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 from E to the exterior product of E with itself. It is possible to extend d uniquely to a graded derivation of degree 1 on the exterior algebra of E:
Then the pair is said to be a Lie coalgebra if d2 = 0,
i.e., if the graded components of the exterior algebra with derivation
form a cochain complex:

Relation to de Rham complex

Just as the exterior algebra of vector fields on a manifold form a Lie algebra, the de Rham complex of differential forms on a manifold form a Lie coalgebra. 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 , nor is the exterior derivative: : 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.

The Lie algebra on the dual

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 and satisfies the so-called cocycle condition
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
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 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 d2 = 0, it follows that
Thus, by the double-duality isomorphism, the Jacobi identity is satisfied.
In particular, note that this proof demonstrates that the cocycle condition d2 = 0 is in a sense dual to the Jacobi identity.