The Details

Symmetry is one of the most important notions in mathematics. In many situations, typically when studying Physics problems, the objects under consideration have many symmetries. An extremely important situation, with far reaching applications, is when the symmetries can be described as a family parametrized by a smooth manifold. A common example is the motions in the Euclidean plane/space. The parametrizing smooth manifolds are called Lie groups, and one fortunate aspect of theirs is that they can be studied by using basic linear algebraic methods through their corresponding Lie algebras. Lie groups and Lie algebras are fundamental notions for any research mathematician and theoretical physicist.

This project aims to teach the selected group of sophomore students with background in math and physics the basic notions of Lie algebras with emphasis on nilpotent algebras (a special type Lie algebras) that arise from graphs. The construction is elementary, using only the definition of graphs and wedge products of elements of a vector space. It is completely within the grasp of our sophomores. The constructed Lie algebras are 2-step nilpotent , and were first introduced in S. Dani, M. Mainkar, “Anosov aoutomorphisms on compact nil-manifolds associated with graphs”, Trans. Amer. Math. Soc. 357(2004) 2235 - 2251. Recently, in M. Mainkar, “A short note on a 2-step nilpotent Lie algebras associated with graphs”, arxiv:1010.40055, it was proved that the construction has a nice property: two graphs are isomorphic if, and only if, the corresponding Lie algebras are isomorphic. In addition to that this construction leads to nil-manifolds with Anosov diffeomorphisms (see the former paper above), and a characterization of graphs for which the corresponding Lie algebra admits a special geometric structure, a symplectic form, is known (H. Pouseele, P. Tirao, “Compact nil-manifolds associated with graphs”, J. Pure Appl. Alg. 213 (2009) 1788 - 1794 ).

This research proposal is centered around two main questions. Question 1. How far can the isomorphism property mentioned above be extended to relate graphs and Lie algebras? More specifically, the Lie algebras form a category. Can the set of graphs be provided with a structure of a category (apart from the obvious one) so that the construction above defines a functor? What Lie algebras correspond under this functor to special types of graphs, such as bipartite, Eulerian, Hamiltonian, k-connected, trees, etc.

The second main question is related to geometric structures which arise from graphs (in the sense mentioned above). Typically, these structures appear in math models of physics theories which makes the question relevant and important for our students. The nilpotent Lie groups and algebras provide an important source of examples (and counterexamples) for interesting geometric structures. These include complex, hyper-complex, para-hyper-complex, and hyper-symplectic structures, and are easily described (locally on a manifold) in terms of Linear Algebra. The interest in these structures within the mathematics community is strong because of their physics applications. The authors of this proposal are involved in an ongoing research studying the para-hyper-complex structures (J.Davidov, G.Grantcharov, O.Muskarov, M.Yotov, “Para-hyper-hermitian surfaces”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 52(100) (2009), no. 3, 281–289. , and J.Davidov, G.Grantcharov, O.Muskarov, M.Yotov, “Generalized Pseudo Kahler structures”, Comm. Math. Phys. (to appear)). For all types of structures described above some of the first examples (or counterexamples) are based nilpotent Lie groups. For instance, one of the first complex non-Kahler manifolds was provided by a 4-dimensional nilpotent Lie group. In fact, the groups and algebras in question are especially appropriate to test the properties of the corresponding geometric structures, and are more effective compared to the “standard” methods used in the area. This leads to

Question 2. What are the graphs for which the corresponding Lie algebra carries one of the following structures: complex, hyper-complex, para-hyper-complex, or hyper-symplectic. Expectations Basic knowledge of Lie algebras and Lie groups (as groups of transformations) is one of the main prerequisites of any professional mathematician or physicists. Our students will learn about these and other topics necessary for the project (linear and tensor algebra transformation groups) during the first phase of the project. Once they master the notions of complex, hyper-complex, symplecitic and quaternion structure on a vector space, further understanding of smooth manifolds and Lie groups with emphasis on particular examples of nil-manifolds will be obtained. A short introduction to Graph Theory will be presented, if necessary. Understanding the relation between graphs and nilpotent Lie algebras will give them, among the other things, a good example about the unity of mathematics. The latter relation is new, not developed yet, but seems to attract the attention of increasing number of researchers. The stated above Questions 1 and 2 are natural and important for the emerging theory. Any advance on them should lead to a publication.Notice that our students are familiar with basic notions from all areas of Mathematics we will be teaching them. This applies to Differential Geometry as well: they have studied and used it in the courses in Physics all of them have taken.

Assessments The first phase achievements will be assessed via homework assignments, as well as via monitoring the students' work in class on a daily basis. The second phase will be assessed via midterm reports on the progress of the participants in their research, and the final presentations. The planned every day discussions with the students on the project will help in that too.

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