Lie Coalgebra - The Lie Algebra On The Dual

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 (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 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

α = dα(x∧y), for all α ∈ E and x,y ∈ E*.

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

, for any α, x, y, and z.

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.