The L_\infty-deformation complex of diagrams of algebras

Details The L_\infty-deformation complex of diagrams of algebras The L_\infty-deformation complex of diagrams of algebras

2008-12-16

arxiv.org/abs/0812.2981v2

New York Journal of Math. 15(2009), 353-392

Mathematics

Algebraic Topology

37 pages. A minor revision

Scientific paper

The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\infty-algebra which induces a graded Lie bracket on cohomology. As an example, the L_\infty-algebra structure on the deformation complex of an associative algebra morphism g, with the underlying cochain complex isomorphic to the Gerstenhaber-Schack complex of g, is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L_\infty-deformation complex has a nontrivial constant term. Explicit formulas for the L_\infty-operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fully-fledged L_\infty-algebra with nontrivial higher operations.

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