In mathematics, the algebraic bracket or Nijenhuis–Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by A. Nijenhuis and R. W. Richardson, Jr (1966, 1967).
The primary motivation for introducing the bracket was to develop a uniform framework for discussing all possible Lie algebra structures on a vector space, and subsequently the deformations of these structures.
The direct sum Alt(V) is a graded vector space.
On homogeneous elements P ∈ Altp(V) and Q ∈ Altq(V), the Nijenhuis–Richardson bracket [P, Q]∧ ∈ Altp+q(V) is given by Here the interior product iP is defined by where
The Nijenhuis–Richardson bracket can be defined on the vector valued forms Ω*(M, T(M)) on a smooth manifold M in a similar way.
This identifies Ω*(M, T(M)) with the algebra of derivations that vanish on smooth functions.