The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.
In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.
We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
