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 ${\displaystyle E}$ be a vector space over a field ${\displaystyle \mathbb {k} }$ equipped with a linear mapping ${\displaystyle d\colon E\to E\wedge E}$ from ${\displaystyle E}$ to the exterior product of ${\displaystyle E}$ with itself. It is possible to extend ${\displaystyle d}$ uniquely to a graded derivation (this means that, for any ${\displaystyle a,b\in E}$ which are homogeneous elements, ${\displaystyle d(a\wedge b)=(da)\wedge b+(-1)^{\deg a}a\wedge (db)}$) of degree 1 on the exterior algebra of ${\displaystyle E}$:

${\displaystyle d\colon \bigwedge ^{\bullet }E\rightarrow \bigwedge ^{\bullet +1}E.}$

Then the pair ${\displaystyle (E,d)}$ is said to be a Lie coalgebra if ${\displaystyle d^{2}=0}$, i.e., if the graded components of the exterior algebra with derivation ${\textstyle (\bigwedge ^{*}E,d)}$ form a cochain complex:

${\displaystyle E\ \xrightarrow {d} \ E\wedge E\ \xrightarrow {d} \ \bigwedge ^{3}E\xrightarrow {d} \ \cdots }$

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 ${\displaystyle \mathbb {k} }$), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field ${\displaystyle \mathbb {k} }$). 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 ${\displaystyle C^{\infty }(M)}$ (the error is the Lie derivative), nor is the exterior derivative: ${\displaystyle d(fg)=(df)g+f(dg)\neq f(dg)}$ (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 ${\displaystyle \Omega ^{1}\to \Omega ^{2}}$, but is also defined for ${\displaystyle C^{\infty }(M)\to \Omega ^{1}(M)}$.

The Lie algebra on the dual

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

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

${\displaystyle \left(d\otimes \mathrm {id} \right)\circ d=\left(\mathrm {id} \otimes d\right)\circ d+\left(\mathrm {id} \otimes \tau \right)\circ \left(d\otimes \mathrm {id} \right)\circ d}$.

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

The dual of the Lie bracket of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ yields a map (the cocommutator)

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

where the isomorphism ${\displaystyle \cong }$ 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 ${\displaystyle E}$ be a Lie coalgebra over a field of characteristic neither 2 nor 3. The dual space ${\displaystyle E^{*}}$ carries the structure of a bracket defined by

${\displaystyle \alpha ([x,y])=d\alpha (x\wedge y)}$, for all ${\displaystyle \alpha \in E}$ and ${\displaystyle x,y\in E^{*}}$.

We show that this endows ${\displaystyle E^{*}}$ with a Lie bracket. It suffices to check the Jacobi identity. For any ${\displaystyle x,y,z\in E^{*}}$ and ${\displaystyle \alpha \in E}$,

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

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

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

Since ${\displaystyle d^{2}=0}$, it follows that

${\displaystyle \alpha ([[x,y],z]+[[y,z],x]+[[z,x],y])=0}$, for any ${\displaystyle \alpha }$, ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$.

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 ${\displaystyle d^{2}=0}$ 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, MR   0594993