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Commutator

In mathematics, the commutator is a binary operation that measures the degree to which two elements fail to commute under a given algebraic structure, typically defined as [a, b] = aba^{-1}b^{-1} in groups or [a, b] = ab - ba in associative algebras and related contexts.^[1] This concept is central to abstract algebra, where the commutator subgroup [G, G] of a group G—generated by all commutators—forms the smallest normal subgroup such that the quotient group G/[G, G] is abelian, providing a key tool for studying non-abelian groups and their derived series.^[2] Elements a and b are said to commute if [a, b] equals the identity, highlighting the operation's role in classifying commutative versus non-commutative structures.^[3] In the theory of Lie algebras, the commutator serves as the Lie bracket, a bilinear, skew-symmetric operation [x, y] = xy - yx on a vector space that also satisfies the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, underpinning the local structure of Lie groups and applications in differential geometry, representation theory, and physics.^[4] Lie algebras derived from matrix commutators, such as \mathfrak{sl}(n), are particularly influential in symmetry studies and quantum field theory.^[5] In quantum mechanics, the commutator of two observables represented by Hermitian operators A and B is [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 [\hat{x}, \hat{p}] = i\hbar for position and momentum.^[6] This framework extends to broader operator algebras and underpins much of modern physics, including quantum field theory and condensed matter systems.^[7] Beyond mathematics and physics, the term commutator also denotes a mechanical component in electrical engineering: a segmented cylindrical switch in direct-current (DC) motors and generators that reverses the flow of electric current in the armature windings to sustain unidirectional torque or output voltage.^[8] This device, essential to brushed DC machines since the early 19th century, enables practical electromechanical energy conversion but introduces challenges like brush wear and sparking.^[9]

Group Theory

Definition

In group theory, the commutator of two elements g, h \in G in a group G is defined as

[g, h] = g^{-1} h^{-1} g h.

^[1] This expression quantifies the extent to which g and h fail to commute, with [g, h] = e (the identity) if and only if g and h commute.

Identities and Properties

In group theory, the commutator operation satisfies several basic identities that highlight its algebraic structure. One fundamental identity is that the inverse of a commutator is obtained by swapping the arguments: [g, h]^{-1} = [h, g] for all elements g, h in a group G. Another key identity concerns the interaction with products: [g, hk] = [g, k][g, h]^k, where ^k denotes conjugation by k, i.e., a^k = k^{-1}ak. These identities can be verified by direct substitution into the definition [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 \in G satisfy x y z = e, then

[[x, y^{-1}], z]^y \cdot [[y, z^{-1}], x]^z \cdot [[z, x^{-1}], y]^x = e.

^[10] This three-variable identity captures intricate dependencies among commutators and plays a crucial role in analyzing group relations and extensions. The commutator subgroup [G, G], generated by all commutators in G, exhibits important structural properties. By definition, every single commutator [g, h] lies in [G, G]. Furthermore, [G, G] is normal in G, since conjugating a commutator yields another commutator: g[g', h']g^{-1} = [g g' g^{-1}, g h' g^{-1}] for all g, g', h' \in G. In nilpotent groups, iterated commutators eventually vanish; specifically, the lower central series G = \gamma_1(G) \triangleright \gamma_2(G) \triangleright \cdots, where \gamma_{i+1}(G) = [\gamma_i(G), G], terminates at the trivial subgroup \{e\} after finitely many steps. As an illustrative example, consider free groups: in the free group F on a finite or countable generating set, the commutators generate the derived subgroup [F, F].^[11]

Derived Subgroup

The derived subgroup of a group G, denoted [G, G] or G', is the subgroup generated by all elements of the form [g, h] = g^{-1} h^{-1} g h for g, h \in G. This subgroup is normal in G, and the quotient G / [G, G] is the largest abelian quotient of G, known as the abelianization.^[12] The derived series of G is the descending chain of subgroups defined recursively by G^{(0)} = G and G^{(n+1)} = [G^{(n)}, G^{(n)}] for n \geq 0. A group G is solvable if its derived series terminates at the trivial subgroup, meaning there exists some finite k such that G^{(k)} = \{e\}; the smallest such k is called the derived length or solvability length of G. This series captures the extent to which G deviates from being abelian, with each step factoring out commutators to progressively simplify the structure.^[12] Nilpotency relates to the derived subgroup through the lower central series, defined by \gamma_1(G) = G and \gamma_{k+1}(G) = [G, \gamma_k(G)] for k \geq 1. A group G is nilpotent if this series reaches the trivial subgroup in finitely many steps, i.e., \gamma_m(G) = \{e\} for some m; the smallest such m 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 G, providing a finer measure of "near-commutativity" that implies solvability but not conversely.^[12] For abelian groups, the derived subgroup is trivial, as all commutators equal the identity. In contrast, the alternating group A_n for n \geq 5 has derived subgroup [A_n, A_n] = A_n, since A_n is a non-abelian simple group and thus admits no proper nontrivial normal subgroups, forcing the normal derived subgroup to coincide with the whole group.^[12]^[13] The derived subgroup and its associated series play a key role in the classification of finite groups, particularly in distinguishing solvable groups—whose composition factors are cyclic of prime order—from nonsolvable ones, aiding the decomposition in the Classification of Finite Simple Groups. Historically, these concepts informed the Burnside problem from the early 1900s, which asks whether finitely generated groups of bounded exponent are finite; negative solutions in the mid-20th century relied on constructing infinite nilpotent groups via commutator relations in free Burnside groups.^[14]^[15]

Ring Theory

Definition

In ring theory, the commutator of two elements a, b \in R in an associative ring R (not necessarily commutative or unital) is defined as

[a, b] = ab - ba.

This expression quantifies the extent to which multiplication in R 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] for all a, b \in R, which follows directly from the definition. It is also bilinear over the integers, meaning [\lambda a + \mu c, b] = \lambda [a, b] + \mu [c, b] and [a, \lambda b + \mu d] = \lambda [a, b] + \mu [a, d] for \lambda, \mu \in \mathbb{Z}, due to the bilinearity of multiplication in associative rings.^[16] A concrete example arises in the ring M_n(F) of n \times n matrices over a field F, where the standard matrix units E_{ij} (with 1 in the (i,j)-position and zeros elsewhere) satisfy

[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - \delta_{li} E_{kj},

with \delta denoting the Kronecker delta.^[17] This computation illustrates how commutators generate the special linear Lie algebra \mathfrak{sl}_n(F) as the trace-zero matrices.^[17]

Basic Identities

In ring theory, the commutator operation [a, b] = ab - ba exhibits linearity with respect to addition in both arguments. Specifically, for all a, b, c \in R,

[a + b, c] = [a, c] + [b, c], \quad [a, b + c] = [a, b] + [a, c].

These identities follow directly from the bilinearity of multiplication over addition in any associative ring R.^[18] The commutator also obeys a product rule resembling the Leibniz rule for derivations:

[ab, c] = a[b, c] + [a, c]b, \quad [a, bc] = [a, b]c + b[a, c]

for all a, b, c \in R. These can be verified by expanding the definitions using the associativity and distributivity axioms of rings.^[18] The centralizer of R, consisting of elements that commute with every element of R, is the center Z(R) = \{ z \in R \mid [z, r] = 0 \ \forall r \in R \}. This subring is always a two-sided ideal of R.^[18] The map ad_a : r \mapsto [a, r] then defines an inner derivation on R for each a \in R.^[18] As an example, consider the polynomial ring k over a field k. Since k is commutative, all commutators vanish, so [f, g] = 0 for any polynomials f, g \in k, regardless of degrees.^[18]

Advanced Identities

In matrix algebras over a field, the trace of a commutator vanishes, i.e., for any matrices A and B, \operatorname{tr}([A, B]) = 0. This follows from the cyclic property of the trace, \operatorname{tr}(AB) = \operatorname{tr}(BA), yielding \operatorname{tr}([A, B]) = \operatorname{tr}(AB - BA) = \operatorname{tr}(AB) - \operatorname{tr}(BA) = 0.^[19] A key identity for powers in associative rings is the generalized Leibniz rule for commutators: for elements a, b \in R and nonnegative integer n,

[a, b^n] = \sum_{k=0}^{n-1} b^k [a, b] b^{n-1-k}.

This holds by induction on n, with the base case n=1 trivial, and the inductive step using the product rule [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, b where higher nested commutators are negligible (e.g., when \|[a, b]\| is small), e^{[a, b]} \approx e^a e^b e^{-a} e^{-b}, with the approximation arising from the series expansion e^a e^b e^{-a} e^{-b} = e^{[a, b] + \frac{1}{2}[a, [a, b]] + \cdots}.^[20] These identities underpin the definition of the commutator ideal [R, R] in a ring R, the two-sided ideal generated by all commutators \{ [a, b] \mid a, b \in R \}. This ideal captures the "noncommutativity" of R, and the quotient R / [R, R] is the largest commutative quotient ring.^[21]

Lie Algebras

Definition as Lie Bracket

In a Lie algebra \mathcal{L} over a field K, the commutator is defined as the Lie bracket [x, y] for elements x, y \in \mathcal{L}, satisfying the antisymmetry condition [x, y] = -[y, x].^[22] This bracket operation turns \mathcal{L} into a non-associative algebra structure, where the commutator captures the failure of elements to commute, distinct from the multiplicative structure in associative algebras.^[4] The Lie bracket often arises from associative algebras by imposing the commutator [x, y] = xy - yx on the underlying vector space, where xy denotes the original associative product.^[23] This construction endows the vector space with a Lie algebra structure, preserving antisymmetry as a direct consequence of the subtraction.^[23] Such Lie algebras from associative origins, like matrix algebras, illustrate how the commutator serves as a fundamental operation in the study of infinitesimal symmetries.^[4] The Lie bracket is bilinear over K, meaning [a x + b y, z] = a [x, z] + b [y, z] and [z, a x + b y] = a [z, x] + b [z, y] for all scalars a, b \in K and elements x, y, z \in \mathcal{L}.^[22] This bilinearity ensures the bracket behaves linearly with respect to vector space operations, facilitating the algebraic manipulations central to Lie theory.^[4] A canonical example is the general linear Lie algebra \mathfrak{gl}(n, K), consisting of n \times n matrices over K with the bracket [A, B] = AB - BA.^[4] Another is the Heisenberg algebra, a three-dimensional Lie algebra over K with basis \{x, y, z\} where [x, y] = z and all other brackets vanish, demonstrating a nilpotent structure arising from the commutator.^[4]

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, z in a Lie algebra \mathcal{L}, it states

[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 infinitesimal symmetries in Lie groups.^[24] When constructing Lie algebras from associative algebras, the Jacobi identity emerges directly from the underlying associativity. In an associative algebra with product ab, the Lie bracket is the commutator [a, b] = ab - ba. Expanding the left-associated term [[x, y], z] = (xy - yx)z - z(xy - yx) yields xyz - yxz - zxy + zyx, and cyclic permutation for the other terms shows that their sum vanishes due to the associative law (ab)c = a(bc), thereby verifying the identity for the induced Lie structure.^[25] Among its key consequences, the Jacobi identity implies that the adjoint representation \mathrm{ad}_x(y) = [x, y] defines a derivation of \mathcal{L}, obeying the Leibniz rule \mathrm{ad}_x[y, z] = [\mathrm{ad}_x y, z] + [y, \mathrm{ad}_x z] for all x, 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, which follows from antisymmetry but is reinforced by the cyclic consistency of the identity.^[26] A concrete illustration occurs in the special linear Lie algebra \mathfrak{sl}(2, K) over a field K of characteristic zero, with standard basis H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. The commutation relations are [H, X] = 2X, [H, Y] = -2Y, [X, Y] = H. To verify the Jacobi identity for these basis elements, compute [H, [X, Y]] + [X, [Y, H]] + [Y, [H, X]]: the first term is [H, H] = 0; the second is [X, 2Y] = 2[X, Y] = 2H; the third is [Y, 2X] = 2[Y, X] = -2H. The sum $0 + 2H - 2H = 0 confirms the identity holds, and similar checks for other combinations follow by bilinearity.^[27]

Adjoint Representation

In Lie algebras, the adjoint map associated to an element x \in \mathfrak{L} is the linear endomorphism \mathrm{ad}_x: \mathfrak{L} \to \mathfrak{L} defined by \mathrm{ad}_x(y) = [x, y] for all y \in \mathfrak{L}, where [\cdot, \cdot] denotes the Lie bracket. This map is a derivation of the Lie algebra, meaning it satisfies \mathrm{ad}_x([y, z]) = [\mathrm{ad}_x(y), z] + [y, \mathrm{ad}_x(z)], which follows directly from the Jacobi identity.^[28] The adjoint representation of the Lie algebra \mathfrak{L} is the homomorphism \rho: \mathfrak{L} \to \mathfrak{gl}(\mathfrak{L}) given by \rho(x) = \mathrm{ad}_x, where \mathfrak{gl}(\mathfrak{L}) is the Lie algebra of linear endomorphisms of \mathfrak{L} equipped with the commutator bracket. The kernel of \rho is the center Z(\mathfrak{L}) = \{ x \in \mathfrak{L} \mid [x, y] = 0 \ \forall y \in \mathfrak{L} \}. A key property is that \mathrm{ad}_{[x, y]} = [\mathrm{ad}_x, \mathrm{ad}_y], where the bracket on the right is the commutator in \mathfrak{gl}(\mathfrak{L}); this confirms that \rho (or \mathrm{ad}) is a Lie algebra homomorphism from \mathfrak{L} to \mathfrak{gl}(\mathfrak{L}).^[28] For a concrete example, consider the Lie algebra \mathfrak{so}(3), which consists of $3 \times 3 skew-symmetric real matrices and can be identified with \mathbb{R}^3 under the cross product as the Lie bracket, so that [x, y] = x \times y. In this identification, the adjoint representation acts as \mathrm{ad}_x(y) = x \times y, corresponding to the natural action of rotations on vectors. The Killing form on \mathfrak{so}(3), defined by 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 \mathbb{R}^3.^[29]^[30]

Advanced Applications

Graded Algebras

In a graded ring R = \bigoplus_{i \in I} R_i, where I is an abelian group, the commutator of homogeneous elements a \in R_i and b \in R_j is defined as [a, b] = ab - ba. Since the multiplication in R respects the grading, with ab \in R_{i+j} and ba \in R_{j+i} = R_{i+j}, the commutator [a, b] also lies in R_{i+j}. This graded commutator preserves the decomposition and forms the basis for Lie structures in graded settings. For \mathbb{Z}/2\mathbb{Z}-graded algebras, known as superalgebras, the structure is A = A_0 \oplus A_1, where A_0 consists of even elements and A_1 of odd elements, with multiplication satisfying A_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, where |a| denotes the degree (0 or 1) of a; 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.^[31] A key property of the supercommutator in Lie superalgebras is the super Jacobi identity, which adapts the classical Jacobi identity to incorporate grading: for homogeneous elements x, 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.

This identity ensures the bracket defines a consistent algebraic structure and holds in any superalgebra where the supercommutator is used. Applications of supercommutators appear prominently in supersymmetry, where supercharges Q_\alpha satisfy graded commutation relations like \{ 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.^[32]^[33] An illustrative example is the Clifford algebra \mathrm{Cl}(V, Q), which is \mathbb{Z}/2\mathbb{Z}-graded with even and odd parts, where the algebra is generated by vectors e_i satisfying the anticommutator relation \{ e_i, e_j \} = 2 g_{ij} \mathbf{1} (with g the metric) for the defining quadratic form Q. The supercommutator of two odd generators is [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 [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 subalgebra and generates the Lie algebra \mathfrak{so}(V), linking to rotations and spin group representations in the even subalgebra.^[34]

Derivations and Adjoint Derivation

In ring theory, a derivation on a ring R is a linear map \delta: R \to R that satisfies the Leibniz rule \delta(ab) = \delta(a)b + a\delta(b) for all a, b \in R, along with additivity \delta(a + b) = \delta(a) + \delta(b).^[35] This structure generalizes the familiar product rule from calculus to abstract algebraic settings.^[35] An adjoint derivation, also known as an inner derivation, arises from the commutator operation within the ring itself; specifically, for a fixed element a \in [R](/page/R), the map \mathrm{ad}_a: [R](/page/R) \to [R](/page/R) defined by \mathrm{ad}_a(b) = [a, b] = ab - ba is a derivation.^[36] The set \mathrm{Inn}(R) of all such inner derivations forms a Lie subring of the Lie ring \mathrm{Der}(R) of all derivations on R.^[35] Derivations that cannot be expressed as inner derivations for any a \in R are termed outer derivations.^[37] The quotient space \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 R.^[35] A notable example occurs in the Weyl algebra, the noncommutative ring generated by a polynomial ring and a partial derivative operator; here, the partial derivative acts as an outer derivation on the underlying commutative polynomial ring, since inner derivations vanish in the commutative case.^[38] This construction via Ore extension using an outer derivation realizes the ring of polynomial differential operators.^[39] In the context of Lie algebras, the adjoint derivation \mathrm{ad}_a serves as a special case restricted to the Lie bracket.^[36]

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[PDF] Derivations and Iterated Skew Polynomial Rings - arXiv
first Weyl algebra over T and is denoted by A1(T). It becomes evident that ... an outer derivation of the polynomial ring R=T[x1] over a ring T ...