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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?

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$[x, y]$ 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.

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  • $\begingroup$ 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 $\Omega^{\bullet}(M)$... $\endgroup$ – Praphulla Koushik Sep 16 at 4:35
  • $\begingroup$ Yes, that’s what this definition is intended to abstract. $\endgroup$ – Qiaochu Yuan Sep 16 at 5:21
  • $\begingroup$ 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... $\endgroup$ – Praphulla Koushik Sep 16 at 5:36
  • $\begingroup$ can you suggest some references $\endgroup$ – Praphulla Koushik Sep 22 at 16:33

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