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 $L_X$ of degree $0$ and $i_X$ of degree $-1$ for all $X\in \mathfrak{g}$, satisfying the relations of contractions, Lie derivative and differential on a manifold with a $\mathfrak{g}$-action:
- $[i_{X}, i_{Y}] = 0$
- $[L_X,i_{Y}] = i_{[X,Y]}$
- $[d, L_\xi] = 0$
- $[L_X,L_Y] = L_{[X,Y]}$
- $[d, d] = 0$
- $[d,i_X] = L_X$
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:
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.