---- > [!definition] Definition. ([[algebra subrepresentation generated by a vector]]) > Let $A$ be a unital associative [[algebra]], $V$ an [[representation of an algebra|algebra representation]] of $A$. Let $v \in V$. The [[Lie algebra subrepresentation|subrepresentation]] of $V$ **generated by** $v$ is $A \cdot v=\{ a \cdot v : a \in A \}.$ The way to think about this is that while $\mathbb{F}v$ is the [[linear subspace|subspace]] generated by $v$, i.e., the $\mathbb{F}$-[[submodule generated by a subset]] of $v$, $A \cdot v$ is the '$A$-span' of $v$. > $A \cdot v$ is the smallest [[algebra subrepresentation|subrepresentation]] containing $v$. ^definition ---- #### ---- #### 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 > ```