---- > [!definition] Definition. ([[Lie algebra]]) > Let $k$ be a [[field|field]]. A **Lie algebra over $k$** is a $k$-[[vector space]] $\mathfrak{g}$ together with an [[alternating multilinear map|alternating]] [[bilinear map|bilinear operation]] $[-,-]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ that satisfies the **Jacobi identity** $\big[ x, [y,z] \big] + \big[ z, [x,y] \big] + \big[ y, [z,x] \big] =0.$ ^definition > [!note] Remarks. > 1. The operation $[-,-]$ is called the **Lie bracket** of $\mathfrak{g}$. Sometimes we write $[xy]$ instead of $[x,y]$. > 2. Despite the name, the collection of Lie algebras is not a subcollection of the collection of [[algebra|associative algebras]]. ^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 > ```