Bracket algebra
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]:
- {w} = 0 if length(w) ≠ n
- {w}{w'}...{w"} = 0 whenever any positive letter a of L occurs more than n times in the monomial {w}{w'}...{w"}.
- 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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References[edit]
- Anick, David; Rota, Gian-Carlo (September 15, 1991), "Higher-Order Syzygies for the Bracket Algebra and for the Ring of Coordinates of the Grassmanian", Proceedings of the National Academy of Sciences, vol. 88, no. 18, pp. 8087–8090, Bibcode:1991PNAS...88.8087A, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546, PMC 52451, PMID 11607210.
- Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), "Supersymmetric Bracket Algebra and Invariant Theory", Acta Applicandae Mathematicae, vol. 21, no. 1–2, Kluwer Academic Publishers, pp. 193–246, doi:10.1007/BF00053298, S2CID 189901418.