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  "body": "\n## Summary\n\nThe Gelfand–Kirillov dimension of an algebra over a field is the exponent of the polynomial rate at which the algebra grows.\nIt is defined through the [[growth|EHY4PYSMGZ]] of finite-dimensional subspaces under multiplication, so it makes sense for noncommutative algebras.\nFor a finitely generated commutative algebra it equals the [[Krull dimension|MVGBS3J1P7]], and the $n$-th [[Weyl algebra|98QE7KY55W]] has Gelfand–Kirillov dimension $2n$.\nIt is a basic invariant in noncommutative ring theory and in the theory of $D$-modules.\n\n## Explanations\n\n### L1 human\n\nTake a finite set of generators and look at all products of at most $n$ of them.\nIf the dimension of their span grows roughly like $n^d$, the algebra has Gelfand–Kirillov dimension $d$.\nThe polynomial ring in $d$ variables has dimension $d$, just like the space it describes.\nAn algebra whose span grows exponentially, like the free algebra on two letters, has infinite Gelfand–Kirillov dimension.\n",
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