en>Headbomb: /* References */Various citation cleanup and WP:AWB general fixes using AWB

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References: Various citation cleanup and WP:AWB general fixes using AWB

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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==
Let ''E'' be a [[vector space]] over a [[field (mathematics)|field]] ''k'' equipped with a linear mapping <math>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 element]]s, <math>d(a \wedge b) = (da)\wedge b + (-1)^{\operatorname{deg} a} a \wedge(db)</math>) of degree 1 on the [[exterior algebra]] of ''E'':
:<math>d\colon \bigwedge^\bullet E\rightarrow \bigwedge^{\bullet+1} E.</math>
Then the pair (''E'', ''d'') is said to be a Lie coalgebra if ''d''2 = 0,
i.e., if the graded components of the [[exterior algebra]] with derivation <math>(\bigwedge^* E, d)</math>
form a [[cochain complex]]:
:<math>E\ \rightarrow^{\!\!\!\!\!\!d}\ E\wedge E\ \rightarrow^{\!\!\!\!\!\!d}\ \bigwedge^3 E\rightarrow^{\!\!\!\!\!\!d}\ \dots</math>

===Relation to de Rham complex===
Just as the exterior algebra (and tensor algebra) of [[vector field]]s 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>C^\infty(M)</math> (the error is the [[Lie derivative]]), nor is the [[exterior derivative]]: <math>d(fg) = (df)g + f(dg) \neq f(dg)</math> (it is a derivation, not linear over functions): they are not [[tensor]]s. 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>\Omega^1 \to \Omega^2</math>, but is also defined for <math>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>[\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>[\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>d\colon E \to E \otimes E</math> which is antisymmetric (this means that it satisfies <math> \tau \circ d = -d </math>, where <math> \tau </math> is the canonical flip <math> E \otimes E \to E \otimes E </math>) and satisfies the so-called ''cocycle condition'' (also known as the ''co-Leibniz rule'')

:<math> \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>d\colon E \to E \otimes E</math> can be also written as a map <math>d\colon E \to E \wedge E</math>.

The dual of the Lie bracket of a Lie algebra <math> \mathfrak g </math> yields a map (the cocommutator)
:<math>[\cdot,\cdot]^*\colon \mathfrak{g}^* \to (\mathfrak{g} \wedge \mathfrak{g})^* \cong \mathfrak{g}^* \wedge \mathfrak{g}^*</math>
where the isomorphism <math>\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'']) = ''d''α(''x''&and;''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>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),</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>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 ''d''2 = 0, it follows that
:<math>\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 ''d''2 = 0 is in a sense dual to the Jacobi identity.

==References==

*{{Citation | last1=Michaelis | first1=Walter | title=Lie coalgebras | url=http://dx.doi.org/10.1016/0001-8708(80)90056-0 | doi=10.1016/0001-8708(80)90056-0 | id={{MR|594993}} | year=1980 | journal=Advances in Mathematics | issn=0001-8708 | volume=38 | issue=1 | pages=1–54}}

[[Category:Coalgebras]]
[[Category:Lie algebras]]

Extensible automorphism - Revision history

en>Headbomb: /* References */Various citation cleanup and WP:AWB general fixes using AWB