# Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003).

In page $$45$$, definition $$5.2$$, author introduce the notion of $$\mathfrak{g}$$-differential graded algebra. Here $$\mathfrak{g}$$ is a Lie algebra ( I am thinking it is Lie algebra of Lie group $$G$$ but not sure)

A differential graded algebra is an graded algebra $$A = \bigoplus_{I=0}^{\infty}A_i$$ with a differential $$d$$ of degree $$1$$, such that $$d$$ is a derivation. It is called a $$\mathfrak{g}$$-differential algebra if, in addition, there are derivations $$L_\xi$$ of degree $$0$$ and $$i_\xi$$ of degree $$-1$$ for all $$\xi\in \mathfrak{g}$$, satisfying the relations of contractions, Lie derivative and differential on a manifold with a $$\mathfrak{g}$$-action:

1. $$[i_{\xi}, i_{\xi ‘}] = 0$$
2. $$[L_\xi,i_{\xi ‘}] = i_{[\xi,\xi’]}$$
3. $$[d, L_\xi] = 0$$
4. $$[L_\xi,L_{\xi ‘}] = L_{[\xi,\xi ‘]}$$
5. $$[d, d] = 0$$
6. $$[d,i_\xi] = L_\xi$$

I don’t understand the notation of $$[d,d]=0$$, what Lie algebra structure are we fixing here? The same confusion for all other conditions. Is this the standard way to define a$$\mathfrak{g}$$-differential graded algebra?

$$[x, y]$$ here is the super or graded commutator (in the graded algebra of graded endomorphisms of $$A$$), which you may not be used to; it refers to

$$xy - (-1)^{\deg x \deg y} yx$$

and so if either $$x$$ or $$y$$ is even it reduces to the ordinary commutator but if $$x, y$$ are both odd it's the anticommutator; in particular $$[d, d] = 2d^2$$ because $$d$$ is odd.

I've never seen this definition before but the motivation is spelled out pretty clearly:

"satisfying the relations of contractions, Lie derivative and differential on a manifold with a $$\mathfrak{g}$$-action"

presumably means this definition is intended to abstract the action of $$\mathfrak{g}$$ on differential forms $$\Omega^{\bullet}(M)$$ coming from a smooth action of a corresponding Lie group $$G$$ on $$M$$.

$$L_{(-)}$$ and $$i_{(-)}$$ are presumably also intended to be linear although this isn't stated explicitly.

• I just want to confirm what you mean by action of $\mathfrak{g}$ on $\Omega^{\bullet}(M)$... Given $A\in \mathfrak{g}$, action of $G$ on $M$ gives a vector field $A^*$ on $M$... Given a vector field $X$ on $M$ and a differential form $\omega$ on $M$, I know two ways to associate another differential form.. One way is to consider $L_X\omega$ (degree does not change in this case). another is to consider the $i_X\omega$ (degree reduce by $1$ in this case)... Is this what you are referring to as actions on $\mathfrak{g}$ on $\Omega^{\bullet}(M)$... – Praphulla Koushik Sep 16 at 4:35
• Yes, that’s what this definition is intended to abstract. – Qiaochu Yuan Sep 16 at 5:21
• Thanks for confirmation... Do you happen to know any references for these... In particular, for "graded algebra of graded endomorphisms of A".. I know the definition but not much properties... – Praphulla Koushik Sep 16 at 5:36
• can you suggest some references – Praphulla Koushik Sep 22 at 16:33