---- > [!definition] Definition. ([[root system isomorphism]]) > An **isomorphism of [[root system|root systems]]** $(\Phi, E)\to (\Phi', E')$ is an [[linear isomorphism|isomorphism of vector spaces]] $f:E \to E'$ such that > **(Root preservation)** $f(\Phi)=\Phi'$ > **(Check-pairing preservation)** $\langle f(\beta), f(\alpha)^{\vee} \rangle=\langle \beta, \alpha^{\vee} \rangle$ for all $\alpha,\beta \in \Phi$. ^definition > [!note] Note. > Note that we *do not* require $f$ to preserve the [[inner product]] $(-,-)$ on $E$. ^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 > ```