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  "body": "\n## Summary\n\nA filtered algebra is an [[associative algebra|3CTW1P4HBP]] with an increasing chain of subspaces $F_0 \\subseteq F_1 \\subseteq \\cdots$ that exhausts it and is compatible with multiplication, $F_i F_j \\subseteq F_{i+j}$.\nIt generalizes a [[graded ring|8KB4SQGC0C]], where elements have an exact degree, to a setting where elements only have a degree \"at most $i$\".\nEach filtered algebra has an [[associated graded|8HFAEGFQT1]] algebra, which is often simpler, and many properties pass from the graded algebra back to the filtered one.\nThe [[universal enveloping algebra|SCZ8PRVY00]] of a Lie algebra and the [[Weyl algebra|98QE7KY55W]] are the classic examples.\n\n## Explanations\n\n### L1 human\n\nIn a polynomial ring every element has a degree, and the degree of a product is at most the sum of the degrees.\nA filtered algebra keeps only this rule, measuring \"size at most $i$\" by the subspace $F_i$.\nIn a noncommutative algebra, such as the algebra of differential operators, there may be no exact degree, but there is still a sensible notion of order.\nPassing to the associated graded algebra forgets the lower-order terms; for differential operators it leaves only the leading symbols, which commute.\n",
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