Comparison of 3-dimensional Z-graded and Z_2-graded L_Infinity algebras

Details Comparison of 3-dimensional Z-graded and Z_2-graded L_Infinity algebras Comparison of 3-dimensional Z-graded and Z_2-graded L_Infinity algebras

2006-06-26

arxiv.org/abs/math/0606644v1

Mathematics

Quantum Algebra

31 pages, 3 figures

Scientific paper

This article explores L_Infinity structures -- also known as 'strongly homotopy Lie algebras' -- on 3-dimensional vector spaces with both Z- and Z_2-gradings. Since the Z-graded L_Infinity algebras are special cases of Z_2-graded algebras in the induced Z_2-grading, there are generally fewer Z-graded L_Infinity 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 Z_2-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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