What is the (Koszul? derived?) interpretation of a pair of Lie algebras with the same cohomology?

Question

There are many words and sentences in mathematics that I basically completely don't understand, including the words "Koszul" and "derived". But rather than ask for a complete description of such words, I will ask about a particular example, and hope that an MOer can spell out concretely what's going on. (Of course, links to good write-ups of Koszul, etc., are always welcome.)

There is a relatively easy theorem, probably due to Chevalley and Eilenberg, with many generalizations due to Koszul:

Let $L$ be a (finite-dimensional) vector space (in characteristic $0$); let $[1] L^*$ denote the dual vector space, "shifted" to a graded vector space supported in degree $1$; and let $([1]L^*)^{\vee \bullet}$ denote the free graded-commutative algebra generated by $[1]L^*$ (I impose the Koszul rule for signs, so that as an algebra $([1]L^*)^{\vee \bullet} = (L^*)^{\wedge\bullet}$ is classically an alternating algebra). Then to give a Lie algebra structure to $L$ is the same as giving to $([1]L^*)^{\vee \bullet}$ a square-zero degree-$1$ derivation.

The construction is (contravariantly) functorial and full and faithful, and so embeds the category of Lie algebras fully-faithfully into the (opposite to the) category of commutative dgas.

Here are two examples. I will call my characteristic-$0$ field $\mathbb R$. The one-dimensional Lie algebra corresponds to the dga $\mathbb R \overset 0 \to \mathbb R \xi$, where $\xi$ is the coordinate function on the one-dimensional vector space, and the algebra is graded-commutative, so $\xi^2 = 0$. The two-dimensional non-commutative Lie algebra corresponds to the dga $\mathbb R \overset 0 \to (\mathbb R \xi \oplus \mathbb R \upsilon) \overset{\xi \mapsto 0, \, \upsilon \mapsto \xi\upsilon}\longrightarrow \mathbb R\xi\upsilon$.

Now, once we're in the land of dgas, it makes sense to talk about their (co?)homology. In the above examples, the cohomology of the one- and two-dimensional Lie algebras are isomorphic as algebras; the isomorphism on cohomology is given in one direction by the "abelianization" map from the two-dimensional Lie algebra to the one-dimensional Lie algebra.

Question: How should I interpret "geometrically" the fact that the cohomologies agree?

An idea that I've heard is that somehow dgas-up-to-? should correspond to some notion of "stack". Now, it is certainly not the case that "point mod one-dimensional" and "point mod two-dimensional" present the same stack. So that's not quite what's going on. Perhaps the problem is that these algebras are not the same as ${\rm A}_\infty$ algebras; we should probably add the word "${\rm A}_\infty$" to the list of words I don't really know. If this is the answer, I hope that some MOer will spell it out.

Another idea that I've heard is that the passage "Lie algebra to dga to cohomology" remembers exactly the "derived category of representations of the Lie algebra". Again, I don't really know what that is, but that's OK. Somehow, a representation of a Lie algebra should be a "quasicoherent sheaf on point mod the Lie algebra", and I know that people like "derived categories of quasicoherent sheaves". So should I understand Koszul duality as remembering not all the data of some "stack-like object" but just some "derived data"? If so, again I hope that some MOer will spell it out pedantically in this example.

Answer 1

I know, basically, two answers to this question. The first one starts from the observation that what you call "a relatively easy theorem" generalizes to infinite-dimensional Lie algebras $L$ by replacing the exterior algebra generated by $L^\ast$ with the exterior coalgebra cogenerated by $L$. Then the construction becomes covariant, but produces a (conilpotent) DG-coalgebra rather than a DG-algebra.

Now it is just a fact that the natural equivalence relation on DG-coalgebras is more delicate than the quasi-isomorphism. There is a certain class of "filtered quasi-isomorphisms" that one is supposed to invert when dealing with (conilpotent) DG-coalgebras. The map of Chevalley-Eilenberg complexes that you describe, when considered as a map of the homological, rather than cohomological, Chevalley-Eilenberg complexes, is not a filtered quasi-isomorphism of DG-coalgebras and cannot be obtained from such using compositions and fractions. This saves the derived Koszul duality as an equivalence between appropriate localizations of the categories of (augmented) DG-algebras and (conilpotent) DG-coalgebras.

The second answer describes what happens if one insists on considering the cohomological Chevalley-Eilenberg complex of a finite-dimensional Lie algebra, viewed as a DG-algebra up to a quasi-isomorphism. One can also apply the derived Koszul duality to it, obtaining a certain DG-coalgebra. This DG-coalgebra is, basically, the k-linear dual to the complete topological DG-algebra of derived adic completion of the enveloping algebra U(L) with respect to its augmentation ideal. So if a morphism of finite-dimensional Lie algebras induces an isomorphism of cohomology, this should be understood to mean that it induces an isomorphism of the derived adic completions of the enveloping algebras at their augmentation ideals. Admittedly, this is not a geometric interpretation.

EDIT: I have been asked in a comment what a filtered quasi-isomorphism of DG-coalgebras is, so let me add an explanation of this to the answer. This notion does not presume a fixed filtration on a DG-coalgebra, but rather a class of admissible filtrations. We consider coaugmented DG-coalgebras, i.e., the basic field $k$ is embedded as a subcoalgebra into a DG-coalgebra $C$ and its image is annihilated by the differential. An increasing filtration $F$ on $C$ is admissible if $F_0C=0$, $F_1C=k$, the components $F_iC$ are preserved by the differential, and the filtration $F$ is compatible with the comultiplication.

Not every coalgebra admits an admissible filtration at all! A coalgebra is called conilpotent if it has an admissible filtration (just as a coalgebra, let us drop the condition of compatibility with the differential for a moment). In this case, it has a certain maximal admissible filtration (which is preserved by the differential automatically). However, filtered quasi-isomorphisms of conilpotent DG-coalgebras are defined in terms of arbitrary admissible filtrations. A morphism of DG-coalgebras is a filtered quasi-isomorphism if admissible filtrations can be chosen on its source and target so that the morphism is compatible with them and is a quasi-isomorphism on all filtration components.

Answer 2

You (and "everyone") may already know this, but there is a really simple answer to a variant of your question, that is when instead of starting with a Lie algebra you start with a connected differential graded Lie algebra over a field of characteristic zero. In that case, Quillen's work on rational homotopy theory says that data determines a rational homotopy type, and Lie algebra cohomology is the cohomology of that homotopy type. So that answer to your question is that two (connected differential graded) Lie algebras with the same cohomology have the same kind of relationship to one another as two spaces which have isomorphic cohomology rings... which is not necessarily much unless you know something more about them. To some small extent Lie algebras when viewed as differential graded Lie algebras in degree zero (with trivial differential) are analogous to K(G,1)'s, but general theory says little there, so I don't think this addresses your main question.

Those interested in an elementary, expository topologist's view of Koszul duality might want to look at this.