L(Infinity) Structures on Spaces of Low Dimension

Abstract

L(Infinity) structures are natural generalizations 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(Infinity) 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 L(3) structure is a differential graded Lie algebra. Additionally, it is shown that some of these differential graded Lie algebras possess a nontrivial L(n) structure for higher n.