A result on exponents of finite-dimensional simple Lie algebras and its application to Kac-Moody algebras (Q1906781)

When generalizing the loop group approach to derive Toda lattice-like equations, one is led to the question of whether the adjoint map of the cyclic element \(E\) of a Kac-Moody algebra \(\mathfrak g\) is injective on the subspace \({\mathfrak q}^{(1)}\) of degree \(-1\) of the natural complement \(\mathfrak q\) of a standard parabolic \(\mathfrak p\) of \(\mathfrak g\). For the untwisted case, this question is answered completely in this paper (Theorem 3.5). In preparation for this a technical result is proven. An exponent \(k\) of \(\mathfrak g\) is called complete if every non-zero element of the centralizer of \(E\) of degree \(k\) contains non-zero components of all root spaces corresponding to roots of height \(k\). The central result of this paper then is Theorem. All but three exponents are complete. The non-complete exponents are precisely (a) the exponent \(\ell-1\) in \(D_\ell\), (b) the exponent 4 in \(E_6\) and (c) the exponent \(g\) in type \(E_7\).