endomorphism dg-Lie algebra
Formal Lie groupoids
Let be a category of chain complexes in a category that is a closed monoidal category. For instance the category of chain complexes in Vect.
For any object – any chain complex – write
for the internal hom object of morphisms from to . This is the chain complex which in degree is
and whose differential
is given by
This complex naturally carries the structure of an internal Lie algebra, hence of a dg-Lie algebra with Lie bracket given by the graded commutator for and
is the endomorphism dg-Lie algebra of . Or, since dg-Lie algebras are special cases of L-∞ algebras: the endomorphism -algebra of .
A representation of an L-∞ algebra on a chain complex is a morphism of -algebras
Sometimes this is called a representation up to homotopy .
See for instance the paragraph above theorem 5.4 in
Revised on April 4, 2011 15:02:04
by Urs Schreiber