Comparison of 3-Dimensional ℤ-Graded and ℤ₂-Graded L_∞ Algebras

Abstract

This article explores L _∞ structures on 3-dimensional vector spaces with both ℤ- and ℤ₂-gradings. Since ℤ-graded L _∞ algebras are special cases of ℤ₂-graded algebras in the induced ℤ₂-grading, there are generally fewer ℤ-graded L _∞ structures on a given space. However, degree zero automorphisms (rather than even automorphisms) determine equivalence in a ℤ-graded space. We therefore find nontrivial examples in which the map from the ℤ-graded moduli space to the ℤ₂-graded moduli space is bijective, injective but not surjective, or surjective but not injective. Additionally, we study how the codifferentials in the moduli spaces deform into other nonequivalent codifferentials.

Key Words:

• Graded algebras
• L _∞ algebras
• Versal deformations

2000 Mathematics Subject Classification:

• 14D15
• 13D10
• 14B12
• 16S80
• 16E40
• 17B55
• 17B70

ACKNOWLEDGMENTS

We would like to thank Jim Stasheff for raising the question, reading various versions of this manuscript and offering suggestions. We also thank Carolyn Otto for creating the figures and helping with the Maple programs. Research of these authors was partially supported by grants from NSF, OTKA T043641, T043034, grants from the University of Wisconsin-Eau Claire and Czech grant ME 603.

Notes

Communicated by K. C. Misra.