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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Commutator</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical concept. For the electrical component, see <a href="Commutator_(electric)" title="Commutator (electric)">Commutator (electric)</a>. For the relation between <a href="Conjugate_variables" title="Conjugate variables">canonical conjugate entities</a>, see <a href="Canonical_commutation_relation" title="Canonical commutation relation">Canonical commutation relation</a>. For other uses, see <a href="Commutation_(disambiguation)" class="mw-redirect mw-disambig" title="Commutation (disambiguation)">Commutation</a>.</div>

<p>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>commutator</b> gives an indication of the extent to which a certain <a href="Binary_operation" title="Binary operation">binary operation</a> fails to be <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>. There are different definitions used in <a href="Group_theory" title="Group theory">group theory</a> and <a href="Ring_theory" title="Ring theory">ring theory</a>.

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<div class="mw-heading mw-heading2"><h2 id="Group_theory">Group theory</h2></div>

<p>The <b>commutator</b> of two elements, <span class="texhtml mvar" style="font-style:italic;">g</span> and <span class="texhtml mvar" style="font-style:italic;">h</span>, of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <span class="texhtml mvar" style="font-style:italic;">G</span>, is the element

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<p>This element is equal to the group's identity if and only if <span class="texhtml mvar" style="font-style:italic;">g</span> and <span class="texhtml mvar" style="font-style:italic;">h</span> commute (that is, if and only if <span class="texhtml"><i>gh</i> = <i>hg</i></span>).

</p><p>The set of all commutators of a group is not in general closed under the group operation, but the <a href="Subgroup" title="Subgroup">subgroup</a> of <i>G</i> <a href="Generating_set_of_a_group" title="Generating set of a group">generated</a> by all commutators is closed and is called the <i>derived group</i> or the <i><a href="Commutator_subgroup" title="Commutator subgroup">commutator subgroup</a></i> of <i>G</i>. Commutators are used to define <a href="Nilpotent_group" title="Nilpotent group">nilpotent</a> and <a href="Solvable_group" title="Solvable group">solvable</a> groups and the largest <a href="Abelian_group" title="Abelian group">abelian</a> <a href="Quotient_group" title="Quotient group">quotient group</a>.

</p><p>The definition of the commutator above is used throughout this article, but many group theorists define the commutator as

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<p>Using the first definition, this can be expressed as <span class="texhtml">[<i>g</i><sup>−1</sup>, <i>h</i><sup>−1</sup>]</span>.

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<div class="mw-heading mw-heading3"><h3 id="Identities_(group_theory)">Identities (group theory)</h3></div>

<p>Commutator identities are an important tool in <a href="Group_theory" title="Group theory">group theory</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The expression <span class="texhtml"><i>a<sup>x</sup></i></span> denotes the <a href="Conjugate_(group_theory)" class="mw-redirect" title="Conjugate (group theory)">conjugate</a> of <span class="texhtml mvar" style="font-style:italic;">a</span> by <span class="texhtml mvar" style="font-style:italic;">x</span>, defined as <span class="texhtml"><i>x</i><sup>−1</sup><i>ax</i></span>.

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<annotation encoding="application/x-tex">{\displaystyle \left[x,y^{-1}\right]=[y,x]^{y^{-1}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left[\left[x,y^{-1}\right],z\right]^{y}\cdot \left[\left[y,z^{-1}\right],x\right]^{z}\cdot \left[\left[z,x^{-1}\right],y\right]^{x}=1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left[\left[x,y\right],z^{x}\right]\cdot \left[[z,x],y^{z}\right]\cdot \left[[y,z],x^{y}\right]=1.}</annotation>

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<p>Identity (5) is also known as the <i>Hall–Witt identity</i>, after <a href="Philip_Hall" title="Philip Hall">Philip Hall</a> and <a href="Ernst_Witt" title="Ernst Witt">Ernst Witt</a>. It is a group-theoretic analogue of the <a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a> for the ring-theoretic commutator (see next section).

</p><p>N.B., the above definition of the conjugate of <span class="texhtml mvar" style="font-style:italic;">a</span> by <span class="texhtml mvar" style="font-style:italic;">x</span> is used by some group theorists.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Many other group theorists define the conjugate of <span class="texhtml mvar" style="font-style:italic;">a</span> by <span class="texhtml mvar" style="font-style:italic;">x</span> as <span class="texhtml"><i>xax</i><sup>−1</sup></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This is often written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{x}a}">

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<annotation encoding="application/x-tex">{\displaystyle {}^{x}a}</annotation>

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</p><p>Many identities that are true modulo certain subgroups are also used. These can be particularly useful in the study of <a href="Solvable_group" title="Solvable group">solvable groups</a> and <a href="Nilpotent_group" title="Nilpotent group">nilpotent groups</a>. For instance, in any group, second powers behave well:

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<annotation encoding="application/x-tex">{\displaystyle (xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].}</annotation>

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<p>If the <a href="Derived_subgroup" class="mw-redirect" title="Derived subgroup">derived subgroup</a> is central, then

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<div class="mw-heading mw-heading2"><h2 id="Ring_theory">Ring theory</h2></div>

<p><a href="Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">Rings</a> often do not support division. Thus, the <b>commutator</b> of two elements <i>a</i> and <i>b</i> of a ring (or any <a href="Associative_algebra" title="Associative algebra">associative algebra</a>) is defined differently by

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<p>The commutator is zero if and only if <i>a</i> and <i>b</i> commute. In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, if two <a href="Endomorphism" title="Endomorphism">endomorphisms</a> of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a <a href="Lie_algebra" title="Lie algebra">Lie bracket</a>, every associative algebra can be turned into a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a>.

</p><p>The <b>anticommutator</b> of two elements <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> of a ring or associative algebra is defined by

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<p>Sometimes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]_{+}}">

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</p><p>The commutator of two operators acting on a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> is a central concept in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, since it quantifies how well the two <a href="Observable" title="Observable">observables</a> described by these operators can be measured simultaneously. The <a href="Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> is ultimately a theorem about such commutators, by virtue of the <a href="Uncertainty_relation" class="mw-redirect" title="Uncertainty relation">Robertson–Schrödinger relation</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In <a href="Phase_space" title="Phase space">phase space</a>, equivalent commutators of function <a href="Moyal_product" title="Moyal product">star-products</a> are called <a href="Moyal_bracket" title="Moyal bracket">Moyal brackets</a> and are completely isomorphic to the Hilbert space commutator structures mentioned.

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<div class="mw-heading mw-heading3"><h3 id="Identities_(ring_theory)">Identities (ring theory)</h3></div>

<p>The commutator has the following properties:

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<div class="mw-heading mw-heading4"><h4 id="Lie-algebra_identities">Lie-algebra identities</h4></div>

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<annotation encoding="application/x-tex">{\displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}</annotation>

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<p>Relation (3) is called <a href="Anticommutativity" class="mw-redirect" title="Anticommutativity">anticommutativity</a>, while (4) is the <a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a>.

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<div class="mw-heading mw-heading4"><h4 id="Additional_identities">Additional identities</h4></div>

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<annotation encoding="application/x-tex">{\displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}</annotation>

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</math></span><img src="./8b79c370dc9984ff5e2720b3e06c6153af7ba572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.124ex; height:2.843ex;" alt="{\displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [AB,C]=A[B,C]+[A,C]B}</annotation>

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</math></span><img src="./71ba3177011b65b292d8dc39dd2e3391f29b0a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.742ex; height:2.843ex;" alt="{\displaystyle [AB,C]=A[B,C]+[A,C]B}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}</annotation>

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</math></span><img src="./b9e755fd1fe3c26c40e007256c6976b9612e5d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.881ex; height:2.843ex;" alt="{\displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}</annotation>

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</math></span><img src="./4e00b8c22f5a63ff3dbbcf8a74f0d01f43994afa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.124ex; height:2.843ex;" alt="{\displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [A,B+C]=[A,B]+[A,C]}</annotation>

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</math></span><img src="./7f50e382d516600cfe9f7a34376ebe694bb62177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.052ex; height:2.843ex;" alt="{\displaystyle [A,B+C]=[A,B]+[A,C]}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}</annotation>

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</math></span><img src="./2e47836a02ce95e65488370410a80446b1ae636c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.532ex; height:2.843ex;" alt="{\displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B}</annotation>

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</math></span><img src="./a8b90ee684250322ac794d8cb193d75c8c1d4db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:108.967ex; height:2.843ex;" alt="{\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}</annotation>

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</math></span><img src="./33e847e641c8d2810846e666be23d2512b08eb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.522ex; height:2.843ex;" alt="{\displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}" loading="lazy"></span></li></ol>

<p>If <span class="texhtml mvar" style="font-style:italic;">A</span> is a fixed element of a ring <i>R</i>, identity (1) can be interpreted as a <a href="Product_rule" title="Product rule">Leibniz rule</a> for the map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{A}:R\rightarrow R}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{A}:R\rightarrow R}</annotation>

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</math></span><img src="./5675bfef6f687609477671fc9d0b5ee4bf438354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.999ex; height:2.509ex;" alt="{\displaystyle \operatorname {ad} _{A}:R\rightarrow R}" loading="lazy"></span> given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{A}(B)=[A,B]}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{A}(B)=[A,B]}</annotation>

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</math></span><img src="./8a40684049019f333e05818bcd1ce5b48f839561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.426ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{A}(B)=[A,B]}" loading="lazy"></span>. In other words, the map ad<sub><i>A</i></sub> defines a <a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> on the ring <i>R</i>. Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation. Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express <b>Z</b>-<a href="Bilinear_map" title="Bilinear map">bilinearity</a>.

</p><p>From identity (9), one finds that the commutator of integer powers of ring elements is:

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<annotation encoding="application/x-tex">{\displaystyle [A^{N},B^{M}]=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}A^{n}B^{m}[A,B]B^{M-m-1}A^{N-n-1}=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}B^{m}A^{n}[A,B]A^{N-n-1}B^{M-m-1}}</annotation>

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</math></span><img src="./d6becce6d1f8d926cacdaa106d7b58dc3b04880e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:89.165ex; height:7.343ex;" alt="{\displaystyle [A^{N},B^{M}]=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}A^{n}B^{m}[A,B]B^{M-m-1}A^{N-n-1}=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}B^{m}A^{n}[A,B]A^{N-n-1}B^{M-m-1}}" loading="lazy"></span></dd></dl>

<p>Some of the above identities can be extended to the anticommutator using the above ± subscript notation.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

For example:

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<annotation encoding="application/x-tex">{\displaystyle [AB,C]_{\pm }=A[B,C]_{-}+[A,C]_{\pm }B}</annotation>

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</math></span><img src="./187aaec93ded1a30cd6ad33d9e323c786d4a39fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.274ex; height:2.843ex;" alt="{\displaystyle [AB,C]_{\pm }=A[B,C]_{-}+[A,C]_{\pm }B}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [AB,CD]_{\pm }=A[B,C]_{-}D+AC[B,D]_{-}+[A,C]_{-}DB+C[A,D]_{\pm }B}</annotation>

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</math></span><img src="./2abfd9c204056463779a2541fa7f5bc02930bf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.8ex; height:2.843ex;" alt="{\displaystyle [AB,CD]_{\pm }=A[B,C]_{-}D+AC[B,D]_{-}+[A,C]_{-}DB+C[A,D]_{\pm }B}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [[A,B],[C,D]]=[[[B,C]_{+},A]_{+},D]-[[[B,D]_{+},A]_{+},C]+[[[A,D]_{+},B]_{+},C]-[[[A,C]_{+},B]_{+},D]}</annotation>

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</math></span><img src="./6c14c4def5d057ec857052e280d0c27658c3796d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:94.609ex; height:2.843ex;" alt="{\displaystyle [[A,B],[C,D]]=[[[B,C]_{+},A]_{+},D]-[[[B,D]_{+},A]_{+},C]+[[[A,D]_{+},B]_{+},C]-[[[A,C]_{+},B]_{+},D]}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \left[A,[B,C]_{\pm }\right]+\left[B,[C,A]_{\pm }\right]+\left[C,[A,B]_{\pm }\right]=0}</annotation>

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</math></span><img src="./7fa38c6ec51fd44c85b6a7dd7820d8ebaee4f2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.26ex; height:2.843ex;" alt="{\displaystyle \left[A,[B,C]_{\pm }\right]+\left[B,[C,A]_{\pm }\right]+\left[C,[A,B]_{\pm }\right]=0}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [A,BC]_{\pm }=[A,B]_{-}C+B[A,C]_{\pm }=[A,B]_{\pm }C\mp B[A,C]_{-}}</annotation>

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</math></span><img src="./cee7d373e2721aa7ec3c50a8f645785d7da7b804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.437ex; height:2.843ex;" alt="{\displaystyle [A,BC]_{\pm }=[A,B]_{-}C+B[A,C]_{\pm }=[A,B]_{\pm }C\mp B[A,C]_{-}}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle [A,BC]=[A,B]_{\pm }C\mp B[A,C]_{\pm }}</annotation>

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</math></span><img src="./2f480a46e5c19fea0ae860621b38e69606a71239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.764ex; height:2.843ex;" alt="{\displaystyle [A,BC]=[A,B]_{\pm }C\mp B[A,C]_{\pm }}" loading="lazy"></span></li></ol>

<div class="mw-heading mw-heading4"><h4 id="Exponential_identities">Exponential identities</h4></div>

<p>Consider a ring or algebra in which the <a href="Exponential_function" title="Exponential function">exponential</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{A}=\exp(A)=1+A+{\tfrac {1}{2!}}A^{2}+\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle e^{A}=\exp(A)=1+A+{\tfrac {1}{2!}}A^{2}+\cdots }</annotation>

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</math></span><img src="./b72148f07ab7bdfde03d77b6c8b17e1aee2c9efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:34.913ex; height:3.676ex;" alt="{\displaystyle e^{A}=\exp(A)=1+A+{\tfrac {1}{2!}}A^{2}+\cdots }" loading="lazy"></span> can be meaningfully defined, such as a <a href="Banach_algebra" title="Banach algebra">Banach algebra</a> or a ring of <a href="Formal_power_series" title="Formal power series">formal power series</a>.

</p><p>In such a ring, <a href="Hadamard's_lemma" title="Hadamard's lemma">Hadamard's lemma</a> applied to nested commutators gives: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle e^{A}Be^{-A}\ =\ B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \ =\ e^{\operatorname {ad} _{A}}(B).}">

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<annotation encoding="application/x-tex">{\textstyle e^{A}Be^{-A}\ =\ B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \ =\ e^{\operatorname {ad} _{A}}(B).}</annotation>

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</math></span><img src="./5e4b0b44677de183969e936bb4cede2bf5515ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:75.279ex; height:3.676ex;" alt="{\textstyle e^{A}Be^{-A}\ =\ B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \ =\ e^{\operatorname {ad} _{A}}(B).}" loading="lazy"></span> (For the last expression, see <i>Adjoint derivation</i> below.) This formula underlies the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#An_important_lemma" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff expansion</a> of log(exp(<i>A</i>) exp(<i>B</i>)).

</p><p>A similar expansion expresses the group commutator of expressions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{A}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{A}}</annotation>

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</math></span><img src="./5ebe1117a4d16dd21dc73da2e92192d4809f17a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.676ex;" alt="{\displaystyle e^{A}}" loading="lazy"></span> (analogous to elements of a <a href="Lie_group" title="Lie group">Lie group</a>) in terms of a series of nested commutators (Lie brackets),

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<annotation encoding="application/x-tex">{\displaystyle e^{A}e^{B}e^{-A}e^{-B}=\exp \!\left([A,B]+{\frac {1}{2!}}[A{+}B,[A,B]]+{\frac {1}{3!}}\left({\frac {1}{2}}[A,[B,[B,A]]]+[A{+}B,[A{+}B,[A,B]]]\right)+\cdots \right).}</annotation>

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</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Graded_rings_and_algebras">Graded rings and algebras</h2></div>

<p>When dealing with <a href="Graded_algebra" class="mw-redirect" title="Graded algebra">graded algebras</a>, the commutator is usually replaced by the <b>graded commutator</b>, defined in homogeneous components as

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<annotation encoding="application/x-tex">{\displaystyle [\omega ,\eta ]_{gr}:=\omega \eta -(-1)^{\deg \omega \deg \eta }\eta \omega .}</annotation>

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</math></span><img src="./4a31537974b529c2386b07f032b0452d7ab70e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.123ex; height:3.343ex;" alt="{\displaystyle [\omega ,\eta ]_{gr}:=\omega \eta -(-1)^{\deg \omega \deg \eta }\eta \omega .}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Adjoint_derivation">Adjoint derivation</h2></div>

<p>Especially if one deals with multiple commutators in a ring <i>R</i>, another notation turns out to be useful. For an element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in R}">

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<annotation encoding="application/x-tex">{\displaystyle x\in R}</annotation>

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</math></span><img src="./0b4a3f5fa1b895f5a40a25ced8581b2152b3c24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.934ex; height:2.176ex;" alt="{\displaystyle x\in R}" loading="lazy"></span>, we define the <a href="Adjoint_representation_of_a_Lie_algebra" class="mw-redirect" title="Adjoint representation of a Lie algebra">adjoint</a> mapping <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ad} _{x}:R\to R}">

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</math></span><img src="./0456f8c501cf86775e9e44e131b39f5d82984e50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.707ex; height:2.509ex;" alt="{\displaystyle \mathrm {ad} _{x}:R\to R}" loading="lazy"></span> by:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{x}(y)=[x,y]=xy-yx.}</annotation>

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</math></span><img src="./06e06c88110f66faa14de2b3fba0644b13e96429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.06ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{x}(y)=[x,y]=xy-yx.}" loading="lazy"></span></dd></dl>

<p>This mapping is a <a href="Derivation_(differential_algebra)" title="Derivation (differential algebra)">derivation</a> on the ring <i>R</i>:

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} _{x}\!(yz)\ =\ \mathrm {ad} _{x}\!(y)\,z\,+\,y\,\mathrm {ad} _{x}\!(z).}</annotation>

</semantics>

</math></span><img src="./32d87229b058624965aa149ed975f8931a987ff5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.175ex; height:2.843ex;" alt="{\displaystyle \mathrm {ad} _{x}\!(yz)\ =\ \mathrm {ad} _{x}\!(y)\,z\,+\,y\,\mathrm {ad} _{x}\!(z).}" loading="lazy"></span></dd></dl>

<p>By the <a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a>, it is also a derivation over the commutation operation:

</p>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} _{x}[y,z]\ =\ [\mathrm {ad} _{x}\!(y),z]\,+\,[y,\mathrm {ad} _{x}\!(z)].}</annotation>

</semantics>

</math></span><img src="./97e49b2f65e16c28bbcad712a40e4bf680ebf909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.962ex; height:2.843ex;" alt="{\displaystyle \mathrm {ad} _{x}[y,z]\ =\ [\mathrm {ad} _{x}\!(y),z]\,+\,[y,\mathrm {ad} _{x}\!(z)].}" loading="lazy"></span></dd></dl>

<p>Composing such mappings, we get for example <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{x}\operatorname {ad} _{y}(z)=[x,[y,z]\,]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{x}\operatorname {ad} _{y}(z)=[x,[y,z]\,]}</annotation>

</semantics>

</math></span><img src="./8c27f6b58b814f258e43d4529f26e9ec0523b58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.13ex; height:3.009ex;" alt="{\displaystyle \operatorname {ad} _{x}\operatorname {ad} _{y}(z)=[x,[y,z]\,]}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{x}^{2}\!(z)\ =\ \operatorname {ad} _{x}\!(\operatorname {ad} _{x}\!(z))\ =\ [x,[x,z]\,].}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{x}^{2}\!(z)\ =\ \operatorname {ad} _{x}\!(\operatorname {ad} _{x}\!(z))\ =\ [x,[x,z]\,].}</annotation>

</semantics>

</math></span></span> We may consider <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ad} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} }</annotation>

</semantics>

</math></span><img src="./4dfabb59497ffe0c094cac2720f0c6a67b33e205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \mathrm {ad} }" loading="lazy"></span> itself as a mapping, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ad} :R\to \mathrm {End} (R)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} :R\to \mathrm {End} (R)}</annotation>

</semantics>

</math></span><img src="./f5316f3377db1b7e8497c7cca943bc119428bdc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.511ex; height:2.843ex;" alt="{\displaystyle \mathrm {ad} :R\to \mathrm {End} (R)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} (R)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {End} (R)}</annotation>

</semantics>

</math></span><img src="./ba76c862ebdf7034bc637cb3e1b4d0002abfb4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.741ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} (R)}" loading="lazy"></span> is the ring of mappings from <i>R</i> to itself with composition as the multiplication operation. Then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ad} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ad} }</annotation>

</semantics>

</p>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{[x,y]}=\left[\operatorname {ad} _{x},\operatorname {ad} _{y}\right].}</annotation>

</semantics>

</math></span><img src="./3c43069feeed684198e360f083c3c004bf3cb2f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.408ex; height:3.176ex;" alt="{\displaystyle \operatorname {ad} _{[x,y]}=\left[\operatorname {ad} _{x},\operatorname {ad} _{y}\right].}" loading="lazy"></span></dd></dl>

<p>By contrast, it is <b>not</b> always a ring homomorphism: usually <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{xy}\,\neq \,\operatorname {ad} _{x}\operatorname {ad} _{y}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{xy}\,\neq \,\operatorname {ad} _{x}\operatorname {ad} _{y}}</annotation>

</semantics>

</math></span><img src="./9e74e61300309c6869910b9125ef3e808150b9f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.836ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{xy}\,\neq \,\operatorname {ad} _{x}\operatorname {ad} _{y}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="General_Leibniz_rule">General Leibniz rule</h3></div>

<p>The <a href="General_Leibniz_rule" title="General Leibniz rule">general Leibniz rule</a>, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:

</p>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{n}y=\sum _{k=0}^{n}{\binom {n}{k}}\operatorname {ad} _{x}^{k}\!(y)\,x^{n-k}.}</annotation>

</semantics>

</math></span><img src="./633c65726cdc414ddd5173451fab8b266c1a5ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.669ex; height:7.009ex;" alt="{\displaystyle x^{n}y=\sum _{k=0}^{n}{\binom {n}{k}}\operatorname {ad} _{x}^{k}\!(y)\,x^{n-k}.}" loading="lazy"></span></dd></dl>

<p>Replacing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> by the differentiation operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial }</annotation>

</semantics>

</math></span><img src="./62b4e7c1cedb9564609aefd2aa2309972f455c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> by the multiplication operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{f}:g\mapsto fg}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{f}:g\mapsto fg}</annotation>

</semantics>

</math></span><img src="./09baf9c0ce6ed043b1212f2b389bcfc55c74b8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.239ex; height:2.843ex;" alt="{\displaystyle m_{f}:g\mapsto fg}" loading="lazy"></span>, we get <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} (\partial )(m_{f})=m_{\partial (f)}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} (\partial )(m_{f})=m_{\partial (f)}}</annotation>

</semantics>

</math></span><img src="./fe22e56a170b8297a6b6f8a943e5f85de5168c33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.055ex; height:3.176ex;" alt="{\displaystyle \operatorname {ad} (\partial )(m_{f})=m_{\partial (f)}}" loading="lazy"></span>, and applying both sides to a function <i>g</i>, the identity becomes the usual Leibniz rule for the <i>n</i>th derivative <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{n}\!(fg)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial ^{n}\!(fg)}</annotation>

</semantics>

</math></span><img src="./7e31b56b72e60c78c2475a0215d725ef0ab8872a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.379ex; height:2.843ex;" alt="{\displaystyle \partial ^{n}\!(fg)}" loading="lazy"></span>.

</p>

<ul><li><a href="Anticommutativity" class="mw-redirect" title="Anticommutativity">Anticommutativity</a></li>

<li><a href="Associator" title="Associator">Associator</a></li>

<li><a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a></li>

<li><a href="Canonical_commutation_relation" title="Canonical commutation relation">Canonical commutation relation</a></li>

<li><a href="Centralizer" class="mw-redirect" title="Centralizer">Centralizer</a> a.k.a. commutant</li>

<li><a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">Derivation (abstract algebra)</a></li>

<li><a href="Moyal_bracket" title="Moyal bracket">Moyal bracket</a></li>

<li><a href="Pincherle_derivative" title="Pincherle derivative">Pincherle derivative</a></li>

<li><a href="Poisson_bracket" title="Poisson bracket">Poisson bracket</a></li>

<li><a href="Ternary_commutator" title="Ternary commutator">Ternary commutator</a></li>

<li><a href="Three_subgroups_lemma" title="Three subgroups lemma">Three subgroups lemma</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerstein1975">Herstein (1975</a>, p. 252)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 108)</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcKay2000">McKay (2000</a>, p. 4)</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerstein1975">Herstein (1975</a>, p. 83)</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 128)</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcMahon2008">McMahon (2008)</a></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiboff2003">Liboff (2003</a>, pp. 140–142)</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFLavrov2014">Lavrov (2014)</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFFraleigh1976" class="citation cs2">Fraleigh, John B. (1976), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hyHvAAAAMAAJ&q=commutator"><i>A First Course In Abstract Algebra</i></a> (2nd ed.), Reading: <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-01984-1</bdi></cite></li>

<li><cite id="CITEREFGriffiths2004" class="citation cs2"><a href="David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2004), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Quantum Mechanics</i></a></span> (2nd ed.), <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-805326-X</bdi></cite></li>

<li><cite id="CITEREFHerstein1975" class="citation cs2"><a href="Israel_Nathan_Herstein" title="Israel Nathan Herstein">Herstein, I. N.</a> (1975), <i>Topics In Algebra</i> (2nd ed.), Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471010901</bdi></cite></li>

<li><cite id="CITEREFLavrov2014" class="citation cs2">Lavrov, P.M. (2014), "Jacobi -type identities in algebras and superalgebras", <i>Theoretical and Mathematical Physics</i>, <b>179</b> (2): <span class="nowrap">550–</span>558, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.5050">1304.5050</a></span>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014TMP...179..550L">2014TMP...179..550L</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11232-014-0161-2">10.1007/s11232-014-0161-2</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119175276">119175276</a></cite></li>

<li><cite id="CITEREFLiboff2003" class="citation cs2"><a href="Richard_L._Liboff" class="mw-redirect" title="Richard L. Liboff">Liboff, Richard L.</a> (2003), <i>Introductory Quantum Mechanics</i> (4th ed.), <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-8714-5</bdi></cite></li>

<li><cite id="CITEREFMcKay2000" class="citation cs2">McKay, Susan (2000), <i>Finite p-groups</i>, Queen Mary Maths Notes, vol. 18, <a href="University_of_London" title="University of London">University of London</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-902480-17-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1802994">1802994</a></cite></li>

<li><cite id="CITEREFMcMahon2008" class="citation cs2">McMahon, D. (2008), <i>Quantum Field Theory</i>, <a href="McGraw_Hill" class="mw-redirect" title="McGraw Hill">McGraw Hill</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-154382-8</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFMcKenzieSnow2005" class="citation cs2"><a href="Ralph_McKenzie" title="Ralph McKenzie">McKenzie, R.</a>; Snow, J. (2005), <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/226377308">"Congruence modular varieties: commutator theory"</a>, in Kudryavtsev, V. B.; Rosenberg, I. G. (eds.), <i>Structural Theory of Automata, Semigroups, and Universal Algebra</i>, NATO Science Series II, vol. 207, Springer, pp. <span class="nowrap">273–</span>329, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-4020-3817-8_11">10.1007/1-4020-3817-8_11</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781402038174</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Commutator">"Commutator"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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