An Algorithm to Compute Abelian Subalgebras in Linear Algebras of Upper-Triangular Matrices (original) (raw)

The computation of abelian subalgebras in the Lie algebra of upper-triangular matrices

2008

This paper deals with the computation of abelian subalgebras of the solvable Lie algebra hn, of n × n upper-triangular matrices. Firstly, we construct an algorithm to find abelian Lie subalgebras in a given Lie algebra hn. This algorithm allows us to compute an abelian subalgebra up to a certain dimension. Such a dimension is proved to be equal to the maximum for abelian subalgebras of hn.

Computational Algorithm for Obtaining Abelian Subalgebras in Lie Algebras

2009

Abstract—The set of all abelian subalgebras is computationally obtained for any given finite-dimensional Lie algebra, starting from the nonzero brackets in its law. More concretely, an algorithm is described and implemented,to compute,a basis for each non- trivial abelian subalgebra with the help of the symbolic computation package MAPLE. Finally, it is also shown a brief computational study for this implementation,

Algorithm to compute the maximal abelian dimension of Lie algebras

Computing, 2009

In this paper, the maximal abelian dimension is computationally obtained for an arbitrary finite-dimensional Lie algebra, defined by its nonzero brackets. More concretely, we describe and implement an algorithm which computes such a dimension by running it in the symbolic computation package MAPLE. Finally, we also show a computational study related to this implementation, regarding both the computing time and the memory used.

Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

International Journal of Computer Mathematics, 2012

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finitedimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. Finally, we determine the maximal dimension of abelian subalgebras and ideals for non-decomposable solvable non-nilpotent Lie algebras of dimension 6 over both the fields R and C, showing the differences between these fields.

The computation of abelian subalgebras in low-dimensional solvable lie algebras

The main goal of this paper is to compute the maximal abelian dimension of each solvable non-decomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky's classical classifications of solvable Lie algebras (see [13] and [19]) and the classification of 6-dimensional nilpo-tent Lie algebras by Goze and Khakimdjanov [7], we have explicitly computed the maximal abelian dimension for the algebras given in those classifications.

Combinatorial structures and Lie algebras of upper triangular matrices

Applied Mathematics Letters, 2012

This work shows how to associate the Lie algebra h n , of upper triangular matrices, with a specific combinatorial structure of dimension 2, for n ∈ N. The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.

Maximal Abelian subalgebras of pseudounitary lie algebras

Linear Algebra Appl, 1990

The task of classifying and constructing all maximal Abelian subalgebras of su(p, y) (p > y > 1) is reduced to that of classifying orthogonally indecomposable MASAs. These are either maximal Abelian nilpotent subalgebras (represented by nilpotent matrices in any finite-dimensional representation), or for p = q they can be (nonorthogonally) decomposable and their study can be reduced to a construction of MANSs of sl(p, C). Two types of MANSs of su(p, y) are shown to exist ("one-rowed" and "non-one-rowed").