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I'm having difficulties understanding what to do on a problem for my exercise class. The problem is as follows:
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Consider a $\mathbb{Z}_2$ graded algebra $\mathcal{A} = \mathcal{A}_0 \oplus \mathcal{A}_1$ together with its graded derivations $$ \delta \in \text{Der}_{\pm} (\mathcal{A}) : \; \mathcal{A}_i \longrightarrow \mathcal{A}_{i+\deg(\delta)} \quad ,$$ $$\delta(ab) \enspace = \enspace \delta(a)b + (-1)^{\deg(a)\deg(\delta)}a\delta(b) \quad , \qquad (\ast)$$ where $$\deg(a) \enspace = \enspace \begin{cases} 0 \; , \quad a \in \mathcal{A}_0 \\ 1 \; , \quad a \in \mathcal{A}_1 \end{cases}$$ and $\deg(\delta) = 0$ for derivations and $\deg(\delta) = 1$ for antiderivations.
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Show that the graded commutator $$\big[ \delta_1, \delta_2 \big]_{\pm} \enspace = \enspace \delta_1 \delta_2 - (-1)^{\deg(\delta_1)\deg(\delta_2)} \delta_1 \delta_2$$ defines again a graded derivation (with $\deg\big( [\delta_1, \delta_2]\big) = \deg(\delta_1) + \deg(\delta_2) \; \text{mod} \; 2$).
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I don't quite get the following: $\;$ My commutator $f(x,y) \equiv [x,y]$ needs two arguments. To show that a map is a graded derivation, I need to show that my map fullfills equation $(\ast)$. However, this equations is defined for a map having only one argument. What do I misunderstand? What do I need to do?
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You are required to prove that the commutator of two derivations is again a derivation, no that the commutator (seen itself as a antisymmetric bilinear map) is a derivation.