Bracket algebra
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A bracket algebra is an algebraic system that connects the notion of a supersymmetric 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.
[edit] 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): 8087–8090, <http://links.jstor.org/sici?sici=0027-8424%2819910915%2988%3A18%3C8087%3AHSFTBA%3E2.0.CO%3B2-8>.
- Huang, Rosa Q.; Rota, Gian-Carlo & Stein, Joel A. (1990), “Supersymmetric Bracket Algebra and Invariant Theory”, Acta Applicandae Mathematicae (Kluwer Academic Publishers) 21: 193–246, <http://www.springerlink.com/content/q821633w3291351g/>.
