Nijenhuis–Richardson bracket
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). It is related to but not the same as the Frölicher–Nijenhuis bracket and the Schouten–Nijenhuis bracket.
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. If V is a vector space and p ≥ −1 is an integer, let
be the space of all skew-symmetric (p + 1)-multilinear mappings of V to itself. The direct sum Alt(V) is a graded vector space. A Lie algebra structure on V is determined by a skew-symmetric bilinear map μ : V × V → V. That is to say, μ is an element of Alt1(V). Furthermore, μ must obey the Jacobi identity. The Nijenhuis–Richardson bracket supplies a systematic manner for expressing this identity in the form [μ, μ] = 0.
In detail, the bracket is a bilinear bracket operation defined on Alt(V) as follows. 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 sum is over all (p, q)-shuffles of the indices. On non-homogeneous elements, the bracket is extended by bilinearity.
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