 

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 


Abstract
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 l_{0}term.
Explicit formulas for the L_{∞}operations in the above examples are
given. A typical deformation complex of a diagram
of algebras is a fullyfledged L_{∞}algebra with nontrivial
higher operations.


Acknowledgements
The first author worked in the frame of grant F1RMTHPUL08GEOQ 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 L1511, Luxembourg, Grand duchy of Luxembourg
yael.fregier@uni.lu
Martin Markl:
Mathematical Institute of the Academy, \v{Z}itná 25, 115 67 Prague 1, The Czech Republic
markl@math.cas.cz
Donald Yau:
Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark, OH 43055, USA
dyau@math.ohiostate.edu

