Elementary gradings on the Lie algebra UTn(−)

The algebras UTn(K) of the 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. In this paper we consider the Lie algebra UTn(K)(−) of all upper triangular matrices of order n. We study the group gradings on this algebra. It turns out that the gradings on the Lie algebra UTn(K) are much more intricate than those in the associative case. In this paper we describe the elementary gradings on the Lie algebra UTn(K)(−). Finally we study the canonical grading on UTn(K)(−) by the cyclic group Zn of order n. We produce a (finite) basis of the graded polynomial identities satisfied by this grading.