Curved infinity-algebras and their characteristic classes

Details Curved infinity-algebras and their characteristic classes Curved infinity-algebras and their characteristic classes

2010-09-30

arxiv.org/abs/1009.6203v2

Mathematics

Quantum Algebra

Final version, to appear in J. Topology. 28 pages

Scientific paper

In this paper we study a natural extension of Kontsevich's characteristic class construction for A-infinity and L-infinity algebras to the case of curved algebras. These define homology classes on a variant of his graph homology which allows vertices of valence >0. We compute this graph homology, which is governed by star-shaped graphs with odd-valence vertices. We also classify nontrivially curved cyclic A-infinity and L-infinity algebras over a field up to gauge equivalence, and show that these are essentially reduced to algebras of dimension at most two with only even-ary operations. We apply the reasoning to compute stability maps for the homology of Lie algebras of formal vector fields. Finally, we explain a generalization of these results to other types of algebras, using the language of operads.

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