Some Considerations about Geometric Algebras in relation with Visibility in Computer Graphics

We give some considerations about the use of geometric algebras in the context of visibility, showing some advantages and disadvantages for their use as the underlying framework. We emphasize the use of conformal geometric algebra since, among other reasons, it allows us to study easily the visibility for flat varieties and, due to the same algebraic expression of hyper-spheres and linear varieties, the results might be generalized to non-flat objects

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 hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn 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

Sophus Lie: un matemático visionario

A diferencia de una mera exposici´on de datos puramente biogr´aficos, este art´ıculo pretende dar a conocer a cualquier persona en general, y a los matem´aticos y cient´ıficos en particular, aquellas an´ecdotas menos conocidas, leyendas m´as o menos ajustadas a la realidad, correspondencia enviada y recibida, y sobre todo, opiniones de otras personas sobre la vida y obra de Sophus Lie

Minimal faithful upper-triangular matrix representations for solvable Lie algebras

A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which is the minimal order of the matrices involved in such representations. Hence, the main goal of this paper is to revisit and implement a method to compute both that minimal order and a matrix representative for a given solvable Lie algebra. As application of this procedure, we compute representatives for each solvable Lie algebra with dimension less than 6.Ministerio de Economía, Industria y CompetitividadFondo Europeo de Desarrollo Regiona

An algorithm to compute abelian subalgebras in linear algebras of upper-triangular matrices

This paper deals with the maximal abelian dimension of the Lie algebra hn, of nxn upper-triangular matrices. Regarding this, we obtain an algorithm which computes abelian subalgebras of hn as well as its implementation (and a computational study) by using the symbolic computation package MAPLE, where the order n of the matrices in hn is the unique input needed. Let us note that the algorithm also allows us to obtain a maximal abelian subalgebra of hn

Application of statistical techniques for comparing Lie algebra algorithms

This paper is devoted to study and compare two algebraic algorithms related to the computation of Lie algebras by using statistical techniques. These techniques allow us to decide which of them is more suitable and less costly depending on several variables, like the dimension of the considered algebra

Relations between combinatorial structures and Lie algebras: centers and derived Lie algebras

In this paper, we study how two important ideals of a given Lie algebra g (namely, the center Z(g) and the derived Lie algebra D(g)) can be translated into the language of Graph Theory. In this way, we obtain some criteria and characterizations of these ideals using Graph Theory.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo Regiona

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 nondecomposable 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] G.M. Mubarakzyanov: Classification of real structures of Lie algebras of fifth order. Izv. Vyss. Ucebn. Zaved. Matematika 3:34, 1963, pp. 99-106. and [19] P. Turkowski: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1990, pp. 1344-1350) and the classification of 6-dimensional nilpotent Lie algebras by Goze and Khakimdjanov [7] M. Goze and Y. Khakimdjanov: Nilpotent and solvable Lie algebras. In M. Hazewinkel (ed.): Handbook of Algebra Vol 2. Elsevier, Amsterdam, 2000, pp. 615–664, we have explicitly computed the maximal abelian dimension for the algebras given in those classifications