nLab Chevalley-Eilenberg algebra in synthetic differential geometry

Urs Schreiber. This is something I thought about in the context of my discussion at Chevalley-Eilenberg algebra.

Idea

We give a description of the Chevalley-Eilenberg algebra of the Lie algebra of a Lie group as the ∞-quantity of functions $C^\infty(\mathbf{B}G_e^{(1)})$ on the simplicial space of infinitesimal neighbourhoods of the identity in the sense of synthetic differential geometry in the simplicial smooth space $\mathbf{B}G$ that is the Lie ∞-groupoid incarnation of the delooping of the Lie group.

The derivation is analogous to and usefully compared with how the deRham algebra of differential forms on a manifold $X$ is the ∞-quantity of functions on the infinitesimal singular simplicial complex $X^{\Delta^\bullet_{diff}} \hookrightarrow \Pi(X)$ of $X$, as described at differential forms in synthetic differential geometry.

We proceed entirely by using theorems and propositions from the book

• Anders Kock, Synthetic Geometry of Manifolds (pdf)

in particular section 6.8 combined with section 4.3. We effectively show that these statements are precisely the ones needed to unwrap what the normalized Moore cochain complex of the cosimplicial algebra $C^\infty(\mathbf{B} G_e^{(1)})$ in the monoidal Dold-Kan correspondence is like.

Definitions and setup

Let $G$ be a Lie group (by which we mean a finite dimensional Lie group). Write $\mathbf{B}G$ for the simplicial smooth space which in degree $k$ is the cartesian product $G^k$ with the standard face and degeneray map (see the examples at nerve for details).

Let $T$ be some topos that models the axioms of synthetic differential geometry and which has a full and faithful embedding Diff $\hookrightarrow T$.

Consider then $\mathbf{B}G$ as a simplicial object in $T$. As usual, we shall call objects in $T$ spaces in the following.

Let $G_e^{(1)} \hookrightarrow G$ be the space that is the first infinitesimal neighbourhood of the neutral element $e$ in $G$. By definition this space is ismorphic to the infinitesimal space

$$D(n) = \{(d_1, \cdots, d_n) \in R^n | \forall i,j : d_i d_j = 0\}$$

for $n$ the dimension of $D$. By the log-exp bijection in synthetic differential geometry? this space is canonically identified with the vector space $g := Lie(G)$ underlying the Lie algebra of $G$.

Moreover, by the Kock-Lawvere axiom morphisms $f : D(n) \simeq G_e^{(1)} \to R$ are necessarily linear $f : d \mapsto f_0 \to f_1 \cdot d$, hence under the $log$–$exp$ bijection are nothing but elements in the dual vector space $g^*$.

Recall that the ordinary Chevalley-Eilenberg algebra of $g$ is the differential graded algebra whose underlying graded-commutative algebra is the Grassmann algebra $\wedge^\bullet g^* = \mathbb{R} \oplus g^* \oplus g^* \wedge g^* \oplus \cdots$.

So the subset of $C^\infty(G_e^{(1)})$ that vanishes at 0 is naturally isomorphic to the degree-$1$ part of the Chevalley–Eilenberg algebra.

Notice now that the multiplication on the group $G$ does not restrict to a multiplication on $G_e^{(1)}$ because the sum $d_1 + d_2$ of two elements that each square to 0 is does in general not square to 0, – $(d_1 + d_2)^2 = 2 d_1 d_2$ but only its cube $(d_1 + d_2)^3 = 0$ does. Therefore the group multiplication induces a composition

$$\cdot : G_e^{(1)}\times G_2^{(1)} \to G_e^{(2)} \,.$$

Consider therefore the space of “infinitesimal $1$-cells of $\mathbf{B}G$ whose composite is again an infinitesimal $1$-cell”, i.e. the pullback

$$\array{ (G \times G)_e^{(1)} &\to& G_e^{(1)} \times G_e^{(1)} \\ \downarrow && \downarrow^{\cdot} \\ G_e^{(1)} &\hookrightarrow& G_e^{(2)} } \,.$$

By item 3) of theorem (6.8.1) this pullback is precisely the space of elements $(x,y) \in G_e^{(1)} \times G_e^{(1)}$ such that not only $x$ and $y$ are infinitesimal neighbours of the neutral element $e$, but also of each other.

By the Kock-Lawvere axiom (entirely analogous to the similar step in the derivation of simplicial differential forms in synthetic differential geometry) it should follow from this that maps $(G \times G)_e^{(1)} \to R$ that vanish on degenerate elements are in bijection with antisymmetric maps that are canonically identified with elements in $g^* \wedge_{\mathbb{R}} g^*$ (need to say this in more detail…)

Continuing in this manner (…details for higher degrees to be filled in…) we define the simplicial space

$$\mathbf{B}G_e^{(1)} := ( \cdots (G \times G)_e^{{1}} \stackrel{\to}{\stackrel{\to}{\to}} G_e^{(1)} \stackrel{\to}{\to} {*} ) \,.$$

The normalized Moore cochain complex $N^\bullet(C^\infty(\mathbf{B}G_e^{(1)}))$ of the cosimplicial algebra

$$C^\infty(\mathbf{B}G_e^{(1)}) := ( \cdots C^\infty((G \times G)_e^{{1}}) \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} C^\infty(G_e^{(1)}) \stackrel{\leftarrow}{\leftarrow} {*} )$$

is in degree $k$ given by the kernel of the joint degeneracy maps. As in the discussion at differential forms in synthetic differential geometry this picks out the functions that vanish on degenerate simplices. So from the above we get

$$N^\bullet(C^\infty(\mathbf{B}G_e^{(1)})) = (\cdots g^* \wedge g^* \stackrel{d = \sum_i (-1)^i d_i}{\leftarrow} g^* \stackrel{0}{\leftarrow} \mathbb{R} ) \,.$$

Recall that the differential of the Chevalley-Eilenberg algebra is on $g^*$ just the dual $[-,-]^* : g^* \to g^* \wedge g^*$ of the Lie bracket $[-,-] : g \otimes g \to g$.

We need to check that this is reproduced by the differential of the Moore cochain complex, which is the alternating sum of the face maps $d = \sum_i (-1)^i d_i$. Let $f \in g^*$. Then we find for all $(x,y) \in (G \times G)_e^{(1)}$ that

$$(d f) (x,y) = f(p_1(x,y)) - f(x \cdot y) + f(p_2(x,y)) = f (x) + f(y) - f(x \cdot y) \,.$$

Now we use the crucial formula (6.8.2) from Anders Kock’s book, which says that the group product on the infinitesimal elements $x,y$ is given by

$$x \cdot y = x + y + \frac{1}{2}\{x,y\} \,,$$

where the last term is the group commutator

$$\{x,y\}:= x \cdot y \cdot x^{-1} \cdot y^{-1} \,.$$

So this is the term that remains in the formula for $d f$:

$$(d f)(x,y) = -\frac{1}{2} f(\{x,y\}) \,.$$

On that we apply theorem 6.6.1 of Kock’s book, which says (in its third item) that under the $log$–$exp$ bijection by which we identified the infinitesimal neighbourhood $G_e^{(1)}$ (and functions on it) with the tangent space $T_e(G)$ (and linear functions on it) the group commutator maps to the Lie algebra commutator. So indeed under the identification of $f$ with an element in $g^*$ we find

$$d f = [-,-]^* f \,.$$

This is indeed the differential of the Chevalley-Eilenberg algebra.

(discussion needs to be completed: situation in higher degree and cup-product mapping to wedge product needs to be discussed…)