Algebraic bracket

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In mathematics, the algebraic bracket or Nijenhuis-Richardson bracket is a graded Lie algebra structure on vector-valued differential forms. It is defined by

[\cdot, \cdot]^{\wedge} : \Omega^{k+1}(M, \mathrm{T}M) \times \Omega^{l+1}(M, \mathrm{T}M) \to \Omega^{k+l+1}(M, \mathrm{T}M) : (K,L) \mapsto [K, L]^{\wedge} := i_K L - (-1)^{kl} i_L K
(i_K L)(X_1, \ldots, X_{k+l+1}) := L(K(X_1, \ldots, X_{k+1}), X_{k+2}, \ldots, X_{k+l+1})

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