Finite order automorphisms on real simple Lie algebras (Q2839953)

Suppose that \(\mathfrak{g}\) is a finite dimensional complex simple Lie algebra with Dynkin diagram \(D.\) A diagram automorphism of order \(r\) leads to the affine Dynkin diagram \(D^r.\) The paper under review is an extension of Kac's work [\textit{V. G. Kac}, Funct. Anal. Appl. 3, 252--254 (1969); translation from Funkts. Anal. Prilozh. 3, No. 3, 94--96 (1969; Zbl 0274.17002)] on representing finite order automorphisms of \(\mathfrak{g}\) from \(D^r\). Indeed, the author uses \(D^r\) to represent finite order automorphisms of real forms of \(\mathfrak{g}.\) Taking the set \(V\) of vertices of \(D^r,\) there is a unique set \(\{a_\alpha\mid \alpha\in V\}\) of positive integers without nontrivial common factors such that \(\sum_{\alpha\in V}a_\alpha\alpha=0\). If we color the vertices of the diagram \(D^r\) by black and white, then it is called painted if \(r\sum_{\alpha\in V_b}a_\alpha=2\) in which \(V_b\) is the set of black vertices. The author considers a real form \(\mathfrak{g}_0\) of \(\mathfrak{g}\) with a Cartan decomposition \(\mathfrak{k}_0+\mathfrak{p}_0\) and defines a painted diagram representing \(\mathfrak{g}_0\). Then he imposes some conditions on a painted diagram representing \(\mathfrak{g}_0\) to get a finite order automorphism of \(\mathfrak{g}_0\) and conversely proves that a finite order automorphism is represented by a pained diagram representing \(\mathfrak{g}_0\) if it satisfies those particular conditions. Using these results, the author studies the extensions of finite order automorphisms of \(\mathfrak{k}_0\) to \(\mathfrak{g}_0\) and also finds the fixed point sets of automorphisms of \(\mathfrak{g}_0\).