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  "body": "\n## Summary\n\nThe growth of a finitely generated [[associative algebra|3CTW1P4HBP]] over a field measures how fast a dimension increases with $n$.\nThis is the dimension of the span of all products of at most $n$ generators.\nUp to a natural equivalence, this rate does not depend on the chosen generators.\nAlgebras then split into algebras of polynomial growth, of exponential growth and of intermediate growth.\nThe notion parallels the [[growth rate of a group|4CMRM3975W]], and its numerical summary is the [[Gelfand–Kirillov dimension|MDVSY2NMD2]].\n\n## Explanations\n\n### L1 human\n\nPick a few generators of an algebra and count how many linearly independent products of length at most $n$ they give.\nFor the polynomial ring $k[x]$ this number is $n + 1$, which grows like a line.\nFor $k[x, y]$ it is about $n^2/2$, which grows like an area.\nFor the free algebra on two generators, where $xy \\ne yx$, it is $2^{n+1} - 1$, which grows exponentially.\nThe growth of an algebra is the shape of this counting function, ignoring constant factors in $n$.\n",
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