---- > [!definition] Definition. ([[dual representation]]) > If $(V, \rho: \mathfrak{g} \to \mathfrak{gl}(V))$ is a [[Lie algebra representation]] of a [[Lie algebra]] $\mathfrak{g}$, then the [[dual vector space|dual space]] $V^{*}=\text{Hom}_{\mathbb{F}}(V,\mathbb{F})$ is a $\mathfrak{g}$-representation via the action $x \cdot \varphi := -\varphi(x \cdot \_)=-(\varphi \circ \rho(x)=-\big(\rho(x)\big)^{*},$ where $\big( \rho(x) \big)^{*}:V^{*} \to V^{*}$ is the [[dual map]] of $\rho(x):V \to V$. ^definition > [!intuition] > Why the minus sign? > 1. Without it, we do not get a $\mathfrak{g}$-representation. > 2. In the group case, you have an inverse... and 'the [[derivative]] of inverse is minus'. ^intuition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```