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  "body": "\n## Summary\n\nAn algebraic group is a group whose elements are the points of an algebraic variety, with multiplication and inversion given by polynomial maps.\nThe two main families are the [[linear algebraic groups|GA81VVQHEP]], such as $\\mathrm{GL}_n$, and the [[abelian varieties|R7W03TXRS9]], such as elliptic curves.\n[[Chevalley's structure theorem|MZ6XDATQKR]] shows that every smooth connected algebraic group over a perfect field is built from one of each.\nAlgebraic groups play for algebraic geometry the role that Lie groups play for differential geometry.\n\n## Explanations\n\n### L1 human\n\nThe invertible $2 \\times 2$ matrices form a group, and both the product and the inverse are given by formulas in the entries.\nSo the group is also a geometric object, the set where the determinant is nonzero.\nAn algebraic group is any group of this kind, where the elements form a variety and the group operations are polynomial maps.\nOther examples are the numbers under addition, the nonzero numbers under multiplication, and the points of an elliptic curve.\n",
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