Desamparados Fernández-Ternero

Perfect discrete Morse functions on 2-complexes

Highlights? We get conditions under which a 2-complex admits a perfect discrete Morse function. ? We study the topology of comlpexes admitting such kind of functions. ? These results are a first step in the study of these functions on 3-manifolds. This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the ith Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties necessary for a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature (Lewiner et al., 2003), since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.

Perfect discrete morse functions on triangulated 3-manifolds

This work is focused on characterizing the existence of a perfect discrete Morse function on a triangulated 3-manifold M, that is, a discrete Morse function satisfying that the numbers of critical simplices coincide with the corresponding Betti numbers. We reduce this problem to the existence of such kind of function on a spine L of M, that is, a 2-subcomplex L such that M−Δ collapses to L, where Δ is a tetrahedron of M. Also, considering the decomposition of every 3-manifold into prime factors, we prove that if every prime factor of M admits a perfect discrete Morse function, then M admits such kind of function.

NILPOTENT LIE ALGEBRAS OF MAXIMAL RANK AND OF KAC-MOODY TYPE: F 4 (1)

Let be the Kac-Moody algebra associated to the affine Cartan matrix F 4 (1). Each nilpotent Lie algebra of type F 4 (1) is isomorphic to a quotient of the positive part of . We determine the isomorphism classes of nilpotent Lie algebras of type F 4 (1).

A graph-theoretical approach to cancelling critical elements

Abstract This work is focused on the links between Formanʼs discrete Morse theory and graph theory. More precisely, we are interested on putting the optimization of a discrete Morse function in terms of matching theory. It can be done by describing the process of cancellation of pairs of critical simplices by means of obtaining Morse matchings on the corresponding Hasse diagram with a greater number of edges using the combinatorial notion of transference.

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.

Graphs associated with nilpotent Lie algebras of maximal rank

In this paper, we use the graphs as a tool to study nilpotent Lie algebras. It implies to set up a link between graph theory and Lie theory. To do this, it is already known that every nilpotent Lie algebra of maximal rank is associated with a generalized Cartan matrix

Nilpotent Lie Algebras of Maximal Rank and of Kac–Moody Type: E6(1)☆

A

Lusternik-Schnirelmann category of simplicial complexes and finite spaces

and it is isomorphic to a quotient of the positive part

Discrete Morse theory on graphs

\mathfrak{n}_+

Simplicial Lusternik-Schnirelmann category

of the Kac-Moody algebra