1. Non-group gradings on simple Lie algebras
- Author
-
Alberto Carlos Elduque Palomo
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Rings and Algebras (math.RA) ,17B70 ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Rings and Algebras ,Geometry and Topology - Abstract
A set grading on the split simple Lie algebra of type $D_{13}$, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern provided by the lines of the projective plane over $GF(3)$. This answers in the negative Question 1.11 in Elduque-Kochetov monograph (2013). Similar non-group gradings are obtained for types $D_n$ with $n$ congruent to 1 modulo 12, by substituting the lines in the projective plane by blocks of suitable Steiner systems., Comment: 11 pages
- Published
- 2022