13 research outputs found
Perfect discrete Morse functions on 2-complexes
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 i-th 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 of a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature [7], 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
Counting excellent discrete Morse functions on compact orientable surfaces
We obtain the number of non-homologically equivalent excellent discrete Morse functions defined on compact orientable surfaces. This work is a continuation of the study which has been done in [2, 4] for graphs
Dealing with the Resolubility of Evolution Algebras
Although since their introduction by Tian in 2004, evolution algebras have been the subject of a very
deep study in the last years due to their numerous applications to other disciplines, this study is not easy since
these algebras lack an identity that characterizes them, such as the identity of Jacobi, for Lie algebras or those of
Leibniz and Malcev for those corresponding algebras. In this paper we deal with the concepts of solvability and
nilpotency of these evolution algebras. Some novel results on them obtained from using the evolution operator of
these algebras are given and some examples illustrating these results are also shown. The main result obtained states
that an evolution algebra is solvable if and only if its structure matrix is nilpotent, which implies, in turn, that the
solvability and the nilpotency indices of that algebra coincide provided the corresponding evolution operator is an
endomorphism of the algebra
Set of evolution operators of an evolution algebra
An automorphism defined on an evolution algebra can provide both a finite
number and an infinite number of evolution operators on it. This question is
dealt with in the paper, as well as others more related to the evolution
operators of evolution algebras. After defining the concept of the set of
evolution operators of an evolution algebra, differences between trivial and
non-trivial sets of evolution operators are also covered. Some properties of
these concepts are studied and several examples of the above issues are shown
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
Using the Evolution Operator to Classify Evolution Algebras
Evolution algebras are currently widely studied due to their importance not only “per se” but also for their many applications to different scientific disciplines, such as Physics or Engineering, for instance. This paper deals with these types of algebras and their applications. A criterion for classifying those satisfying certain conditions is given and an algorithm to obtain degenerate evolution algebras starting from those of smaller dimensions is also analyzed and constructed
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 A and it is isomorphic to a quotient of the positive part n+ of the KacMoody algebra g(A). Then, if A is affine, we can associate n+ with a directed graph (from now on, we use the term digraph) and we can also associate a subgraph of this digraph with every isomorphism class of nilpotent Lie algebras of maximal rank and of type A. Finally, we show an algorithm which obtains these subgraphs and also groups them in isomorphism classes
Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem
We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory
for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik–
Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott
functions
Some advances in the research on Lie algebras
The main goal of this poster, which is written in the form of a survey and tries to show some aspects of the research of authors on Lie algebras, is to pay homage to the memory of Pilar Pisón Casares, who was firstly teacher of some of them and later colleague of all of them during different stages of her stay as a member of the Departamento de Algebra, Computación, Geometría y Topología de la Universidad de Sevilla