---- Let $\mathfrak{g}$ be a [[Lie algebra]] over a [[field|field]] $\mathbb{F}$. > [!definition] Definition. ([[trivial Lie algebra representation]]) > Consider the one-dimensional [[general linear Lie algebra]] $\mathfrak{gl}(\mathbb{F})$. The **trivial representation** of $\mathfrak{g}$ is given by choosing $\rho: \mathfrak{g} \to \mathfrak{gl}_{1}(\mathbb{F})=\text{End}(\mathbb{F})$ to be the zero [[Lie algebra homomorphism|homomorphism]], or equivalently, the $\mathfrak{g}$-action to be $(x,v) \mapsto x \cdot v = 0$. ^definition > [!note] Remark. > When discussing the [[group representation|representation theory of finite groups]], the [[trivial group representation]] sends elements to the [[identity map]], *not* zero. Why the discrepancy? Think about derivatives... ^note ---- #### ---- #### 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 > ```