Lie coalgebra

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.