Florida International University

Title: Lie Algebras and Geometric Structures for Undergraduates

Directors:

Gueo Grantcharov
Mirroslav Yotov

Email:

Dates of Program: May 9 - June 17, 2011

Summary:

This proposed project will study a relation between Linear Algebra and Graph Theory with a view toward Differential Geometric applications. In 2004, S. Dani and M. Miankar associated to a directed graph a two-step nilpotent Lie algebra in a pretty easy way: For any such graph with vertices V = {v₁, . . . , v_n} and edges E, we assign vector spaces W and U, such that W has a basis w₁, ...,w_n corresponding to the elements in V and U ’ Î?²V is generated by w_ij = w_i â?§ w_j , where w_iw_j ’ E. A Lie bracket on W ’ U is defined in the following way:

[w_i,w_j] = w_ij = -[w_j ,w_i]

if v_iv_j is an edge and all other brackets of the generators are zero.
M. Miankar proved recently that two such 2-step nilpotent Lie algebras are isomorphic iff their corresponding graphs are isomorphic. The students participating in the project will explore further the construction above concentrating on two main problems. The first one is about extending the categorical properties of the relation beyond the statement above. A natural question here is whether there is an additional category structure on the set of graphs for which the correspondence is a functor into the category of Lie algebras. Also how to characterize the algebras corresponding to particular types of graphs, such as bipartite, Eulerian, Hamiltonian, k-connected etc. graphs. The second problem is to relate various geometric structures on Lie algebras and the corresponding nilmanifolds to their graph counterparts. The geometric structures to be considered are complex, symplectic, hypercomplex, para-hypercomplex and hypersymplectic. These structures appear in theoretical physics and are objects of constant interest for mathematicians and physicists. On the other hand, they are easily described in terms of tensors on the Lie algebra of vector fields of the manifold, and are appropriate objects for investigation within the context of the project.