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]:

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.

See also

• Bracket ring

References

• 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 88 (18), pp. 8087–8090, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546.
• Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), "Supersymmetric Bracket Algebra and Invariant Theory", Acta Applicandae Mathematicae (Kluwer Academic Publishers) 21, pp. 193–246, doi:10.1007/BF00053298.