Lie coalgebra

Definition

Let E be a vector space over a field k equipped with a linear mapping d : E → E ∧ E 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, d ( a ∧ b ) = ( d a ) ∧ b + ( − 1 ) deg ⁡ a a ∧ ( d b ) ) of degree 1 on the exterior algebra of E:

Then the pair (E, d) is said to be a Lie coalgebra if d² = 0, i.e., if the graded components of the exterior algebra with derivation ( ⋀ ∗ E , d ) 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 C ∞ ( M ) (the error is the Lie derivative), nor is the exterior derivative: d ( f g ) = ( d f ) g + f ( d g ) ≠ f ( d g ) (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 Ω 1 → Ω 2 , but is also defined for C ∞ ( M ) → Ω 1 ( M ) .

The Lie algebra on the dual

A Lie algebra structure on a vector space is a map [ ⋅ , ⋅ ] : g × g → g which is skew-symmetric, and satisfies the Jacobi identity. Equivalently, a map [ ⋅ , ⋅ ] : g ∧ g → g that satisfies the Jacobi identity.

Dually, a Lie coalgebra structure on a vector space E is a linear map d : E → E ⊗ E which is antisymmetric (this means that it satisfies τ ∘ d = − d , where τ is the canonical flip E ⊗ E → E ⊗ E ) and satisfies the so-called cocycle condition (also known as the co-Leibniz rule)

Due to the antisymmetry condition, the map d : E → E ⊗ E can be also written as a map d : E → E ∧ E .

The dual of the Lie bracket of a Lie algebra g 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 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 (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 d² = 0 is in a sense dual to the Jacobi identity.