Multibracket

Multibracket is a concept used in algebra and geometry to denote a family of multilinear operations that generalize the binary Lie bracket. A multibracket of arity k is a linear map that takes k arguments from a vector space (or module) and outputs another element of the same space. In many frameworks, brackets of several different arities are included, collectively forming a higher algebraic structure. The defining identities extend the idea of Jacobi compatibility to higher arities, often in a graded or homotopy-theoretic setting.

Two common frameworks for multibrackets are n-Lie algebras and L-infinity algebras. An n-Lie algebra (Filippov algebra) has a single n-ary bracket that is antisymmetric and satisfies a fundamental identity, a higher-arity analogue of the Jacobi identity. In contrast, L-infinity algebras (strong homotopy Lie algebras) carry a hierarchy of brackets l_n for n ≥ 1, each multilinear and graded antisymmetric, which satisfy a sequence of higher Jacobi relations up to homotopy. This broader perspective allows the construction of complex higher-bracket structures in geometry and mathematical physics.

Examples and applications include the Nambu bracket, a ternary multibracket on smooth functions that generalizes the Poisson bracket and satisfies a fundamental identity. Higher multibracket structures feature in deformation theory, differential graded algebras, and theoretical physics, where they model extended symmetries and field theories. See also Nambu bracket, Lie algebra, Filippov algebra, L-infinity algebra, and A-infinity algebra.