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.

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

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