Group Theory: Definitions and Axioms

The Lorentz, Poincaré, and internal symmetry groups (e.g. $U (1)$ , $S U (2)$ , $S U (3)$ ) used throughout physics are all instances of Lie groups with associated Lie algebras and representations. This document collects the abstract definitions; specific examples appear in the next section. It is intended as a reference companion to QFT/preliminaries.md, which uses these definitions to set up the Lorentz and Poincaré groups and Wigner's classification.

For the manifold/topology prerequisites referenced below — manifold, tangent space, smooth map — see topology-manifolds.md.

Group axioms

A group $(G, \cdot)$ is a set $G$ together with a binary operation $\cdot : G \times G \to G$ satisfying:

(G1) Closure: $g_{1}, g_{2} \in G \Rightarrow g_{1} \cdot g_{2} \in G$ .
(G2) Associativity: $(g_{1} \cdot g_{2}) \cdot g_{3} = g_{1} \cdot (g_{2} \cdot g_{3})$ for all $g_{1}, g_{2}, g_{3} \in G$ .
(G3) Identity: there exists $e \in G$ such that $e \cdot g = g \cdot e = g$ for all $g \in G$ .
(G4) Inverses: for every $g \in G$ there exists $g^{- 1} \in G$ such that $g \cdot g^{- 1} = g^{- 1} \cdot g = e$ .

If additionally $g_{1} \cdot g_{2} = g_{2} \cdot g_{1}$ for all elements, $G$ is abelian (commutative).

Subgroups, normal subgroups, quotients

$H \subseteq G$ is a subgroup if $H$ is itself a group under the inherited operation.
For $g \in G$ and $H \subseteq G$ , the conjugate of $H$ by $g$ is the set $g H g^{- 1} \equiv {g h g^{- 1} : h \in H} \subseteq G,$ obtained by sandwiching every element of $H$ between $g$ and $g^{- 1}$ . It is itself a subgroup of $G$ , isomorphic to $H$ .
The left coset of $H$ by $g$ is the set $g H \equiv {g h : h \in H}$ ; the right coset is $H g \equiv {h g : h \in H}$ . Cosets partition $G$ into disjoint pieces of equal size.
$H \subseteq G$ is a normal subgroup ( $H ◃ G$ ) if $g H g^{- 1} = H$ for all $g \in G$ — equivalently, every left coset equals the corresponding right coset ( $g H = H g$ ). Normal subgroups are exactly the kernels of group homomorphisms.
For $H ◃ G$ , the quotient group $G / H$ has elements the cosets $g H$ , with multiplication $(g_{1} H) (g_{2} H) = (g_{1} g_{2}) H$ . (Normality is what makes this product well-defined independently of which representatives $g_{i}$ are chosen.)

Example: in $S U (2)$ , the centre ${\pm 1}$ is a normal subgroup; the quotient $S U (2) / {\pm 1} ≅ SO (3)$ .

Homomorphisms and isomorphisms

A homomorphism $ϕ : G \to H$ is a map preserving the group operation: $ϕ (g_{1} g_{2}) = ϕ (g_{1}) ϕ (g_{2})$ . It automatically satisfies $ϕ (e_{G}) = e_{H}$ and $ϕ (g^{- 1}) = ϕ (g)^{- 1}$ . A bijective homomorphism is an isomorphism, written $G ≅ H$ .

The kernel of $ϕ$ is the preimage of the identity:

$ker ϕ \equiv ϕ^{- 1} (e_{H}) = {g \in G : ϕ (g) = e_{H}} \subseteq G .$

It is the set of elements that $ϕ$ collapses to the identity, and is always a normal subgroup of $G$ . Properties:

$ϕ$ is injective iff $ker ϕ = {e_{G}}$ (only the identity gets sent to the identity).
The image $im ϕ = ϕ (G) \subseteq H$ is a subgroup of $H$ (not necessarily normal).
First isomorphism theorem: $G / ker ϕ ≅ im ϕ$ .

Example: the determinant $det : G L (n, R) \to R^{*}$ is a homomorphism with kernel $S L (n, R)$ (matrices of determinant 1). The covering map $S L (2, C) \to S O^{+} (1, 3)$ has kernel ${\pm 1}$ .

Group actions and representations

A (left) group action of $G$ on a set $X$ is a map $G \times X \to X$ , $(g, x) \mapsto g \cdot x$ , satisfying $e \cdot x = x$ and $(g_{1} g_{2}) \cdot x = g_{1} \cdot (g_{2} \cdot x)$ .

A representation of $G$ on a vector space $V$ is a homomorphism $ρ : G \to G L (V)$ (the group of invertible linear maps on $V$ ). Equivalently, $G$ acts linearly on $V$ :

$ρ (g_{1} g_{2}) = ρ (g_{1}) ρ (g_{2}), ρ (e) = 1_{V} .$

$ρ$ is unitary if $V$ has an inner product preserved by $ρ (g)$ for all $g$ (i.e. $ρ (g)^{†} ρ (g) = 1_{V}$ ). Unitary representations are what physical Hilbert-space transformations must be (probabilities are conserved).
$ρ$ is irreducible (irrep) if no proper non-trivial subspace $W \subset V$ is invariant under all $ρ (g)$ . Irreps are the elementary building blocks; reducible reps decompose as direct sums of irreps (Schur's lemma).
A projective representation satisfies $ρ (g_{1}) ρ (g_{2}) = ω (g_{1}, g_{2}) ρ (g_{1} g_{2})$ for some phase $ω$ . Quantum mechanics uses projective reps because physical states are rays, not vectors. Projective reps of $G$ correspond to ordinary reps of the universal cover $\tilde{G}$ (this is why $S L (2, C)$ rather than $S O^{+} (1, 3)$ shows up in physics — see QFT/preliminaries.md § Universal cover $S L (2, C)$ ).

Standard matrix groups (notation)

The following classical matrix Lie groups appear throughout this repository and in the rest of the QFT literature. Let $F$ denote either $R$ or $C$ .

Symbol	Name	Definition	Dimension over $R$
$G L (n, F)$	General linear group	invertible $n \times n$ matrices over $F$ (equivalently, $det \neq = 0$ )	$n^{2}$ ( $R$ ) or $2 n^{2}$ ( $C$ )
$S L (n, F)$	Special linear group	${M \in G L (n, F) : det M = 1}$	$n^{2} - 1$ or $2 (n^{2} - 1)$
$O (n)$	Orthogonal group	${M \in G L (n, R) : M^{T} M = 1}$ (preserves Euclidean inner product)	$n (n - 1) /2$
$SO (n)$	Special orthogonal group	$O (n) \cap S L (n, R)$ (rotations: orientation-preserving orthogonal)	$n (n - 1) /2$
$O (p, q)$	Indefinite orthogonal group	preserves a metric of signature $(p, q)$ ; e.g. $O (1, 3)$ is the Lorentz group	$\frac{1}{2} (p + q) (p + q - 1)$
$S O^{+} (p, q)$	Proper orthochronous component	identity component of $O (p, q)$	same
$U (n)$	Unitary group	${M \in G L (n, C) : M^{†} M = 1}$ (preserves Hermitian inner product)	$n^{2}$
$S U (n)$	Special unitary group	$U (n) \cap S L (n, C)$ ( $det M = 1$ )	$n^{2} - 1$
$Sp (2 n, F)$	Symplectic group	preserves a symplectic form $ω$	$n (2 n + 1)$
$Spin (n)$	Spin group	universal double cover of $SO (n)$ for $n \geq 3$	$n (n - 1) /2$

Inclusion lattice (for the most-used cases): $S U (n) \subset U (n) \subset G L (n, C)$ , and $SO (n) \subset O (n) \subset G L (n, R)$ , with $S U (n), SO (n)$ being the determinant-1 subgroups.

Compactness summary (matters for representation theory, see Compact vs. non-compact below):

Compact: $U (n), S U (n), O (n), SO (n), Spin (n), Sp (n)$ .
Non-compact: $G L (n, F), S L (n, F), O (p, q)$ with $p, q > 0$ , $Sp (2 n, R)$ .

Lie groups and Lie algebras

A Lie group is a group that is also a smooth manifold, with the group operations $g \cdot h$ and $g^{- 1}$ being smooth maps. (Here "smooth" means $C^{\infty}$ — infinitely differentiable in local coordinates; for matrix Lie groups it just means the matrix entries depend smoothly on parameters. Full definitions of manifold, topological space, and smooth map are collected in topology-manifolds.md.) Examples: $R^{n}$ (translations), $U (1) = S^{1}$ , $SO (n)$ , $S U (n)$ , $S L (n, R)$ , $S L (n, C)$ , the Lorentz group $L_{+}^{↑}$ , the Poincaré group $P_{+}^{↑}$ .

The Lie algebra $g$ of a Lie group $G$ is the tangent space at the identity, $T_{e} G$ , equipped with the Lie bracket $[\cdot, \cdot] : g \times g \to g$ — bilinear, antisymmetric, and satisfying the Jacobi identity

$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.$

For matrix Lie groups (the only kind we use), the bracket is the matrix commutator $[X, Y] = X Y - Y X$ .

The exponential map $exp : g \to G$ takes a Lie algebra element to a one-parameter subgroup of $G$ :

$exp (tX) \in G, exp (0) = e, \frac{d}{d t}_{t = 0} exp (tX) = X .$

For matrix groups, $exp (X) = \sum_{n = 0}^{\infty} X^{n} / n!$ . In the connected component of $G$ , every element lies in some one-parameter subgroup, so $exp$ is locally bijective and gives a near-identity parameterization.

Generators and structure constants. A generator of $G$ is an element of the Lie algebra $g = T_{e} G$ (the tangent space at the identity element $e \in G$ — see topology-manifolds.md § Tangent Spaces). Choose a basis ${T^{a}}_{a = 1}^{d i m G}$ of $g$ — i.e. a linearly independent set spanning $g$ , so that every $X \in g$ has a unique expansion $X = X^{a} T^{a}$ with real coefficients $X^{a}$ . Every group element near the identity can then be written as

$g = exp (i θ^{a} T^{a}) \in G, θ^{a} \in R,$

so the $T^{a}$ "generate" group elements via the exponential map — hence the name. The Lie algebra structure is encoded in the structure constants $f^{ab c}$ :

$[T^{a}, T^{b}] = i f^{ab c} T^{c} .$

(The factor of $i$ is a physics convention; mathematicians omit it. With this convention, Hermitian $T^{a}$ produce unitary $exp (i θ^{a} T^{a}) \in G$ .) The structure constants are antisymmetric in $a, b$ , satisfy a Jacobi identity inherited from the Lie bracket, and determine the entire group structure near the identity.

Casimir operators. A Casimir is an element of the universal enveloping algebra of $g$ that commutes with all generators. By Schur's lemma, it acts as a scalar multiple of the identity on every irrep, providing a labelling of irreps. The number of independent Casimirs equals the rank of $g$ (= dimension of a Cartan subalgebra).

Examples:

$su (2)$ has rank 1; the single Casimir is $J^{2} = J_{x}^{2} + J_{y}^{2} + J_{z}^{2}$ with eigenvalue $j (j + 1)$ .
$su (3)$ has rank 2; two Casimirs label irreps by Dynkin labels $(p, q)$ .
The Poincaré algebra has two Casimirs: $P^{2}$ (mass-squared) and $W^{2}$ (Pauli–Lubanski-squared, related to spin). See QFT/preliminaries.md § Casimir invariants.

Direct products and semidirect products

Two ways of building a group from smaller pieces.

Direct product

The direct product $G_{1} \times G_{2}$ is the set of ordered pairs $(g_{1}, g_{2})$ with componentwise multiplication:

$(g_{1}, g_{2}) \cdot (g_{1}^{'}, g_{2}^{'}) = (g_{1} g_{1}^{'}, g_{2} g_{2}^{'}) .$

Both factors are normal subgroups, and they commute with each other inside the product (elements of $G_{1}$ and $G_{2}$ , viewed as subgroups, satisfy $g_{1} g_{2} = g_{2} g_{1}$ ). The Lie algebra is the direct sum $g_{1} \oplus g_{2}$ with $[g_{1}, g_{2}] = 0$ .

Semidirect product

The semidirect product $N ⋊_{ϕ} H$ combines a normal factor $N$ and a non-normal factor $H$ that acts on $N$ by automorphisms. The data is a homomorphism

$ϕ : H \to Aut (N), h \mapsto ϕ_{h},$

so each $h \in H$ defines an automorphism $ϕ_{h}$ of $N$ . The underlying set is pairs $(n, h)$ , but multiplication is twisted by $ϕ$ :

$(n_{1}, h_{1}) \cdot (n_{2}, h_{2}) = (n_{1} \cdot ϕ_{h_{1}} (n_{2}), h_{1} h_{2}) .$

The second entry $n_{2}$ is first transformed by $ϕ_{h_{1}}$ before being combined with $n_{1}$ . The inverse is $(n, h)^{- 1} = (ϕ_{h^{- 1}} (n^{- 1}), h^{- 1})$ .

Key structural properties:

$N$ embedded as ${(n, e_{H})}$ is a normal subgroup (the kernel of the projection $(n, h) \mapsto h$ ).
$H$ embedded as ${(e_{N}, h)}$ is a subgroup, but generally not normal.
Inside the product, $N$ and $H$ do not commute: $hn h^{- 1} = ϕ_{h} (n)$ . When $ϕ$ is the trivial action ( $ϕ_{h} = id_{N}$ for all $h$ ), the semidirect product degenerates to the direct product $N \times H$ .
The Lie algebra is $n ⋊ h$ with $[h, n] \subseteq n$ given by the derivative of $ϕ$ at the identity of $H$ .

Examples.

Poincaré group $P = R^{1, 3} ⋊ L$ : Lorentz transformations $Λ$ act on translations $a$ by $ϕ_{Λ} (a) = Λ a$ . The twisted multiplication $(Λ_{1}, a_{1}) \cdot (Λ_{2}, a_{2}) = (Λ_{1} Λ_{2}, a_{1} + Λ_{1} a_{2})$ encodes the physical fact that translating then boosting is not the same as boosting then translating — the translation vector itself gets rotated. Translations form the normal subgroup; the Lorentz group is not normal.
Euclidean group $ISE (n) = R^{n} ⋊ O (n)$ : the same structure with rotations acting on translations. The little group of a massless particle is the 3D analogue $ISO (2) = R^{2} ⋊ SO (2)$ (see QFT/preliminaries.md § Little groups).
Dihedral group $D_{n} = Z_{n} ⋊ Z_{2}$ : reflection acts on rotation by inversion ( $ϕ (r) = r^{- 1}$ ). Symmetries of a regular $n$ -gon.
Affine group $Aff (n, F) = F^{n} ⋊ G L (n, F)$ : translations and linear transformations of $F^{n}$ .
Trivial direct product is the special case where $ϕ$ is trivial: $N ⋊_{triv} H = N \times H$ .

When does a group $G$ decompose as a semidirect product? If $G$ has a normal subgroup $N$ and a subgroup $H$ with $N \cap H = {e}$ and $N H = G$ (a split extension), then $G ≅ N ⋊ H$ with $ϕ_{h} (n) = hn h^{- 1}$ (conjugation inside $G$ ). Not every extension splits: e.g. $Z_{4}$ has the normal subgroup $Z_{2}$ , but there is no complementary $Z_{2}$ giving $Z_{4} ≅ Z_{2} ⋊ Z_{2}$ (such a product would be the Klein four-group instead).

Compact vs. non-compact

A Lie group is compact if its underlying manifold is compact (closed and bounded for matrix groups). Compactness has dramatic consequences for representation theory:

Compact groups ( $U (n)$ , $S U (n)$ , $SO (n)$ , $Sp (n)$ ): all finite-dimensional irreps are unitary; every unitary rep decomposes as a direct sum of finite-dimensional irreps.
Non-compact groups (Lorentz $L_{+}^{↑}$ , Poincaré $P_{+}^{↑}$ , $S L (2, C)$ , $S L (n, R)$ ): finite-dimensional reps are generically non-unitary; unitary reps are infinite-dimensional.

This is why field components transform under finite-dimensional non-unitary $(j_{+}, j_{-})$ reps of $L_{+}^{↑}$ , while Hilbert-space states must transform under infinite-dimensional unitary reps of $P_{+}^{↑}$ (Wigner's classification).

Connectedness and discrete components

A Lie group can have multiple connected components (e.g. the Lorentz group has four — see QFT/preliminaries.md § The Lorentz group). The component containing the identity is the identity component $G^{0}$ , a normal subgroup; the component group $G / G^{0}$ is discrete.

A connected Lie group can be simply connected or multiply connected, depending on whether all loops contract to a point. The universal cover $G$ is the simply connected Lie group with the same Lie algebra $g$ , related to $G$ by a covering homomorphism $G \to G$ whose kernel is a discrete (central) subgroup. Examples:

$G$	$\tilde{G}$	$ker (\tilde{G} \to G)$
$SO (3)$	$S U (2)$	$Z_{2} = {\pm 1}$
$S O^{+} (1, 3)$	$S L (2, C)$	$Z_{2}$
$U (1)$	$R$	$Z$
$SO (n), n \geq 3$	$Spin (n)$	$Z_{2}$

Universal covers matter in physics because, by the projective-representation theorem, projective reps of $G$ are ordinary reps of $G$ . Quantum mechanics admits projective reps (states are rays), so the relevant symmetry group is always $G$ , not $G$ itself.

Worked examples: $SO (2)$ and $S U (2)$

Two compact Lie groups that exercise nearly all of the machinery above without notational overhead.

$SO (2)$ — rotations in the plane

$SO (2)$ is the group of $2 \times 2$ rotation matrices

$R (θ) = (cos θ sin θ - sin θ cos θ), θ \in [0, 2 π) .$

Group axioms: $R (θ_{1}) R (θ_{2}) = R (θ_{1} + θ_{2})$ (closure + associativity); $R (0) = 1$ (identity); $R (θ)^{- 1} = R (- θ)$ (inverses). Abelian.
Topology: $SO (2)$ is diffeomorphic to the circle $S^{1}$ — compact and connected, but not simply connected (the loop $θ \in [0, 2 π]$ does not contract). Its universal cover is $R$ (the additive group) with covering map $θ \mapsto e^{i θ}$ and kernel $2 π Z$ .
Lie algebra: $so (2) ≅ R$ , 1-dimensional. Single generator $J = (01 - 1 0)$ (anti-Hermitian); the physics-convention generator is $T = - i J = (0 - i i 0)$ . Then $R (θ) = exp (i θT)$ . No structure constants (1-dim algebra is automatically abelian, $[T, T] = 0$ ).
Representations: All irreps of $SO (2)$ are 1-dimensional (compact + abelian ⇒ irreps are characters). They are labelled by an integer $n \in Z$ : $ρ_{n} (R (θ)) = e^{in θ} .$ Integer because $R (2 π) = 1$ . Non-integer $n$ would give multi-valued reps — these are projective reps of $SO (2)$ , equivalently ordinary reps of the universal cover $R$ .
Casimir: $T^{2} = 1$ , eigenvalue 1 on every irrep — trivial, since rank = 1 and the algebra is 1-dimensional. The genuine label is $n$ .
Physics use: rotations in the plane (e.g. for 2D systems), the gauge group of QED ( $U (1) ≅ SO (2)$ as Lie groups, with a different identification: $U (1)$ is parameterized by $e^{i θ}$ , which is also a circle).

$S U (2)$ — the simplest non-abelian compact group

$S U (2)$ is the group of $2 \times 2$ complex unitary matrices with determinant 1:

$U = (a - \overset{ˉ}{b} b \overset{a}{ˉ}), ∣ a ∣^{2} + ∣ b ∣^{2} = 1.$

Group axioms: standard matrix multiplication; identity $1_{2}$ ; inverse $U^{- 1} = U^{†}$ . Non-abelian (matrix multiplication doesn't commute in general).
Topology: the constraint $∣ a ∣^{2} + ∣ b ∣^{2} = 1$ identifies $S U (2)$ with the unit 3-sphere $S^{3} \subset C^{2}$ — compact, connected, and simply connected (so $S U (2)$ is its own universal cover). It double-covers $SO (3)$ : $S U (2) 2 : 1 SO (3), ker = {\pm 1} = Z_{2} .$
Lie algebra: $su (2) = {X \in M_{2} (C) : X^{†} = - X, tr X = 0}$ — anti-Hermitian traceless $2 \times 2$ matrices, real-3-dimensional. Conventional generators are $T^{a} = σ^{a} /2$ for $a = 1, 2, 3$ , with $σ^{a}$ the Pauli matrices: $σ^{1} = (0110), σ^{2} = (0 i - i 0), σ^{3} = (10 0 - 1) .$ Hermitian and traceless, so $U = exp (i θ^{a} T^{a}) \in S U (2)$ . Structure constants: $[T^{a}, T^{b}] = i ϵ^{ab c} T^{c}$ , so $f^{ab c} = ϵ^{ab c}$ (the Levi-Civita symbol).
Casimir: rank 1 ⇒ one Casimir, $T^{2} = T^{a} T^{a} = (σ^{a} σ^{a}) /4 = 3/4 \cdot 1$ on the defining rep. On a general spin- $j$ irrep it acts as $j (j + 1) 1$ .
Representations: irreps labelled by $j \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ , of dimension $2 j + 1$ . The defining rep ( $j = \frac{1}{2}$ ) is the spinor rep; $j = 1$ is the vector rep that descends to the defining rep of $SO (3)$ (since $SO (3)$ admits only integer- $j$ reps). Half-integer $j$ reps are projective reps of $SO (3)$ , equivalently ordinary reps of $S U (2)$ — this is the deep reason a $2 π$ rotation flips the sign of a spin- $\frac{1}{2}$ wavefunction.
Physics use: spin in non-relativistic QM, isospin, the $S U (2)_{L}$ factor of the electroweak gauge group, and (in complexified form $su (2)_{C} \oplus su (2)_{C}$ ) the Lorentz algebra.

Side-by-side summary

	$SO (2)$	$S U (2)$
Dimension	1	3
Topology	$S^{1}$	$S^{3}$
Connected	yes	yes
Simply connected	no	yes
Universal cover	$R$	$S U (2)$ itself
Abelian	yes	no
Compact	yes	yes
Generators	$T$ (1)	$T^{1}, T^{2}, T^{3}$ (3)
Structure constants	trivial	$ϵ^{ab c}$
Rank / # Casimirs	1 / 1	1 / 1
Casimir eigenvalue (irrep)	$1$	$j (j + 1)$
Irreps	1-dim, labelled by $n \in Z$	$(2 j + 1)$ -dim, labelled by $j \in \frac{1}{2} Z_{\geq 0}$

Why this matters

Group theory is the language in which every QFT symmetry is expressed:

Spacetime symmetries (Lorentz, Poincaré, conformal, supersymmetry) — see QFT/preliminaries.md § Lorentz and Poincaré Groups.
Internal symmetries ( $U (1)_{em}$ , $S U (2)_{L} \times U (1)_{Y}$ , $S U (3)_{c}$ ) — gauge groups of the Standard Model.
Discrete symmetries ( $C, P, T$ and their products).
Spontaneous symmetry breaking is the breaking of the global / gauge symmetry group $G$ to a subgroup $H$ , with the coset $G / H$ parameterizing the Goldstone modes.
Anomalies are obstructions to lifting a classical symmetry to the quantum theory, classified by group cohomology of $G$ .

References

Hall, Lie Groups, Lie Algebras, and Representations — modern mathematical reference.
Fulton & Harris, Representation Theory: A First Course — finite groups and Lie algebras with concrete examples.
Tung, Group Theory in Physics — physics-oriented, with detailed Poincaré-group treatment.
Georgi, Lie Algebras in Particle Physics — pragmatic computational reference.
Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 2 — the Poincaré-group classification using the structure laid out here.

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