In mathematics a Lie coalgebra is the dual structure to a Lie algebra.
In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely.
be a vector space over a field
equipped with a linear mapping
uniquely to a graded derivation (this means that, for any
) of degree 1 on the exterior algebra of
, i.e., if the graded components of the exterior algebra with derivation
form a cochain complex: Just as the exterior algebra (and tensor algebra) of vector fields on a manifold form a Lie algebra (over the base field
), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field
Further, there is a pairing between vector fields and differential forms.
However, the situation is subtler: the Lie bracket is not linear over the algebra of smooth functions
(it is a derivation, not linear over functions): they are not tensors.
They are not linear over functions, but they behave in a consistent way, which is not captured simply by the notion of Lie algebra and Lie coalgebra.
Further, in the de Rham complex, the derivation is not only defined for
A Lie algebra structure on a vector space is a map
which is skew-symmetric, and satisfies the Jacobi identity.
that satisfies the Jacobi identity.
Dually, a Lie coalgebra structure on a vector space E is a linear map
which is antisymmetric (this means that it satisfies
) and satisfies the so-called cocycle condition (also known as the co-Leibniz rule) Due to the antisymmetry condition, the map
yields a map (the cocommutator) where the isomorphism
holds in finite dimension; dually for the dual of Lie comultiplication.
In this context, the Jacobi identity corresponds to the cocycle condition.
be a Lie coalgebra over a field of characteristic neither 2 nor 3.
carries the structure of a bracket defined by
It suffices to check the Jacobi identity.
, where the latter step follows from the standard identification of the dual of a wedge product with the wedge product of the duals.
, it follows that Thus, by the double-duality isomorphism (more precisely, by the double-duality monomorphism, since the vector space needs not be finite-dimensional), the Jacobi identity is satisfied.
In particular, note that this proof demonstrates that the cocycle condition
is in a sense dual to the Jacobi identity.