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 [math]\displaystyle{ d\colon E \to E \wedge E }[/math] 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, [math]\displaystyle{ d(a \wedge b) = (da)\wedge b + (-1)^{\deg a} a \wedge(db) }[/math]) of degree 1 on the exterior algebra of E:

[math]\displaystyle{ d\colon \bigwedge^\bullet E\rightarrow \bigwedge^{\bullet+1} E. }[/math]

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 [math]\displaystyle{ (\bigwedge^* E, d) }[/math] form a cochain complex:

[math]\displaystyle{ E\ \xrightarrow{d}\ E\wedge E\ \xrightarrow{d}\ \bigwedge^3 E\xrightarrow{d}\ \cdots }[/math]

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 [math]\displaystyle{ C^\infty(M) }[/math] (the error is the Lie derivative), nor is the exterior derivative: [math]\displaystyle{ d(fg) = (df)g + f(dg) \neq f(dg) }[/math] (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 [math]\displaystyle{ \Omega^1 \to \Omega^2 }[/math], but is also defined for [math]\displaystyle{ C^\infty(M) \to \Omega^1(M) }[/math].

The Lie algebra on the dual

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

Dually, a Lie coalgebra structure on a vector space E is a linear map [math]\displaystyle{ d\colon E \to E \otimes E }[/math] which is antisymmetric (this means that it satisfies [math]\displaystyle{ \tau \circ d = -d }[/math], where [math]\displaystyle{ \tau }[/math] is the canonical flip [math]\displaystyle{ E \otimes E \to E \otimes E }[/math]) and satisfies the so-called cocycle condition (also known as the co-Leibniz rule)

[math]\displaystyle{ \left(d\otimes \mathrm{id}\right)\circ d = \left(\mathrm{id}\otimes d\right)\circ d+\left(\mathrm{id} \otimes \tau\right)\circ\left(d\otimes \mathrm{id}\right)\circ d }[/math].

Due to the antisymmetry condition, the map [math]\displaystyle{ d\colon E \to E \otimes E }[/math] can be also written as a map [math]\displaystyle{ d\colon E \to E \wedge E }[/math].

The dual of the Lie bracket of a Lie algebra [math]\displaystyle{ \mathfrak g }[/math] yields a map (the cocommutator)

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

where the isomorphism [math]\displaystyle{ \cong }[/math] 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,

[math]\displaystyle{ \begin{align} d^2\alpha (x\wedge y\wedge z) &= \frac{1}{3} d^2\alpha(x\wedge y\wedge z + y\wedge z\wedge x + z\wedge x\wedge y) \\ &= \frac{1}{3} \left(d\alpha([x, y]\wedge z) + d\alpha([y, z]\wedge x) +d\alpha([z, x]\wedge y)\right), \end{align} }[/math]

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

[math]\displaystyle{ d^2\alpha (x\wedge y\wedge z) = \frac{1}{3} \left(\alpha([[x, y], z]) + \alpha([[y, z], x])+\alpha([[z, x], y])\right). }[/math]

Since d² = 0, it follows that

[math]\displaystyle{ \alpha([[x, y], z] + [[y, z], x] + [[z, x], y]) = 0 }[/math], 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.

References

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