L(Infinity) structures on spaces of low dimension

by Daily, Marilyn Elizabeth.

Daily, Marilyn. L? Structures on Spaces of Low Dimension (Under the direction of Tom Lada.) L? structures are a natural generalization of Lie algebras, which need satisfy the standard graded Jacobi identity only up to homotopy. They have also been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This dissertation classifies all possible L? structures which can be constructed on a Z-graded (characteristic 0) vector space of dimension three or less. It also includes necessary and sufficient conditions under which a space with an L3 structure is a differential graded Lie algebra. Additionally, it is shown that some of these differential graded Lie algebras possess a nontrivial Ln structure for higher n. L? Structures on Spaces of Low Dimension by Marilyn Daily Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics North Carolina State University 2004 Dr. Tom Lada Chair, Advisory Committee Dr. Ron Fulp Committee Member Dr. Kailash Misra Committee Member Dr. Jim Stasheff Committee Member Biography After receiving an undergraduate degree in Computer Science from Indiana University, Marilyn Daily worked for IBM as a programmer, writing software to manage computer networks in real time. When this job no longer seemed sufficiently interesting or challenging, she decided to focus her attention on mathematics. She became a full time graduate student at North Carolina State University in the fall of 1997, supporting herself by teaching calculus classes. In her final year of graduate school, she also had the opportunity to co-teach the graduate-level sequence in Topology and Algebraic Topology as a “Preparing the Professoriate” Fellow. After graduation, she will become a postdoctoral researcher at the Max Planck Institute for Gravitational Physics in Potsdam, Germany. She looks forward to a long and fruitful academic career. ii