---- > [!definition] Definition. ([[Lie algebra subrepresentation generated by a vector]]) > Let $\mathfrak{g}$ be a [[Lie algebra]], $V$ a [[Lie algebra representation|representation]] of $\mathfrak{g}$. Let $v \in V$. > The set $\mathfrak{g} \cdot v$ does not naturally carry the structure of a [[Lie algebra subrepresentation|subrepresentation]] of $\mathfrak{g}$.[^2] We have to pass to the [[universal enveloping algebra]]: define the [[algebra subrepresentation|subrepresentation]] of $V$ **generated by** $v$ to be[^1] $\mathcal{U}(\mathfrak{g}) \cdot v= \{ x \cdot v : x \in \mathcal{U}(\mathfrak{g}) \}.$ > It is the smallest subrepresentation of $V$ containing $v$. ^definition > [!note] Note. > Since $\mathfrak{g} \subset \mathcal{U}(\mathfrak{g})$, $\mathfrak{g}\cdot v \subset \mathcal{U}(\mathfrak{g}) \cdot v$. One can often say a lot about $\mathcal{U}(\mathfrak{g}) \cdot v$ based on the copy of $\mathfrak{g} \cdot v$ inside it. ---- #### [^1]: To be precise about notation here, the representation $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ extends to an [[algebra homomorphism]] $\rho':\mathcal{U}(\mathfrak{g}) \to \text{End }V$ by the [[universal property]], and $x \cdot v$ means by definition $\rho'(x)(v)$ if $x \in \mathcal{U}(\mathfrak{g})$. We still get a representation *of $\mathfrak{g}$* by restriction of $\rho'$ to the copy of $\mathfrak{g}$ living inside $\mathcal{U}(\mathfrak{g})$. [^2]: To see this, write down the justification for why $A \cdot v$ *is* an [[representation of an algebra|algebra representation]] for $A$ a [[algebra|unital associative algebra]], and notice that associativity was crucial to doing so. $\mathfrak{g}$ is generally not associative! ---- #### 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 > ```