Maximal Abelian Sets of Roots. - Lawther, R. (Memoirs of the American Mathematical Society., 2018)
In this work the author lets Φ be an irreducible root system, with Coxeter group W. He considers subsets of Φ which are abelian, meaning that no two roots in the set have sum in Φ \cup \{ 0 \}. He classifies all maximal abelian sets (i.e., abelian sets properly contained in no other) up to the action of W: for each W-orbit of maximal abelian sets we provide an explicit representative X, identify the (setwise) stabilizer W_X of X in W, and decompose X into W_X-orbits. Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelia.