This page provides lists of elements of finite order of simple Lie groups. For each complex simple Lie algebra \(\mathfrak{g}\) of rank \(\le 8\), we consider its group of automorphisms \(\textrm{Aut}(\mathfrak{g})\) and describe its elements of order \(\le 8\). This project is carried out by Sara Lombardo, Jan Sanders, and Vincent Knibbeler, with support from the London Mathematical Society through an Emmy Noether Fellowship.
The project is in progress. If you see anything that is not right, please send us a line. We aim to add database functionalities in the future.
A symmetry of a Dynking diagram defines an automorphism of the associated Lie algebra \(\mathfrak{g}\). The subalgebra of this Lie algebra consisting of elements fixed by this automorphism is again simple. Therefore it has Chevalley generators \(E_0,\ldots,E_\ell\).
An automorphism \(\theta_{\alpha^\star,k}\) of \(\mathfrak{g}\) of order \(\nu\) will be defined by the order \(k\) of the induced Dynkin symmetry, and a sequence of integers \(\alpha^\star=(s_0,\ldots,s_\ell)\), called Kac coordinates, by the relations
\[\theta_{\alpha^\star,k}(E_j)=\exp\left(\frac{2\pi i s_j}{\nu}\right)E_j,\quad j=0,\ldots,\ell.\]
We tabulate one set of Kac coordinates \(\alpha^\star\) for each conjugacy class of automorphisms up to order \(8\), together with its multiplicities \(\mu\) of eigenvalues \(\exp\left(\frac{2\pi i j}{\nu}\right)\), \(j=0,\ldots,\nu-1\). For the kernel \(\mathcal{Z}\) of each automorphism we include its Lie type (with a hat for long roots and a check for short roots) and its number of positive roots.
For the multiplicities we use a special notation to avoid duplicates. For example, at order \(4\) we find expressions for \(\mu\) of the form \( m_0\langle m_1 | m_2 |\) indicating that the multiplicities are \((m_0,m_1,m_2,m_1)\). At order \(5\) we see \( m_0\langle m_1, m_2 |\) for multiplicities \((m_0,m_1,m_2,m_2, m_1)\).