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  "body": "\n## Summary\n\nThe adjoint representation of a [[Lie group|7JBRGCVJRB]] $G$ is an action of $G$ on its own Lie algebra $\\mathfrak{g}$.\nIt comes from differentiating conjugation at the identity.\nIt is a [[representation|WYGRXCG2RN]] $\\operatorname{Ad} \\colon G \\to GL(\\mathfrak{g})$ that every Lie group carries.\nIts derivative is the map $X \\mapsto [X, \\cdot\\,]$ on the Lie algebra, and its kernel, for a connected group, is the centre of $G$.\nIt is the starting point for root systems, the Killing form and the classification of semisimple groups.\n\n## Explanations\n\n### L1 human\n\nEach element $g$ of a group acts on the group by conjugation, $h \\mapsto g h g^{-1}$, and this map fixes the identity.\nIts effect on tangent vectors at the identity is a linear map of the Lie algebra, called $\\operatorname{Ad}(g)$.\nFor a group of matrices this is simply $X \\mapsto g X g^{-1}$, a change of basis applied to the matrix $X$.\nFor a commutative group, conjugation does nothing, so every $\\operatorname{Ad}(g)$ is the identity.\n",
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