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Let $\mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $\mathfrak{g}$ defines a Set-valued functor on differential graded commutative algebras (also, with some finiteness and boundedness assumption) mapping each cgda $\Omega$ to the set $MC(\Omega\otimes \mathfrak{g})$ of Maurer-Cartan elements of the dgla $\Omega\otimes \mathfrak{g}$ (with the natural dgla structure on the tensor product of a dgla with a cdga). It is well known that this functor is representable: $MC(\Omega\otimes \mathfrak{g})\cong Hom_{dgca}(CE(\mathfrak{g}),\Omega)$, where the Chevalley-Eilenberg dgca $CE(\mathfrak{g})$ is the free graded commutative algebra on the shifted linear dual of $\mathfrak{g}$ endowed with the differential induced by the dgla structure on $\mathfrak{g}$.

If one looks at the category of commutative graded algebras instead (i.e., one forgets the differential), then $CE(\mathfrak{g})$ represents the functor $\Omega\mapsto (\Omega\otimes\mathfrak{g})^1$.

My question is: is this latter functor representable also in the category of differential graded commutative algebras? if yes, by which dgca?

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up vote 5 down vote accepted

By the Weil algebra of $\mathfrak{g}$.

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