Bracket algebra

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title: "Bracket algebra" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["invariant-theory", "algebras"] topic_path: "general/invariant-theory" source: "https://en.wikipedia.org/wiki/Bracket_algebra" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants.

Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L] of dimension n over the field K is the quotient of the algebra Brace{L} obtained by imposing the congruence relations below, where w, *w''', ..., w" are any monomials in Super[*L'']:

  1. {w} = 0 if length(w) ≠ n
  2. {w}{w'}...{w"} = 0 whenever any positive letter a of L occurs more than n times in the monomial {w}{*w'''}...{*w''"}.
  3. Let {w}{*w'''}...{w"} be a monomial in Brace{L} in which some positive letter a occurs more than n times, and let b, c, d, e, ..., f, g be any letters in *L''.

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