---- > [!definition] Definition. ([[general linear Lie algebra]]) > Let $V$ be a finite-dimensional [[vector space]] over a [[field|field]] $\mathbb{F}$. Let $\mathfrak{gl}(V)=\text{End}(V)$. If we fix a [[basis]] of $V$ then we can [[hom and mat are isomorphic|identify]] $\mathfrak{gl}(V)$ with $\text{Mat}_{n}(\mathbb{F})$. In this case we write $\mathfrak{gl}_{n}$ or $\mathfrak{gl}_{n}(\mathbb{F})$. With the [[commutator]] as its bracket, $\mathfrak{gl}(V)$ becomes a [[Lie algebra]] over $\mathbb{F}$. ^definition > [!justification] [[vector space of linear maps between two vector spaces|We already know]] that $\text{End}(V)$ has evident [[vector space]] structure. So we just need to verify that the [[commutator]] operation on it is [[bilinear map|bilinear]], [[alternating multilinear map|alternating]], and satisfies the Jacobi identity. Bilinearity follows e.g. from the computation $\begin{align} [a_{1}A_{1} + a_{2}A_{2}, B] = & (a_{1}A_{1} + a_{2}A_{2}) B - B(a_{1}A_{1} + a_{2}A_{2}) \\ = & a_{1}A_{1} B + a_{2}A_{2} B - a_{1} BA_{1} + a_{2}BA_{2} \\ = & a_{1} [A_{1}B - BA_{1}] + a_{2} [A_{2}B - BA_{2}] \\ = & a_{1} [A_{1}, B] + a_{2} [A_{2}, B]. \end{align}$ Alternating is clear because $[A,A]=A^{2} - A^{2} = 0$. The Jacobi identity follows because $\begin{align} &\big[ A, [B,C] \big] + \big[ C, [A,B] \big] + \big[ B, [C,A] \big] \\= & \big[ A, BC-CA \big] + [C, AB-BA] + [B, CA - AC] \\= & A(BC-CB) - (BC-CB)A \\+& C(AB-BA) - (AB-BA)C \\+ & B(CA-AC) - (CA-AC)B \\= & \textcolor{Thistle}{ABC} \textcolor{LimeGreen}{- ACB}\textcolor{Apricot}{-BCA} \textcolor{Skyblue}{+ CBA} \\ \textcolor{Red}{+} & \textcolor{Red}{CAB} \textcolor{Skyblue}{- CBA} \textcolor{Thistle}{-ABC} \textcolor{blue}{+ BAC} \\ \textcolor{Apricot}{+} & \textcolor{Apricot}{BCA} \textcolor{blue}{- BAC} \textcolor{Red}{- CAB} \textcolor{LimeGreen}{+ ACB} \\ = & 0. \end{align}$ ---- #### ---- #### 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 > ```