L-Infinity Structures on Spaces with 3 One-Dimensional Components

Abstract

L-Infinity structures have been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This paper provides a class of easily constructible examples of $L_n$ and $L_{\infty}$ structures on graded vector spaces with three one-dimensional components. In particular, it demonstrates a way to classify ALL possible $L_n$ and $L_{\infty}$ structures on $V = V_m \oplus V_{m+1} \oplus V_{m+2}$ when each of the three components is one-dimensional. Included are necessary and sufficient conditions under which a space with an $L_3$ structure is a differential graded Lie algebra. It is also shown that some of these d.g. Lie algebras possess a nontrivial $L_n$ structure for higher n.