---- > [!definition] Definition. ([[commutator]]) > Let $G$ be a [[group]] and $x,y \in G$. The **commutator** of $x$ and $y$ is the element $xyx ^{-1}y^{-1} \in G$. It is denoted $[x,y]$. > > > Let $R$ be a [[ring]] and $a,b \in R$. The **commutator** of $a$ and $b$ is the element $[a,b]:=ab-ba$.The **anticommutator** is $\{ A ,B \}:=ab+ba$. ---- #### (Sakurai 1.2) Expanding the RHS, easy to check that: $[ab, cd]=-ac \{ d, b \}+a \{ c,b \}d-c\{ d, a \}b+\{ c,a \}db$ ---- #### 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 > ```