Juan C. Benjumea

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown.

A method to obtain the lie group associated with a nilpotent lie algebra

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.

Computational calculus of the law of a family of solvable Lie algebras

This paper studies the law of the Lie algebras hn associated with a particular type of Lie groups: the Lie groups Hn formed by all the n × n upper-triangular matrices without zeros in their main diagonal. Indeed, these laws are obtained by means of a computational algorithm which we have constructed and particularly implemented by using the symbolic computation package MAPLE. Besides, the complexity of this algorithm is studied by considering the number of computations carried out with this implementation.

New properties of filiform Lie algebras and its computational processing

In this paper we study a characterization for a complex filiform Lie algebra to be characteristically nilpotent and we also give two new ways of defining complex filiform Lie algebras, by using proper subsets of the set of commutators of the basis elements. Besides, we present three polynomial time algorithms suitable to be used in the study of these results. The first of them allow us to know if a complex filiform Lie algebra is characteristically nilpotent. The other two ones are useful to give two new ways of defining complex filiform Lie algebras by using less non-null brackets than usual.