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  "body": "\n## Summary\n\nA Lie pseudogroup is a [[pseudogroup|3017J7P6JD]] of local diffeomorphisms of a manifold that is cut out by a system of partial differential equations.\nMembership of a local diffeomorphism is decided by its [[jets|1QE0Z0756D]] up to a fixed order at each point.\nIsometries, conformal maps, symplectomorphisms, holomorphic maps and volume-preserving maps all form Lie pseudogroups.\nLie pseudogroups were introduced by Lie and studied by Cartan as the infinite-dimensional analogues of Lie groups.\n\n## Explanations\n\n### L1 human\n\nMany geometric structures are defined by the maps that preserve them.\nOften those maps are the solutions of differential equations: a map of the plane preserves area when its Jacobian determinant equals $1$.\nThe set of all local solutions of such an equation forms a pseudogroup, and this is a Lie pseudogroup.\nUnlike a Lie group, it can be infinite-dimensional, as in the area-preserving example, where there are as many solutions as there are functions of two variables.\n",
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