L_∞ Structures on Spaces with Three One-Dimensional Components

Abstract

L _∞ 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 _∞ structures on graded vector spaces with three one-dimensional components. In particular, it demonstrates a way to classify all possible L _n and L _∞ structures on V = V _m  ⊕ V _m+1 ⊕ 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 differential graded Lie algebras possess a nontrivial L _n structure for higher n.

Key Words:

• L-Infinity algebras
• Strongly homotopy Lie algebras

Acknowledgments

The results in this paper are a portion of my Ph.D. research under the supervision of Tom Lada. I would also like to thank Jim Stasheff for many helpful suggestions and comments.

Notes

^#Communicated by K. Misra.