Graded identities for Lie algebras

Gradings on associative and Lie algebras have long been objects of intensive study. During the last two decades graded identities of associative algebras emerged as a new and very important branch of PI theory. Surprisingly the theory of graded identities for Lie algebras is not as well developed as that in the associative case, and the known results are sparse. Here we discuss several recent results concerning gradings and the corresponding graded identities for important; Lie algebras. Furthermore we provide a new, elementary description of the 2-graded identities for the Lie algebra sl(2)(K) over an infinite field of characteristic not equal 2. Recall that recently these have been described by using representations of the symmetric group (in characteristic 0), and invariant theory (characteristic-free).