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.

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Definition

Let E be a vector space over a field k equipped with a linear mapping d\colon E \to E \wedge E from E to the exterior product of E with itself. It is possible to extend d uniquely to a graded derivation[1] of degree 1 on the exterior algebra of E:

d\colon \bigwedge^\bullet E\rightarrow \bigwedge^{\bullet%2B1} 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 (\bigwedge^* E, d) form a cochain complex:

E\ \rightarrow^{\!\!\!\!\!\!d}\ E\wedge E\ \rightarrow^{\!\!\!\!\!\!d}\ \bigwedge^3 E\rightarrow^{\!\!\!\!\!\!d}\ \dots

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^\infty(M) (the error is the Lie derivative), nor is the exterior derivative: d(fg) = (df)g %2B f(dg) \neq f(dg) (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 \Omega^1 \to \Omega^2, but is also defined for C^\infty(M) \to \Omega^1(M).

The Lie algebra on the dual

A Lie algebra structure on a vector space is a map [\cdot,\cdot]\colon \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g} which is skew-symmetric, and satisfies the Jacobi identity. Equivalently, a map [\cdot,\cdot]\colon \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g} that satisfies the Jacobi identity.

Dually, a Lie coalgebra structure on a vector space is a map d\colon E \to E \wedge E which satisfies the cocycle condition. The dual of the Lie bracket yields a map (the cocommutator)

[\cdot,\cdot]^*\colon \mathfrak{g}^* \to (\mathfrak{g} \wedge \mathfrak{g})^* \cong \mathfrak{g}^* \wedge \mathfrak{g}^*

where the isomorphism \cong 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([x, y]) = 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,

d^2\alpha (x\wedge y\wedge z) = \frac{1}{3} d^2\alpha(x\wedge y\wedge z %2B y\wedge z\wedge x %2B z\wedge x\wedge y) = \frac{1}{3} \left(d\alpha([x, y]\wedge z) %2B d\alpha([y, z]\wedge x) %2Bd\alpha([z, x]\wedge y)\right),

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

d^2\alpha (x\wedge y\wedge z) = \frac{1}{3} \left(\alpha([[x, y], z]) %2B \alpha([[y, z], x])%2B\alpha([[z, x], y])\right).

Since d2 = 0, it follows that

\alpha([[x, y], z] %2B [[y, z], x] %2B [[z, x], y]) = 0, 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.

Notes

  1. ^ This means that, for any a, b ∈ E which are homogeneous elements, d(a \wedge b) = (da)\wedge b %2B (-1)^{\operatorname{deg} a} a \wedge(db).