Exercise 25, § 1

Lie Groups and Lie Algebras, IV

Let $I$ be a set and let $\mathscr{F}$ be the set of finite subsets of $I$ . For any $A \in \mathscr{F}$ , put $\varepsilon(A) = (-1)^{\mathrm{Card}(A)}$ . Let $G$ be an abelian group written additively, and let $\varphi$ and $\psi$ be two maps from $\mathscr{F}$ to $G$ . Show that the following properties are equivalent:

\varphi(A) = \sum_{B \subset A} \psi(B) \quad \text{for all } A \in \mathscr{F}; \tag{i}

\psi(A) = \sum_{B \subset A} \varepsilon(A - B) \varphi(B) \quad \text{for all } A \in \mathscr{F}. \tag{ii}

Solution

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