Juan Carlos Hernández Núñez

Relations among invariants of complex filiform Lie algebras

Apart from the own dimension n of a complex filiform Lie algebra, the first invariant known for these algebras was the lower central sequence of the algebra, obtained by Ancochea and Goze in 1989. Later, Echarte, Gomez and Nunez, in 1996, and ourselves, in the same year, obtained the invariants i and j, respectively. In this paper, we prove new properties of the invariant j and we use them to set some relations among these invariants and the dimension n. These results allow us to classify complex filiform Lie algebras according to some values of those invariants.

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.

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying r × s partial Latin rectangles based on n symbols of a given size, shape, type or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all r, s, n ≤ 6. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of size up to six. We focus then on the study of non-compressible regular partial Latin squares and their equivalent incidence structure called seminet, whose distribution into main classes is explicitly determined for point rank up to eight. We prove in particular the existence of two new configurations of point rank eight.