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 {\displaystyle E} be a vector space over a field k {\displaystyle \mathbb {k} } equipped with a linear mapping d : E → E ∧ E {\displaystyle d\colon E\to E\wedge E} from E {\displaystyle E} to the exterior product of E {\displaystyle E} with itself. It is possible to extend d {\displaystyle d} uniquely to a graded derivation (this means that, for any a , b ∈ E {\displaystyle a,b\in E} which are homogeneous elements, d ( a ∧ b ) = ( d a ) ∧ b + ( − 1 ) deg a a ∧ ( d b ) {\displaystyle d(a\wedge b)=(da)\wedge b+(-1)^{\deg a}a\wedge (db)} ) of degree 1 on the exterior algebra of E {\displaystyle E} :
d : ⋀ ∙ E → ⋀ ∙ + 1 E . {\displaystyle d\colon \bigwedge ^{\bullet }E\rightarrow \bigwedge ^{\bullet +1}E.}
Then the pair ( E , d ) {\displaystyle (E,d)} is said to be a Lie coalgebra if d 2 = 0 {\displaystyle d^{2}=0} , i.e., if the graded components of the exterior algebra with derivation ( ⋀ ∗ E , d ) {\textstyle (\bigwedge ^{*}E,d)} form a cochain complex:
E → d E ∧ E → d ⋀ 3 E → d ⋯ {\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 k {\displaystyle \mathbb {k} } ), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field k {\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 C ∞ ( M ) {\displaystyle C^{\infty }(M)} (the error is the Lie derivative), nor is the exterior derivative: d ( f g ) = ( d f ) g + f ( d g ) ≠ f ( d g ) {\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 Ω 1 → Ω 2 {\displaystyle \Omega ^{1}\to \Omega ^{2}} , but is also defined for C ∞ ( M ) → Ω 1 ( M ) {\displaystyle C^{\infty }(M)\to \Omega ^{1}(M)} .
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