad [{\mathfrak p}^{\pm},{\mathfrak p}^{\pm}]= 0,NEWLINE\]NEWLINE where \({\mathfrak p}^{\pm}= {\mathfrak p}\cap\sum_{\alpha\in\Delta^{\pm}}{\mathfrak g}_\alpha\). Also a real form of \({\mathfrak g}\) is said to be Hermitian if \({\mathfrak g}\) has a Cartan subalgebra \({\mathfrak h}\) such that \(\theta_{|\mathfrak h}= 1\), \(\theta_{\delta}=1\), and \(\dot\theta\) is Hermitian.NEWLINENEWLINE The authors show that a real form has admissible positive system if and only if it is Hermitian. They use the Vogan diagrams to classify the Hermitian real forms, and show that their symmetric spaces carry complex structures.