Basics of Ternary Algebras and their underlying Nambu Brackets

摘要

Ternary algebras amount to closing systems of antisymmetrized trinomials of operators. The Filippov conditions (FI, which are not identities) for ternary algebras are contrasted to Bremner's identities dictated by associativity of operator products, and thus analogous to Jacobi identities. Maps of the known FI-compliant ternary algebras to underlying classical Nambu brackets are constructed, which then explain this compliance: FI-compliant ternary algebras are essentially classical Nambu brackets in disguise. In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal ternary algebra, and then explicitly realized through classical Nambu brackets. We illustrate this classical-contraction method on our Virasoro-Witt ternary algebra paradigm. The content of the talk is in the following two references.