Computing abelian subalgebras for linear algebras of upper-triangular matrices from an algorithmic perspective

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra h_n, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of h_n besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of h_n and, hence, its maximal abelian dimension. The order n of the matrices h_n is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.