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Lie coalgebra

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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

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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:

Relation to de Rham complex

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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 .

The Lie algebra on the dual

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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

, for any , , , and .

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.

References

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  • Michaelis, Walter (1980), "Lie coalgebras", Advances in Mathematics, 38 (1): 1–54, doi:10.1016/0001-8708(80)90056-0, ISSN 0001-8708, MR 0594993