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 [3, Question 1.11]. Similar non-group gradings are obtained for types D-n with n = 1(mod12), by substituting the lines in the projective plane by blocks of suitable Steiner systems. (C) 2022 The Author(s). Published by Elsevier Inc.
Set gradingGroup gradingPure gradingOrthogonal Lie algebraSteiner system
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