Group gradings on the Lie algebra of upper triangular matrices

The algebras UTn of the n×n upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. We classify the group gradings on the Lie algebra UTn(−). It was proved by Valenti and Zaicev in 2007 that every group grading on the associative algebra UTn is isomorphic to an elementary grading. The elementary gradings on UTn are also well understood, see [6]. It follows from our results that there are nonelementary gradings on UTn(−). Thus the gradings on the Lie algebra UTn(−) are much more intricate than those in the associative case.