The structure of symmetric Lie algebras (Q2639945)

The main result of this paper is showing the existence of a decomposition of symmetric Lie algebras over a field of characteristic \(p>7\) corresponding to some extent to that of Levi for characteristic 0. In particular, the algebras considered are defined to have Cartan decompositions \(L=\sum L_ a\) with \(L_ 0\) a split Cartan subalgebra and \(a([L^ 1_{-a},L^ 1_ a])\neq 0\) for \(a\neq 0\), where \(L^ 1_ a=\{x\in L_ a |\) \([h,x]=a(h)x\) for all \(h\in L_ 0\}\). In the decomposition obtained, \(L=L^ W+I\), and \(I=L_ S\oplus Solv I\), \(L_ S\) is classical, and \(L^ W\) has a Witt rootsystem (i.e. all integral multiples of roots are roots). The proof makes use of the classification of Lie root systems of low rank to study the structure of the automorphism group of L and then employs the theory of algebraic groups and the theory of Lie root systems. The hypotheses of the theorem exclude root systems containing a certain type of 2-section not known to exist for symmetric Lie algebras.