Non-group gradings on simple Lie algebras

TL;DR

This paper constructs specific set gradings on simple Lie algebras of type D that cannot be realized as group gradings, using combinatorial structures from finite geometry, thus answering a previously open question.

Contribution

It introduces new non-group gradings on simple Lie algebras of certain types, expanding the understanding of algebra gradings beyond group-based structures.

Findings

Constructed non-group gradings for type D_{13} Lie algebra.

Extended non-group gradings to types D_n with n ≡ 1 mod 12.

Utilized combinatorial designs from finite geometry to achieve these gradings.

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.