Comparison of 3-dimensional Z-graded and Z2-graded LInfinity algebras
Abstract
This article explores LInfinity structures -- also known as 'strongly homotopy Lie algebras' -- on 3-dimensional vector spaces with both Z- and Z2-gradings. Since the Z-graded LInfinity algebras are special cases of Z2-graded algebras in the induced Z2-grading, there are generally fewer Z-graded LInfinity structures on a given space. On the other hand, only degree zero automorphisms, rather than just even automorphisms, are used to 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, which gives each moduli space a sort of topology.
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