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.
Definition.
Let "E" be a vector space over a field "k" equipped with a linear mapping formula_1 from "E" to the exterior product of "E" with itself. It is possible to extend "d" uniquely to a graded derivation (this means that, for any "a", "b" ∈ "E" which are homogeneous elements, formula_2) of degree 1 on the exterior algebra of "E":
Then the pair ("E", "d") is said to be a Lie coalgebra if "d"2 = 0,
i.e., if the graded components of the exterior algebra with derivation formula_4
form a cochain complex:
Relation to de Rham complex.
Just as the exterior algebra (and tensor algebra) of vector fields on a manifold form a Lie algebra (over the base field "K"), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field "K"). 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 formula_6 (the error is the Lie derivative), nor is the exterior derivative: formula_7 (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 formula_8, but is also defined for formula_9.
The Lie algebra on the dual.
A Lie algebra structure on a vector space is a map formula_10 which is skew-symmetric, and satisfies the Jacobi identity. Equivalently, a map formula_11 that satisfies the Jacobi identity.
Dually, a Lie coalgebra structure on a vector space "E" is a linear map formula_12 which is antisymmetric (this means that it satisfies formula_13, where formula_14 is the canonical flip formula_15) and satisfies the so-called "cocycle condition" (also known as the "co-Leibniz rule")
Due to the antisymmetry condition, the map formula_12 can be also written as a map formula_1.
The dual of the Lie bracket of a Lie algebra formula_19 yields a map (the cocommutator)
where the isomorphism formula_21 holds in finite dimension; dually for the dual of Lie comultiplication. In this context, the Jacobi identity corresponds to the cocycle condition.
More explicitly, let "E" be a Lie coalgebra over a field of characteristic neither "2" nor "3". The dual space "E"* carries the structure of a bracket defined by
We show that this endows "E"* with a Lie bracket. It suffices to check the Jacobi identity. For any "x", "y", "z" ∈ "E"* and α ∈ "E", 
where the latter step follows from the standard identification of the dual of a wedge product with the wedge product of the duals. Finally, this gives
Since "d"2 = 0, 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 (algebraic topology)|cocycle]] condition "d"2 = 0 is in a sense dual to the Jacobi identity.
References.
[[Category:Coalgebras]]
[[Category:Lie algebras]]