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.
Let ' be a vector space over a field ' equipped with a linear mapping from ' to the exterior product of ' with itself. It is possible to extend ' uniquely to a graded derivation (this means that, for any ' which are homogeneous elements, ) of degree 1 on the exterior algebra of ':
Then the pair ' is said to be a Lie coalgebra if ', 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 (the error is the Lie derivative), nor is the exterior derivative: (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 , but is also defined for .
A Lie algebra structure on a vector space is a map which is skew-symmetric, and satisfies the Jacobi identity. Equivalently, a map 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 , where is the canonical flip ) and satisfies the so-called cocycle condition (also known as the co-Leibniz rule)
Due to the antisymmetry condition, the map can be also written as a map .
The dual of the Lie bracket of a Lie algebra 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.
More explicitly, let ' be a Lie coalgebra over a field of characteristic neither 2 nor 3. The dual space ' carries the structure of a bracket defined by
, for all ' and '.
We show that this endows ' with a Lie bracket. It suffices to check the Jacobi identity. For any ' and ',
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 ', 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.