The Classical Matrix Groups

The classical groups are the matrix Lie groups defined by preserving a bilinear or sesquilinear form. They are the concrete Lie groups that appear throughout physics — the Lorentz group $O (1, 3)$ , the gauge groups $U (1), S U (2), S U (3)$ , the spin groups — and the smallest interesting examples for every general theorem in lie-groups.md and lie-algebras.md. This page is the notation reference the rest of the folder and the QFT tree link back to. Let $F$ denote either $R$ or $C$ .

The families

Symbol	Name	Definition	$dim_{R}$
$G L (n, F)$	General linear	invertible $n \times n$ over $F$ ( $det \neq = 0$ )	$n^{2}$ ( $R$ ) / $2 n^{2}$ ( $C$ )
$S L (n, F)$	Special linear	$det M = 1$	$n^{2} - 1$ / $2 (n^{2} - 1)$
$O (n)$	Orthogonal	$M^{T} M = 1$ (preserves Euclidean form)	$n (n - 1) /2$
$SO (n)$	Special orthogonal	$O (n) \cap S L (n, R)$ (rotations)	$n (n - 1) /2$
$O (p, q)$	Indefinite orthogonal	preserves signature $(p, q)$ ; $O (1, 3)$ = Lorentz	$\frac{1}{2} (p + q) (p + q - 1)$
$S O^{+} (p, q)$	Proper orthochronous	identity component of $O (p, q)$	same
$U (n)$	Unitary	$M^{†} M = 1$ (preserves Hermitian form)	$n^{2}$
$S U (n)$	Special unitary	$U (n) \cap S L (n, C)$ ( $det M = 1$ )	$n^{2} - 1$
$Sp (2 n, F)$	Symplectic	preserves a symplectic form $ω$	$n (2 n + 1)$
$Sp (n)$	Compact symplectic	$Sp (2 n, C) \cap U (2 n)$ (quaternionic unitary)	$n (2 n + 1)$
$Spin (n)$	Spin	universal double cover of $SO (n)$ , $n \geq 3$	$n (n - 1) /2$

Inclusion lattice (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 $S U (n), SO (n)$ the determinant- $1$ subgroups.

Their Lie algebras

Each classical group $G$ is a smooth manifold; its Lie algebra $g = T_{e} G$ (see lie-algebras.md) is the matrices $X$ with $exp (tX) \in G$ for all $t$ . Differentiating the defining condition at $t = 0$ :

$G$	$g$	Condition on $X$
$G L (n, F)$	$gl (n, F)$	none (all matrices)
$S L (n, F)$	$sl (n, F)$	$tr X = 0$
$O (n), SO (n)$	$so (n)$	$X^{T} = - X$ (antisymmetric)
$O (p, q), SO (p, q)$	$so (p, q)$	$X^{T} η + η X = 0$ ( $η$ the metric)
$U (n)$	$u (n)$	$X^{†} = - X$ (anti-Hermitian)
$S U (n)$	$su (n)$	$X^{†} = - X, tr X = 0$
$Sp (2 n, F)$	$sp (2 n, F)$	$X^{T} Ω + Ω X = 0$

Note $O (n)$ and $SO (n)$ share a Lie algebra: they differ only in the disconnected piece ( $det = - 1$ ), invisible to the tangent space at the identity. The physics convention writes group elements as $exp (i θ^{a} T^{a})$ with Hermitian generators $T^{a}$ , absorbing the anti-Hermitian $i$ into the exponent — see lie-algebras.md § Structure constants.

Compactness

Compactness (closed and bounded as a matrix set) controls representation theory — see compact-groups.md:

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 $p, q > 0$ (e.g. Lorentz), $Sp (2 n, R)$ .

Compact groups have finite-dimensional unitary irreps and a fully reducible representation theory; non-compact groups (Lorentz, Poincaré) have no finite-dimensional unitary reps — their unitary reps are infinite-dimensional, which is why relativistic quantum fields carry finite-dimensional non-unitary Lorentz reps while states carry infinite-dimensional unitary ones.

Ranks (for the classification)

The rank — the dimension of a maximal torus / Cartan subalgebra (see lie-classification.md) — indexes the four infinite families:

Cartan type	Group	Rank
$A_{n}$	$S U (n + 1)$	$n$
$B_{n}$	$SO (2 n + 1)$	$n$
$C_{n}$	$Sp (2 n)$	$n$
$D_{n}$	$SO (2 n)$	$n$

plus the five exceptional groups $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ .

Accidental isomorphisms

In low dimensions the families overlap — the accidental (exceptional) isomorphisms, mostly at the level of the covering groups:

$S U (2) ≅ Spin (3) ≅ Sp (1)$ ; and $S U (2) / Z_{2} ≅ SO (3)$ .
$S L (2, C) ≅ Spin (1, 3)$ (double-covers the Lorentz group) — the reason spinors exist; see physics-bridge.md.
$S U (2) \times S U (2) ≅ Spin (4)$ , and $so (1, 3)_{C} ≅ sl (2, C) \oplus sl (2, C)$ — the $(j_{+}, j_{-})$ labelling of Lorentz reps.
$Spin (5) ≅ Sp (2)$ , $Spin (6) ≅ S U (4)$ .
$SO (3) ≅ RP^{3}$ as a manifold; $S U (3)$ is the QCD colour group.

These coincidences stop at rank $≲ 3$ ; the classification in lie-classification.md explains why.

References

Hall, Lie Groups, Lie Algebras, and Representations, Ch. 1–3.
Gilmore, Lie Groups, Physics, and Geometry.
Georgi, Lie Algebras in Particle Physics — the physics-facing catalogue.

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