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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]].\nLet $\\Omega(M) = \\bigoplus_k \\Omega^k(M)$ be the graded algebra of [[differential forms|0E3G0GRPZ6]] with the [[wedge product|0GRAATMGP1]].\nLet $\\Omega^k(M, TM)$ be the space of $k$-forms with values in $TM$, that is, smooth sections of $\\Lambda^k T^*M \\otimes TM$.\nA graded [[derivation|3J89REV9WQ]] of degree $j$ of $\\Omega(M)$ is a linear map $D$ with $D(\\Omega^k) \\subseteq \\Omega^{k+j}$ and $D(\\alpha \\wedge \\beta) = D\\alpha \\wedge \\beta + (-1)^{jk} \\alpha \\wedge D\\beta$ for $\\alpha \\in \\Omega^k(M)$.\nThe graded commutator of derivations $D_1, D_2$ of degrees $j_1, j_2$ is $[D_1, D_2] = D_1 D_2 - (-1)^{j_1 j_2} D_2 D_1$.\n\nFor $K \\in \\Omega^k(M, TM)$, let $\\iota_K$ be the derivation of degree $k - 1$ given on decomposable $K = \\alpha \\otimes X$ by $\\iota_K \\omega = \\alpha \\wedge \\iota_X \\omega$.\nLet $\\mathcal{L}_K = [\\iota_K, d] = \\iota_K d - (-1)^{k-1} d\\, \\iota_K$, a derivation of degree $k$, where $d$ is the [[exterior derivative|0ZHSTAG8ZY]].\n\nFor $K \\in \\Omega^k(M, TM)$ and $L \\in \\Omega^l(M, TM)$, the *Frölicher–Nijenhuis bracket* $[K, L] \\in \\Omega^{k+l}(M, TM)$ is the unique element with\n$$\\mathcal{L}_{[K, L]} = [\\mathcal{L}_K, \\mathcal{L}_L].$$\n\n## Remarks\n\nThe bracket exists and is unique because every graded derivation of $\\Omega(M)$ that commutes with $d$ in the graded sense is $\\mathcal{L}_K$ for exactly one vector-valued form $K$.\nIt is graded antisymmetric, $[K, L] = -(-1)^{kl} [L, K]$, and satisfies the graded Jacobi identity.\n\nFor $\\alpha \\in \\Omega^k(M)$, $\\beta \\in \\Omega^l(M)$ and vector fields $X, Y$, an explicit formula is\n$$[\\alpha \\otimes X, \\beta \\otimes Y] = \\alpha \\wedge \\beta \\otimes [X, Y] + \\alpha \\wedge \\mathcal{L}_X \\beta \\otimes Y - \\mathcal{L}_Y \\alpha \\wedge \\beta \\otimes X + (-1)^k \\big( d\\alpha \\wedge \\iota_X \\beta \\otimes Y + \\iota_Y \\alpha \\wedge d\\beta \\otimes X \\big).$$\n\nExample: for $k = l = 0$, take $\\alpha = \\beta = 1$.\nThe formula gives $[X, Y]$, the [[Lie bracket of vector fields|68FGQEBD6J]], and $\\mathcal{L}_X$ is the usual [[Lie derivative|PJWK0PKN6Q]].\nAnother example: the identity $\\mathrm{Id} \\in \\Omega^1(M, TM)$ has $\\iota_{\\mathrm{Id}} \\omega = j \\omega$ for $\\omega \\in \\Omega^j(M)$.\nHence $\\mathcal{L}_{\\mathrm{Id}} \\omega = (j + 1) d\\omega - j\\, d\\omega = d\\omega$, so $\\mathcal{L}_{\\mathrm{Id}} = d$ and $[\\mathrm{Id}, L] = 0$ for every $L$.\n\nNon-example: the naive rule $[\\alpha \\otimes X, \\beta \\otimes Y] = \\alpha \\wedge \\beta \\otimes [X, Y]$ does not define a bracket on $\\Omega(M, TM)$.\nFor a function $f$, the elements $f\\alpha \\otimes X$ and $\\alpha \\otimes fX$ are equal, but the rule gives different results for them, since $[fX, Y] \\ne f[X, Y]$ in general.\n",
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