New York Journal of Mathematics
Volume 15 (2009) 353-392


Yael Frégier, Martin Markl, and Donald Yau

The L-deformation complex of diagrams of algebras

Published: August 9, 2009
Keywords: Deformation, colored PROP, diagram of algebras, strongly homotopy Lie algebra
Subject: 14D15, 20G10

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-algebra which induces a graded Lie bracket on cohomology.

As an example, the L-algebra structure on the deformation complex of an associative algebra morphism 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-deformation complex has nontrivial l0-term.

Explicit formulas for the L-operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fully-fledged L-algebra with nontrivial higher operations.


The first author worked in the frame of grant F1R-MTH-PUL-08GEOQ of Professor Schlichenmaier. The second author was supported by the grant GA ČR 201/08/0397 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.

Author information

Yael Frégier:
Mathematics Research Unit, 162A, avenue de la faiencerie L-1511, Luxembourg, Grand duchy of Luxembourg

Martin Markl:
Mathematical Institute of the Academy, \v{Z}itná 25, 115 67 Prague 1, The Czech Republic

Donald Yau:
Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark, OH 43055, USA