Gradings by finite groups on Lie algebras of type D₄

Abstract

In this thesis we explore the gradings by finite groups on Lie algebras of type D₄ over the field of complex numbers. For gradings on simple Lie algebras several approaches have been studied. In (9], Onishchik and Vinberg give an exposition of the results of V. Kac who had classified all automorphisms of finite order in all simple Lie algebras, hence classified the gradings of such algebras by finite cyclic groups. J. Patera and co-authors (5], (6], (7] have focused on "fine" gradings and approach this with the help of maximal Abelian subgroups (MAD-subgroups) of diagonizable automorphisms in Aut(gl(n, C)). More recently Y. Bahturin, I. Shestakov, M. Zaicev (1] have approached gradings on simple Lie algebras by finite groups by looking at the dual group action which will be the main approach used in this paper. The gradings on simple Lie algebras of type D₁, l > 4, have been described by Y. Bahturin and M. Zaicev in (4]. This was done by looking at gradings on the full matrix algebras and noting that for a realisation of a Lie algebra of type D₁, l > 4, as K (M₂₁ , *), the skew-symmetric matrices with respect to a transpose involution *, the automorphisms of K(M21 , *) can be lifted to automorphisms of the full matrix algebra. The gradings on Lie algebras of type D₄ were not described in [1] or [6] because some of the automorphisms of these Lie algebras cannot be lifted to the full matrix algebra. In this thesis we apply the same approach as [1], [2], [3], [4] to describe all gradings that can be lifted to the full matrix algebra and all gradings that are isomorphic to these gradings. We also develop an approach inspired by [9] which may be fruitful for describing the remaining gradings. We give examples of some of these gradings.