Automorphisms of finite order of the affine Lie algebra \(A_{\ell}^{(1)}\)
Let \({\mathfrak g}\) be the affine Lie algebra over \({\mathbb{C}}\) of type \(A_{n- 1}(1)\) (for \(n\geq 2)\). The author explicitly determines all the automorphisms of prime order of the Lie algebra \({\mathfrak g}\) upto conjugacy in Aut(\({\mathfrak g})\). By the universality of the central extension \[ 0 \to {\mathbb{C}}Z \to {\mathfrak g} \to s\ell_ n({\mathbb{C}}[t,t^{- 1}]) \to 0, \] it can be seen that Aut(\({\mathfrak g})\approx Aut_{{\mathbb{C}}}(s\ell_ n({\mathbb{C}}[t,t^{-1}]))\). Now the author reduces the study of elements of prime order k in \(Aut_{{\mathbb{C}}}(s\ell_ n({\mathbb{C}}[t,t^{-1}]))\) (upto conjugacy) to the study of the group cohomology \(H^ 1({\mathbb{Z}}_ k, PGL_ n({\mathbb{C}}[t,t^{-1}]))\) with \({\mathbb{Z}}_ k\) acting (in general) nontrivially on \(PGL_ n\). The large part of the paper is devoted to determine \(H^ 1({\mathbb{Z}}_ k,PGL_ n({\mathbb{C}}[t,t^{-1}]))\) explicitly, which is done case by case and which uses Grothendieck's ''theory of descent''.
scientific article; zbMATH DE number 4002338