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Commutator
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In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

Group theory

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The commutator of two elements, g and h, of a group G, is the element

[g, h] = g−1h−1gh.[1]

This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg).

The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group.

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

[g, h] = ghg−1h−1.[2]

Using the first definition, this can be expressed as [g−1, h−1].

Identities (group theory)

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Commutator identities are an important tool in group theory.[3] The expression ax denotes the conjugate of a by x, defined as x−1ax.

  3. and
  4. and
  5. and

Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).

N.B., the above definition of the conjugate of a by x is used by some group theorists.[4] Many other group theorists define the conjugate of a by x as xax−1.[5] This is often written . Similar identities hold for these conventions.

Many identities that are true modulo certain subgroups are also used. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:

If the derived subgroup is central, then

Ring theory

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Rings often do not support division. Thus, the commutator of two elements a and b of a ring (or any associative algebra) is defined differently by

The commutator is zero if and only if a and b commute. In linear algebra, if two endomorphisms 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 Lie bracket, every associative algebra can be turned into a Lie algebra.

The anticommutator of two elements a and b of a ring or associative algebra is defined by

Sometimes is used to denote anticommutator, while is then used for commutator.[6] The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras and in the derivation of the Dirac equation in particle physics.

The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation.[7] In phase space, equivalent commutators of function star-products are called Moyal brackets and are completely isomorphic to the Hilbert space commutator structures mentioned.

Identities (ring theory)

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The commutator has the following properties:

Lie-algebra identities

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Relation (3) is called anticommutativity, while (4) is the Jacobi identity.

Additional identities

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If A is a fixed element of a ring R, identity (1) can be interpreted as a Leibniz rule for the map given by . In other words, the map adA defines a derivation on the ring R. 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 Z-bilinearity.

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

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.[8] For example:

Exponential identities

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Consider a ring or algebra in which the exponential can be meaningfully defined, such as a Banach algebra or a ring of formal power series.

In such a ring, Hadamard's lemma applied to nested commutators gives: (For the last expression, see Adjoint derivation below.) This formula underlies the Baker–Campbell–Hausdorff expansion of log(exp(A) exp(B)).

A similar expansion expresses the group commutator of expressions (analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets),

Graded rings and algebras

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When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

Adjoint derivation

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Especially if one deals with multiple commutators in a ring R, another notation turns out to be useful. For an element , we define the adjoint mapping by:

This mapping is a derivation on the ring R:

By the Jacobi identity, it is also a derivation over the commutation operation:

Composing such mappings, we get for example and We may consider itself as a mapping, , where is the ring of mappings from R to itself with composition as the multiplication operation. Then is a Lie algebra homomorphism, preserving the commutator:

By contrast, it is not always a ring homomorphism: usually .

General Leibniz rule

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The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:

Replacing by the differentiation operator , and by the multiplication operator , we get , and applying both sides to a function g, the identity becomes the usual Leibniz rule for the nth derivative .

See also

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Notes

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Commutator

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In , the commutator is a that measures the degree to which two elements fail to commute under a given , typically defined as [a,b]=aba−1b−1[a, b] = aba^{-1}b^{-1} in groups or [a,b]=ab−ba[a, b] = ab - ba in associative algebras and related contexts. This concept is central to , where the [G,G][G, G] of a group GG—generated by all commutators—forms the smallest such that the G/[G,G]G/[G, G] is abelian, providing a key tool for studying non-abelian groups and their derived series. Elements aa and bb are said to commute if [a,b][a, b] equals the identity, highlighting the operation's role in classifying commutative versus non-commutative structures. In the theory of Lie algebras, the commutator serves as the Lie bracket, a bilinear, skew-symmetric operation [x,y]=xy−yx[x, y] = xy - yx on a that also satisfies the [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, underpinning the local structure of Lie groups and applications in , , and physics. Lie algebras derived from matrix commutators, such as sl(n)\mathfrak{sl}(n), are particularly influential in symmetry studies and . In , the commutator of two observables represented by Hermitian operators AA and BB is [A,B]=AB−BA[A, B] = AB - BA; if this vanishes, the observables commute and can be simultaneously measured with arbitrary precision, whereas non-zero commutators imply fundamental limits via the Heisenberg uncertainty principle, such as [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar for position and . This framework extends to broader operator algebras and underpins much of , including and condensed matter systems. Beyond and physics, the term commutator also denotes a mechanical component in : a segmented cylindrical switch in direct-current (DC) motors and generators that reverses the flow of in the armature windings to sustain unidirectional or output voltage. This device, essential to brushed DC machines since the early , enables practical electromechanical conversion but introduces challenges like wear and sparking.

Group Theory

Definition

In group theory, the commutator of two elements g,h∈Gg, h \in G in a group GG is defined as [g,h]=g−1h−1gh.[g, h] = g^{-1} h^{-1} g h. This expression quantifies the extent to which gg and hh fail to commute, with [g,h]=e[g, h] = e (the identity) if and only if gg and hh commute.

Identities and Properties

In , the commutator operation satisfies several basic identities that highlight its . One fundamental identity is that the inverse of a commutator is obtained by swapping the arguments: [g,h]−1=[h,g][g, h]^{-1} = [h, g] for all elements g,hg, h in a group GG. Another key identity concerns the interaction with products: [g,hk]=[g,k][g,h]k[g, hk] = [g, k][g, h]^k, where k^k denotes conjugation by kk, i.e., ak=k−1aka^k = k^{-1}ak. These identities can be verified by direct substitution into the definition [g,h]=g−1h−1gh[g, h] = g^{-1}h^{-1}gh and expanding using the group axioms. A significant higher-order relation is the Hall–Witt identity, which extends commutator properties to three elements. If x,y,z∈Gx, y, z \in G satisfy xyz=ex y z = e, then [[x,y−1],z]y⋅[[y,z−1],x]z⋅[[z,x−1],y]x=e.[[x, y^{-1}], z]^y \cdot [[y, z^{-1}], x]^z \cdot [[z, x^{-1}], y]^x = e. This three-variable identity captures intricate dependencies among commutators and plays a crucial role in analyzing group relations and extensions. The [G,G][G, G], generated by all commutators in GG, exhibits important structural properties. By definition, every single commutator [g,h][g, h] lies in [G,G][G, G]. Furthermore, [G,G][G, G] is normal in GG, since conjugating a commutator yields another commutator: g[g′,h′]g−1=[gg′g−1,gh′g−1]g[g', h']g^{-1} = [g g' g^{-1}, g h' g^{-1}] for all g,g′,h′∈Gg, g', h' \in G. In groups, iterated commutators eventually vanish; specifically, the lower central series G=γ1(G)▹γ2(G)▹⋯G = \gamma_1(G) \triangleright \gamma_2(G) \triangleright \cdots, where γi+1(G)=[γi(G),G]\gamma_{i+1}(G) = [\gamma_i(G), G], terminates at the trivial {e}\{e\} after finitely many steps. As an illustrative example, consider s: in the FF on a finite or countable generating set, the commutators generate the derived [F,F][F, F].

Derived Subgroup

The derived of a group GG, denoted [G,G][G, G] or G′G', is the generated by all elements of the form [g,h]=g−1h−1gh[g, h] = g^{-1} h^{-1} g h for g,h∈Gg, h \in G. This is normal in GG, and the G/[G,G]G / [G, G] is the largest abelian of GG, known as the abelianization. The derived series of GG is the descending chain of subgroups defined recursively by G(0)=GG^{(0)} = G and G(n+1)=[G(n),G(n)]G^{(n+1)} = [G^{(n)}, G^{(n)}] for n≥0n \geq 0. A group GG is solvable if its derived series terminates at the trivial subgroup, meaning there exists some finite kk such that G(k)={e}G^{(k)} = \{e\}; the smallest such kk is called the derived length or solvability length of GG. This series captures the extent to which GG deviates from being abelian, with each step factoring out commutators to progressively simplify the structure. Nilpotency relates to the derived subgroup through the lower central series, defined by γ1(G)=G\gamma_1(G) = G and γk+1(G)=[G,γk(G)]\gamma_{k+1}(G) = [G, \gamma_k(G)] for k≥1k \geq 1. A group GG is nilpotent if this series reaches the trivial subgroup in finitely many steps, i.e., γm(G)={e}\gamma_m(G) = \{e\} for some mm; the smallest such mm is the nilpotency class. Unlike the derived series, which iterates commutators within the previous term, the lower central series incorporates commutators with the full group GG, providing a finer measure of "near-commutativity" that implies solvability but not conversely. For abelian groups, the derived subgroup is trivial, as all commutators equal the identity. In contrast, the AnA_n for n≥5n \geq 5 has derived subgroup [An,An]=An[A_n, A_n] = A_n, since AnA_n is a non-abelian and thus admits no proper nontrivial normal s, forcing the normal derived subgroup to coincide with the whole group. The derived subgroup and its associated series play a key role in the of finite groups, particularly in distinguishing solvable groups—whose composition factors are cyclic of prime order—from nonsolvable ones, aiding the decomposition in the . Historically, these concepts informed the from the early , which asks whether finitely generated groups of bounded exponent are finite; negative solutions in the mid-20th century relied on constructing infinite groups via commutator relations in free Burnside groups.

Ring Theory

Definition

In ring theory, the commutator of two elements a,b∈Ra, b \in R in an associative ring RR (not necessarily commutative or unital) is defined as [a,b]=ab−ba.[a, b] = ab - ba. This expression quantifies the extent to which multiplication in RR fails to be commutative, analogous to the role of the group commutator in measuring deviations from multiplicativity in groups. The commutator operation is skew-symmetric, satisfying [a,b]=−[b,a][a, b] = -[b, a] for all a,b∈Ra, b \in R, which follows directly from the definition. It is also bilinear over the integers, meaning [λa+μc,b]=λ[a,b]+μ[c,b][\lambda a + \mu c, b] = \lambda [a, b] + \mu [c, b] and [a,λb+μd]=λ[a,b]+μ[a,d][a, \lambda b + \mu d] = \lambda [a, b] + \mu [a, d] for λ,μ∈Z\lambda, \mu \in \mathbb{Z}, due to the bilinearity of multiplication in associative rings. A concrete example arises in the ring Mn(F)M_n(F) of n×nn \times n matrices over a field FF, where the standard matrix units EijE_{ij} (with 1 in the (i,j)(i,j)-position and zeros elsewhere) satisfy [Eij,Ekl]=δjkEil−δliEkj,[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - \delta_{li} E_{kj}, with δ\delta denoting the Kronecker delta. This computation illustrates how commutators generate the special linear Lie algebra sln(F)\mathfrak{sl}_n(F) as the trace-zero matrices.

Basic Identities

In , the commutator operation [a,b]=ab−ba[a, b] = ab - ba exhibits linearity with respect to in both arguments. Specifically, for all a,b,c∈Ra, b, c \in R, [a+b,c]=[a,c]+[b,c],[a,b+c]=[a,b]+[a,c].[a + b, c] = [a, c] + [b, c], \quad [a, b + c] = [a, b] + [a, c]. These identities follow directly from the bilinearity of over in any associative ring RR. The commutator also obeys a resembling the Leibniz rule for derivations: [ab,c]=a[b,c]+[a,c]b,[a,bc]=[a,b]c+b[a,c][ab, c] = a[b, c] + [a, c]b, \quad [a, bc] = [a, b]c + b[a, c] for all a,b,c∈Ra, b, c \in R. These can be verified by expanding the definitions using the associativity and distributivity axioms of rings. The centralizer of RR, consisting of elements that commute with every element of RR, is the center Z(R)={z∈R∣[z,r]=0 ∀r∈R}Z(R) = \{ z \in R \mid [z, r] = 0 \ \forall r \in R \}. This is always a two-sided ideal of RR. The map ada:r↦[a,r]ad_a : r \mapsto [a, r] then defines an inner derivation on RR for each a∈Ra \in R. As an example, consider the polynomial ring kk over a field kk. Since kk is commutative, all commutators vanish, so [f,g]=0[f, g] = 0 for any polynomials f,g∈kf, g \in k, regardless of degrees.

Advanced Identities

In matrix algebras over a field, the trace of a commutator vanishes, i.e., for any matrices AA and BB, tr⁡([A,B])=0\operatorname{tr}([A, B]) = 0. This follows from the cyclic property of the trace, tr⁡(AB)=tr⁡(BA)\operatorname{tr}(AB) = \operatorname{tr}(BA), yielding tr⁡([A,B])=tr⁡(AB−BA)=tr⁡(AB)−tr⁡(BA)=0\operatorname{tr}([A, B]) = \operatorname{tr}(AB - BA) = \operatorname{tr}(AB) - \operatorname{tr}(BA) = 0. A key identity for powers in associative rings is the generalized Leibniz rule for commutators: for elements a,b∈Ra, b \in R and nonnegative integer nn, [a,bn]=∑k=0n−1bk[a,b]bn−1−k.[a, b^n] = \sum_{k=0}^{n-1} b^k [a, b] b^{n-1-k}. This holds by induction on nn, with the base case n=1n=1 trivial, and the inductive step using the [a,bc]=[a,b]c+b[a,c][a, bc] = [a, b]c + b[a, c], which commutators satisfy in any associative ring. In rings admitting exponentials, such as those over fields of characteristic zero or matrix rings, the exponential of a commutator relates to conjugations via the Baker-Campbell-Hausdorff formula. Specifically, for elements a,ba, b where higher nested commutators are negligible (e.g., when ∥[a,b]∥\|[a, b]\| is small), e[a,b]≈eaebe−ae−be^{[a, b]} \approx e^a e^b e^{-a} e^{-b}, with the arising from the eaebe−ae−b=e[a,b]+12[a,[a,b]]+⋯e^a e^b e^{-a} e^{-b} = e^{[a, b] + \frac{1}{2}[a, [a, b]] + \cdots}. These identities underpin the definition of the commutator [R,R][R, R] in a ring RR, the two-sided ideal generated by all commutators {[a,b]∣a,b∈R}\{ [a, b] \mid a, b \in R \}. This captures the "noncommutativity" of RR, and the R/[R,R]R / [R, R] is the largest commutative .

Lie Algebras

Definition as Lie Bracket

In a L\mathcal{L} over a field KK, the commutator is defined as the Lie bracket [x,y][x, y] for elements x,y∈Lx, y \in \mathcal{L}, satisfying the antisymmetry condition [x,y]=−[y,x][x, y] = -[y, x]. This bracket operation turns L\mathcal{L} into a non-associative structure, where the commutator captures the failure of elements to commute, distinct from the multiplicative structure in associative algebras. The Lie bracket often arises from associative algebras by imposing the commutator [x,y]=xy−yx[x, y] = xy - yx on the underlying vector space, where xyxy denotes the original associative product. This construction endows the vector space with a structure, preserving antisymmetry as a direct consequence of the . Such from associative origins, like matrix algebras, illustrate how the commutator serves as a fundamental operation in the study of symmetries. The Lie bracket is bilinear over KK, meaning [ax+by,z]=a[x,z]+b[y,z][a x + b y, z] = a [x, z] + b [y, z] and [z,ax+by]=a[z,x]+b[z,y][z, a x + b y] = a [z, x] + b [z, y] for all scalars a,b∈Ka, b \in K and elements x,y,z∈Lx, y, z \in \mathcal{L}. This bilinearity ensures the bracket behaves linearly with respect to operations, facilitating the algebraic manipulations central to . A canonical example is the general linear gl(n,K)\mathfrak{gl}(n, K), consisting of n×nn \times n matrices over KK with the bracket [A,B]=AB−BA[A, B] = AB - BA. Another is the Heisenberg algebra, a three-dimensional over KK with basis {x,y,z}\{x, y, z\} where [x,y]=z[x, y] = z and all other brackets vanish, demonstrating a nilpotent structure arising from the commutator.

Jacobi Identity

The Jacobi identity serves as a defining associator for Lie algebras, encapsulating the non-associative nature of the Lie bracket while ensuring structural integrity. For all elements x,y,zx, y, z in a Lie algebra L\mathcal{L}, it states [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. This axiom, alongside bilinearity and antisymmetry, distinguishes Lie algebras from more general non-associative algebras and is essential for modeling symmetries in Lie groups. When constructing algebras from s, the emerges directly from the underlying associativity. In an with product abab, the is the commutator [a,b]=ab−ba[a, b] = ab - ba. Expanding the left-associated term [[x,y],z]=(xy−yx)z−z(xy−yx)[[x, y], z] = (xy - yx)z - z(xy - yx) yields xyz−yxz−zxy+zyxxyz - yxz - zxy + zyx, and for the other terms shows that their sum vanishes due to the associative law (ab)c=a(bc)(ab)c = a(bc), thereby verifying the identity for the induced structure. Among its key consequences, the Jacobi identity implies that the adjoint representation adx(y)=[x,y]\mathrm{ad}_x(y) = [x, y] defines a derivation of L\mathcal{L}, obeying the Leibniz rule adx[y,z]=[adxy,z]+[y,adxz]\mathrm{ad}_x[y, z] = [\mathrm{ad}_x y, z] + [y, \mathrm{ad}_x z] for all x,y,z∈Lx, y, z \in \mathcal{L}. This derivation property facilitates the study of ideals, subalgebras, and representations, while also ensuring alternativity in iterated brackets, such as [x,[x,y]]=0[x, [x, y]] = 0, which follows from antisymmetry but is reinforced by the cyclic consistency of the identity. A concrete illustration occurs in the special linear Lie algebra sl(2,K)\mathfrak{sl}(2, K) over a field KK of characteristic zero, with H=(100−1)H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, X=(0100)X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y=(0010)Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. The commutation relations are [H,X]=2X[H, X] = 2X, [H,Y]=−2Y[H, Y] = -2Y, [X,Y]=H[X, Y] = H. To verify the for these basis elements, compute [H,[X,Y]]+[X,[Y,H]]+[Y,[H,X]][H, [X, Y]] + [X, [Y, H]] + [Y, [H, X]]: the first term is [H,H]=0[H, H] = 0; the second is [X,2Y]=2[X,Y]=2H[X, 2Y] = 2[X, Y] = 2H; the third is [Y,2X]=2[Y,X]=−2H[Y, 2X] = 2[Y, X] = -2H. The sum 0+2H−2H=00 + 2H - 2H = 0 confirms the identity holds, and similar checks for other combinations follow by bilinearity.

Adjoint Representation

In Lie algebras, the adjoint map associated to an element x∈Lx \in \mathfrak{L} is the linear adx:L→L\mathrm{ad}_x: \mathfrak{L} \to \mathfrak{L} defined by adx(y)=[x,y]\mathrm{ad}_x(y) = [x, y] for all y∈Ly \in \mathfrak{L}, where [⋅,⋅][\cdot, \cdot] denotes the bracket. This map is a derivation of the , meaning it satisfies adx([y,z])=[adx(y),z]+[y,adx(z)]\mathrm{ad}_x([y, z]) = [\mathrm{ad}_x(y), z] + [y, \mathrm{ad}_x(z)], which follows directly from the . The adjoint representation of the Lie algebra L\mathfrak{L} is the homomorphism ρ:L→gl(L)\rho: \mathfrak{L} \to \mathfrak{gl}(\mathfrak{L}) given by ρ(x)=adx\rho(x) = \mathrm{ad}_x, where gl(L)\mathfrak{gl}(\mathfrak{L}) is the Lie algebra of linear endomorphisms of L\mathfrak{L} equipped with the commutator bracket. The kernel of ρ\rho is the center Z(L)={x∈L∣[x,y]=0 ∀y∈L}Z(\mathfrak{L}) = \{ x \in \mathfrak{L} \mid [x, y] = 0 \ \forall y \in \mathfrak{L} \}. A key property is that ad[x,y]=[adx,ady]\mathrm{ad}_{[x, y]} = [\mathrm{ad}_x, \mathrm{ad}_y], where the bracket on the right is the commutator in gl(L)\mathfrak{gl}(\mathfrak{L}); this confirms that ρ\rho (or ad\mathrm{ad}) is a Lie algebra homomorphism from L\mathfrak{L} to gl(L)\mathfrak{gl}(\mathfrak{L}). For a concrete example, consider the so(3)\mathfrak{so}(3), which consists of 3×33 \times 3 skew-symmetric real matrices and can be identified with R3\mathbb{R}^3 under the as the Lie bracket, so that [x,y]=x×y[x, y] = x \times y. In this identification, the acts as adx(y)=x×y\mathrm{ad}_x(y) = x \times y, corresponding to the natural action of rotations on vectors. The Killing form on so(3)\mathfrak{so}(3), defined by B(x,y)=tr(adxady)B(x, y) = \mathrm{tr}(\mathrm{ad}_x \mathrm{ad}_y), is non-degenerate and proportional to the negative of the standard Euclidean inner product on R3\mathbb{R}^3.

Advanced Applications

Graded Algebras

In a R=⨁i∈IRiR = \bigoplus_{i \in I} R_i, where II is an , the commutator of homogeneous elements a∈Ria \in R_i and b∈Rjb \in R_j is defined as [a,b]=ab−ba[a, b] = ab - ba. Since the multiplication in RR respects the grading, with ab∈Ri+jab \in R_{i+j} and ba∈Rj+i=Ri+jba \in R_{j+i} = R_{i+j}, the commutator [a,b][a, b] also lies in Ri+jR_{i+j}. This graded commutator preserves the decomposition and forms the basis for structures in graded settings. For Z/2Z\mathbb{Z}/2\mathbb{Z}-graded algebras, known as superalgebras, the structure is A=A0⊕A1A = A_0 \oplus A_1, where A0A_0 consists of even elements and A1A_1 of odd elements, with multiplication satisfying AiAj⊆Ai+jmod  2A_i A_j \subseteq A_{i+j \mod 2}. The supercommutator, which generalizes the usual commutator to account for parity, is defined for homogeneous elements as [a,b]s=ab−(−1)∣a∣∣b∣ba[a, b]_s = ab - (-1)^{|a||b|} ba, where ∣a∣|a| denotes the degree (0 or 1) of aa; this extends bilinearly to all elements. In Lie superalgebras, the supercommutator serves as the Lie bracket, ensuring antisymmetry up to sign: [b,a]s=−(−1)∣a∣∣b∣[a,b]s[b, a]_s = - (-1)^{|a||b|} [a, b]_s. A key property of the supercommutator in Lie superalgebras is the super Jacobi identity, which adapts the classical to incorporate grading: for homogeneous elements x,y,zx, y, z, (−1)∣x∣∣y∣[[x,y]s,z]s+(−1)∣y∣∣z∣[[y,z]s,x]s+(−1)∣z∣∣x∣[[z,x]s,y]s=0.(-1)^{|x||y|} [[x, y]_s, z]_s + (-1)^{|y||z|} [[y, z]_s, x]_s + (-1)^{|z||x|} [[z, x]_s, y]_s = 0. This identity ensures the bracket defines a consistent and holds in any superalgebra where the supercommutator is used. Applications of supercommutators appear prominently in , where supercharges QαQ_\alpha satisfy graded commutation relations like {Qα,Qˉβ}=2σαβμPμ\{ Q_\alpha, \bar{Q}_\beta \} = 2 \sigma^\mu_{\alpha \beta} P_\mu, closing the algebra on spacetime translations and the Hamiltonian, thus unifying bosonic and fermionic symmetries. An illustrative example is the Cl(V,Q)\mathrm{Cl}(V, Q), which is Z/2Z\mathbb{Z}/2\mathbb{Z}-graded with even and odd parts, where the algebra is generated by vectors eie_i satisfying the anticommutator relation {ei,ej}=2gij1\{ e_i, e_j \} = 2 g_{ij} \mathbf{1} (with gg the metric) for the defining QQ. The supercommutator of two odd generators is [ei,ej]s=eiej+ejei={ei,ej}=2gij1[e_i, e_j]_s = e_i e_j + e_j e_i = \{ e_i, e_j \} = 2 g_{ij} \mathbf{1}, which is central (even). However, the ordinary commutator [ei,ej]=eiej−ejei=2(eiej−gij1)[e_i, e_j] = e_i e_j - e_j e_i = 2 (e_i e_j - g_{ij} \mathbf{1}) lies in the even and generates the so(V)\mathfrak{so}(V), linking to rotations and representations in the even .

Derivations and Adjoint Derivation

In ring theory, a derivation on a ring RR is a linear map δ:R→R\delta: R \to R that satisfies the Leibniz rule δ(ab)=δ(a)b+aδ(b)\delta(ab) = \delta(a)b + a\delta(b) for all a,b∈Ra, b \in R, along with additivity δ(a+b)=δ(a)+δ(b)\delta(a + b) = \delta(a) + \delta(b). This structure generalizes the familiar product rule from calculus to abstract algebraic settings. An adjoint derivation, also known as an inner derivation, arises from the commutator operation within the ring itself; specifically, for a fixed element a∈[R](/page/R)a \in [R](/page/R), the map ada:[R](/page/R)→[R](/page/R)\mathrm{ad}_a: [R](/page/R) \to [R](/page/R) defined by ada(b)=[a,b]=ab−ba\mathrm{ad}_a(b) = [a, b] = ab - ba is a derivation. The set Inn(R)\mathrm{Inn}(R) of all such inner derivations forms a subring of the ring Der(R)\mathrm{Der}(R) of all derivations on RR. Derivations that cannot be expressed as inner derivations for any a∈Ra \in R are termed outer derivations. The quotient space Der(R)/Inn(R)\mathrm{Der}(R)/\mathrm{Inn}(R) provides a classification of outer derivations up to equivalence by inner ones, capturing the "essential" non-inner behavior of derivations on RR. A notable example occurs in the Weyl algebra, the generated by a and a operator; here, the acts as an outer derivation on the underlying commutative , since inner derivations vanish in the commutative case. This construction via Ore extension using an outer derivation realizes the ring of polynomial differential operators. In the context of Lie algebras, the adjoint derivation ada\mathrm{ad}_a serves as a special case restricted to the Lie bracket.

References

  1. https://mathworld.wolfram.com/Commutator.html
  2. https://people.brandeis.edu/~igusa/Math101aF07/Math101a_notesA2.pdf
  3. https://www.math.wm.edu/~vinroot/430S11Commutators.pdf
  4. https://www2.math.upenn.edu/~brweber/Courses/2012/Math650/Notes/L1_DefsandExamples.pdf
  5. https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Turzillo.pdf
  6. http://eng-web1.eng.famu.fsu.edu/~dommelen/quantum/style_a/commute.html
  7. https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf
  8. https://ocw.mit.edu/courses/6-061-introduction-to-electric-power-systems-spring-2011/22956cb93817f9edabecd0731b51dd1b_MIT6_061S11_ch11.pdf
  9. https://geosci.uchicago.edu/~moyer/GEOS24705/Readings/ElecReadingII_Motors.pdf
  10. https://math.berkeley.edu/~tb65536/Commuting_Probability.pdf
  11. https://academicweb.nd.edu/~andyp/papers/Commutator.pdf
  12. http://math.stanford.edu/~conrad/210BPage/handouts/SOLVandNILgroups.pdf
  13. http://ramanujan.math.trinity.edu/rdaileda/teach/s19/m3362/alternating.pdf
  14. https://math.mit.edu/research/highschool/primes/circle/documents/2022/Gracie.pdf
  15. https://marcelgoh.ca/misc/expo/burnside.pdf
  16. https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/commutators-in-associative-rings/A0171691FB08F152595225FA6CE2858E
  17. https://beckassets.blob.core.windows.net/product/readingsample/10216396/9789814335317_excerpt_001.pdf
  18. https://books.google.com/books/about/A_First_Course_in_Noncommutative_Rings.html?id=f15FyZuZ3-4C
  19. https://arxiv.org/pdf/2111.04884
  20. https://webhome.phy.duke.edu/~mehen/760/ProblemSets/BCH.pdf
  21. https://academics.uccs.edu/zmesyan/papers/Commutators.pdf
  22. https://pi.math.cornell.edu/~dmehrle/notes/old/promys14/liealgebras.pdf
  23. https://people.brandeis.edu/~igusa/Math223aF11/Notes223a01.pdf
  24. https://prclare.people.wm.edu/m410f23/Liealg_chap02.pdf
  25. https://webusers.imj-prg.fr/~michael.harris/Day1.pdf
  26. https://www.math.uci.edu/~brusso/humphreys.pdf
  27. https://www.imsc.res.in/~knr/past/laten/sl2.pdf
  28. https://books.physics.oregonstate.edu/GELG/adjointn.html
  29. https://courses.physics.ucsd.edu/2018/Spring/physics220/LECTURES/CH08.pdf
  30. https://www.physik.uni-hamburg.de/th2/ag-louis/dokumente/lectures/ws-15-16/ws-15-16-proseminar-10.pdf
  31. https://arxiv.org/abs/1710.02706
  32. https://arxiv.org/pdf/1304.5050
  33. https://zzxianyu.com/wp-content/uploads/2017/01/susy_basics.pdf
  34. https://www.math.toronto.edu/mein/teaching/LieClifford/cl2.pdf
  35. https://www.researchgate.net/publication/261402165_ON_DERIVATIONS_IN_RINGS_AND_THEIR_APPLICATIONS
  36. https://escholarship.org/content/qt5sg0d8wd/qt5sg0d8wd.pdf
  37. https://www.sciepub.com/portal/downloads?doi=10.12691/ajams-7-2-4&filename=ajams-7-2-4.pdf
  38. https://files.vipulnaik.com/exposition/flavourofnca.pdf
  39. https://arxiv.org/pdf/1210.1476
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