Graded contractions of the [formula omitted]-grading on [formula omitted]

Abstract

Graded contractions of the -grading on the complex exceptional Lie algebra are classified up to equivalence and up to strong equivalence. The non-toral fine -grading is highly symmetric, with all the homogeneous components Cartan subalgebras. This makes possible a combinatorial treatment based on certain nice subsets of the set of 21 edges of the Fano plane. There are 24 such nice sets up to collineation. Each of these is the support of an admissible graded contraction, one of which is present in every equivalence class of graded contractions. Each nice set gives rise to a single Lie algebra, except for three of the cases in which families depending on one or two parameters are found. In particular, a large family of 14-dimensional Lie algebras arise, most of which are solvable. The properties of each of these Lie algebras are studied.