The mathematical reference
Lie groups and their forms.
Separate the global group, real Lie algebra, complexification, and chosen map. Then follow the representations and topology.
49finite group records41Lie group records38Lie algebras22specified maps
163 released records. Individual groups and parameterized families are counted separately.
| Object | Kind | Essential data | Coverage |
|---|---|---|---|
| SU(2) Special unitary group | Group Lie theory | dimℝ 3 | executable 7 linked workbenchs |
| SO(3) Rotation group in three dimensions | Group Lie theory | dimℝ 3 | executable 2 linked workbenchs |
| SU(3) Special unitary group | Group Lie theory | dimℝ 8 | executable 8 linked workbenchs |
| PSU(3) Projective special unitary group | Group Lie theory | dimℝ 8 | executable 1 linked workbench |
| U(1) Circle group | Group Lie theory | dimℝ 1 | reference Sourced record |
| Spin(4) Euclidean spin group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SL(2,ℂ) Lorentz spin group | Group Lie theory | dimℝ 6 | reference Sourced record |
| Aᵣ Simple Lie type Aᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Bᵣ Simple Lie type Bᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Cᵣ Simple Lie type Cᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Dᵣ Simple Lie type Dᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| E₆ Simple Lie type E₆ | Family Lie theory | Parameterized family | classification Sourced record |
| E₇ Simple Lie type E₇ | Family Lie theory | Parameterized family | classification Sourced record |
| E₈ Simple Lie type E₈ | Family Lie theory | Parameterized family | classification Sourced record |
| F₄ Simple Lie type F₄ | Family Lie theory | Parameterized family | classification Sourced record |
| G₂ Simple Lie type G₂ | Family Lie theory | Parameterized family | classification Sourced record |
| sl(2, C) sl(2, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 3 | reference Sourced record |
| sl(3, C) sl(3, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 8 | reference Sourced record |
| sl(4, C) sl(4, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 15 | reference Sourced record |
| sl(5, C) sl(5, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 24 | reference Sourced record |
| so(5, C) ≅ sp(4, C) so(5, C) ≅ sp(4, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 10 | reference Sourced record |
| sp(6, C) sp(6, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 21 | reference Sourced record |
| so(8, C) so(8, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 28 | reference Sourced record |
| so(10, C) so(10, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 45 | reference Sourced record |
| e₆(C) e₆(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 78 | reference Sourced record |
| e₇(C) e₇(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 133 | reference Sourced record |
| e₈(C) e₈(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 248 | reference Sourced record |
| f₄(C) f₄(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 52 | reference Sourced record |
| g₂(C) g₂(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 14 | reference Sourced record |
| sl(2, C) ⊕ sl(2, C) sl(2, C) ⊕ sl(2, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 6 | reference Sourced record |
| C (zero bracket) C (zero bracket) — complex Lie algebra | Lie algebra Lie theory | dimℂ 1 | reference Sourced record |
| sl(2, C) ⊕ C sl(2, C) ⊕ C — complex Lie algebra | Lie algebra Lie theory | dimℂ 4 | reference Sourced record |
| su(2) su(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 3 | reference Sourced record |
| su(3) su(3) — real Lie algebra | Lie algebra Lie theory | dimℝ 8 | reference Sourced record |
| su(4) ≅ so(6) su(4) ≅ so(6) — real Lie algebra | Lie algebra Lie theory | dimℝ 15 | reference Sourced record |
| su(5) su(5) — real Lie algebra | Lie algebra Lie theory | dimℝ 24 | reference Sourced record |
| sp(2) ≅ so(5) sp(2) ≅ so(5) — real Lie algebra | Lie algebra Lie theory | dimℝ 10 | reference Sourced record |
| sp(3) sp(3) — real Lie algebra | Lie algebra Lie theory | dimℝ 21 | reference Sourced record |
| so(8) so(8) — real Lie algebra | Lie algebra Lie theory | dimℝ 28 | reference Sourced record |
| so(10) so(10) — real Lie algebra | Lie algebra Lie theory | dimℝ 45 | reference Sourced record |
| e₆ compact e₆ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 78 | reference Sourced record |
| e₇ compact e₇ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 133 | reference Sourced record |
| e₈ compact e₈ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 248 | reference Sourced record |
| f₄ compact f₄ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 52 | reference Sourced record |
| g₂ compact g₂ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 14 | reference Sourced record |
| su(2) ⊕ su(2) su(2) ⊕ su(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 6 | reference Sourced record |
| u(1) u(1) — real Lie algebra | Lie algebra Lie theory | dimℝ 1 | reference Sourced record |
| u(2) u(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 4 | reference Sourced record |
| sl(2, R) sl(2, R) — real Lie algebra | Lie algebra Lie theory | dimℝ 3 | reference Sourced record |
| sl(3, R) sl(3, R) — real Lie algebra | Lie algebra Lie theory | dimℝ 8 | reference Sourced record |
| su(2,2) ≅ so(2,4) su(2,2) ≅ so(2,4) — real Lie algebra | Lie algebra Lie theory | dimℝ 15 | reference Sourced record |
| sp(4, R) ≅ so(2,3) sp(4, R) ≅ so(2,3) — real Lie algebra | Lie algebra Lie theory | dimℝ 10 | reference Sourced record |
| sl(2, C) viewed over R sl(2, C) viewed over R — real Lie algebra | Lie algebra Lie theory | dimℝ 6 | reference Sourced record |
| su(2) su(2) real form | Real form Lie theory | Compact | reference Sourced record |
| su(3) su(3) real form | Real form Lie theory | Compact | reference Sourced record |
| su(4) ≅ so(6) su(4) ≅ so(6) real form | Real form Lie theory | Compact | reference Sourced record |
| su(5) su(5) real form | Real form Lie theory | Compact | reference Sourced record |
| sp(2) ≅ so(5) sp(2) ≅ so(5) real form | Real form Lie theory | Compact | reference Sourced record |
| sp(3) sp(3) real form | Real form Lie theory | Compact | reference Sourced record |
| so(8) so(8) real form | Real form Lie theory | Compact | reference Sourced record |
| so(10) so(10) real form | Real form Lie theory | Compact | reference Sourced record |
| e₆ compact e₆ compact real form | Real form Lie theory | Compact | reference Sourced record |
| e₇ compact e₇ compact real form | Real form Lie theory | Compact | reference Sourced record |
| e₈ compact e₈ compact real form | Real form Lie theory | Compact | reference Sourced record |
| f₄ compact f₄ compact real form | Real form Lie theory | Compact | reference Sourced record |
| g₂ compact g₂ compact real form | Real form Lie theory | Compact | reference Sourced record |
| su(2) ⊕ su(2) su(2) ⊕ su(2) real form | Real form Lie theory | Compact | reference Sourced record |
| sl(2, R) sl(2, R) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sl(3, R) sl(3, R) real form | Real form Lie theory | Noncompact | reference Sourced record |
| su(2,2) ≅ so(2,4) su(2,2) ≅ so(2,4) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sp(4, R) ≅ so(2,3) sp(4, R) ≅ so(2,3) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sl(2, C) viewed over R sl(2, C) viewed over R real form | Real form Lie theory | Noncompact | reference Sourced record |
| SU(4) Special unitary group in four dimensions | Group Lie theory | dimℝ 15 | executable 4 linked workbenchs |
| PSU(4) Adjoint compact group of type A₃ | Group Lie theory | dimℝ 15 | reference Sourced record |
| SU(5) Special unitary group in five dimensions | Group Lie theory | dimℝ 24 | executable 4 linked workbenchs |
| U(2) Unitary group in two dimensions | Group Lie theory | dimℝ 4 | reference Sourced record |
| SO(4) Rotation group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SO(5) Rotation group in five dimensions | Group Lie theory | dimℝ 10 | reference Sourced record |
| SO(6) Rotation group in six dimensions | Group Lie theory | dimℝ 15 | reference Sourced record |
| Sp(2) Compact symplectic group of quaternionic rank two | Group Lie theory | dimℝ 10 | executable 1 linked workbench |
| Sp(3) Compact symplectic group of quaternionic rank three | Group Lie theory | dimℝ 21 | reference Sourced record |
| Spin(8) Simply connected compact spin group in eight dimensions | Group Lie theory | dimℝ 28 | executable 1 linked workbench |
| SO(8) Rotation group in eight dimensions | Group Lie theory | dimℝ 28 | reference Sourced record |
| PSO(8) Adjoint compact group of type D₄ | Group Lie theory | dimℝ 28 | reference Sourced record |
| Spin(10) Simply connected compact spin group in ten dimensions | Group Lie theory | dimℝ 45 | executable 1 linked workbench |
| SO(10) Rotation group in ten dimensions | Group Lie theory | dimℝ 45 | reference Sourced record |
| G₂ compact Compact group of octonion automorphisms | Group Lie theory | dimℝ 14 | executable 1 linked workbench |
| F₄ compact Compact exceptional group of type F₄ | Group Lie theory | dimℝ 52 | executable 1 linked workbench |
| E₆ compact, simply connected Simply connected compact exceptional group of type E₆ | Group Lie theory | dimℝ 78 | executable 1 linked workbench |
| E₇ compact, simply connected Simply connected compact exceptional group of type E₇ | Group Lie theory | dimℝ 133 | executable 1 linked workbench |
| E₈ compact Compact exceptional group of type E₈ | Group Lie theory | dimℝ 248 | executable 1 linked workbench |
| E₆ compact, adjoint Adjoint compact exceptional group of type E₆ | Group Lie theory | dimℝ 78 | reference Sourced record |
| E₇ compact, adjoint Adjoint compact exceptional group of type E₇ | Group Lie theory | dimℝ 133 | reference Sourced record |
| SL(2, R) Real special linear group in two dimensions | Group Lie theory | dimℝ 3 | reference Sourced record |
| PSL(2, R) Adjoint real group of type A₁ | Group Lie theory | dimℝ 3 | reference Sourced record |
| Universal cover of SL(2, R) Simply connected group integrating sl(2,R) | Group Lie theory | dimℝ 3 | reference Sourced record |
| SL(3, R) Real special linear group in three dimensions | Group Lie theory | dimℝ 8 | reference Sourced record |
| SU(2,2) Indefinite special unitary group of signature (2,2) | Group Lie theory | dimℝ 15 | reference Sourced record |
| Sp(4, R) Real symplectic group on a four-dimensional space | Group Lie theory | dimℝ 10 | reference Sourced record |
| SO₀(1,3) Proper orthochronous Lorentz group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SO₀(2,4) Identity component of the orthogonal group of signature (2,4) | Group Lie theory | dimℝ 15 | reference Sourced record |
| SO₀(2,3) Identity component of the orthogonal group of signature (2,3) | Group Lie theory | dimℝ 10 | reference Sourced record |
| SU(2) → SO(3) Spin cover of three-dimensional rotations | Map Lie theory | SO(3) | executable 1 linked workbench |
| SU(3) → PSU(3) Adjoint quotient of SU(3) | Map Lie theory | PSU(3) | executable 1 linked workbench |
| SU(2) ↪ SU(3) Regular block inclusion in SU(3) | Map Lie theory | {diag(A,1) : A ∈ SU(2)} ≅ SU(2) | executable 2 linked workbenchs |
| SU(2) → SU(3), spin 1 Principal spin-one homomorphism into SU(3) | Map Lie theory | An SU(3)-conjugate of the real SO(3) subgroup | executable 2 linked workbenchs |
| U(1) ↪ SU(2) Normalized maximal torus of SU(2) | Map Lie theory | {diag(z,z⁻¹) : |z|=1} | executable 1 linked workbench |
| U(1) ≅ SO(2) ↪ SO(3) Normalized axial rotation subgroup of SO(3) | Map Lie theory | Rotations fixing a chosen oriented axis, isomorphic to SO(2) | executable 1 linked workbench |
| Spin(4) → SO(4) Quaternionic spin cover in four Euclidean dimensions | Map Lie theory | SO(4) | reference Sourced record |
| SU(4) → SO(6) Exterior-square realization of the Spin(6) cover | Map Lie theory | SO(6) | reference Sourced record |
| SU(4) → PSU(4) Full central quotient of SU(4) | Map Lie theory | PSU(4) | reference Sourced record |
| Sp(2) → SO(5) Quaternionic construction of the Spin(5) cover | Map Lie theory | SO(5) | reference Sourced record |
| Spin(8) → SO(8) Vector spin cover in eight Euclidean dimensions | Map Lie theory | SO(8) | reference Sourced record |
| Spin(10) → SO(10) Vector spin cover in ten Euclidean dimensions | Map Lie theory | SO(10) | reference Sourced record |
| SL(2, C) → SO₀(1,3) Hermitian-matrix Lorentz spin cover | Map Lie theory | SO₀(1,3) | reference Sourced record |
| SL(2, R) → PSL(2, R) Adjoint central quotient of SL(2,R) | Map Lie theory | PSL(2,R) ≅ SO₀(2,1) | reference Sourced record |
| Universal SL(2,R) cover → SL(2,R) Universal covering map of SL(2,R) | Map Lie theory | SL(2,R) | reference Sourced record |
| SU(2,2) → SO₀(2,4) Exterior-square spin cover in signature (2,4) | Map Lie theory | SO₀(2,4) | reference Sourced record |
| Sp(4, R) → SO₀(2,3) Primitive-exterior-square spin cover in signature (2,3) | Map Lie theory | SO₀(2,3) | reference Sourced record |
| SU(2)/U(1) ≅ S² Hopf sphere as SU(2)/U(1) | Space Lie theory | Reference | executable 1 linked workbench |
| SO(3)/SO(2) ≅ S² Rotation sphere as SO(3)/SO(2) | Space Lie theory | Reference | executable 1 linked workbench |
| A1 · (1) Doublet of SU(2) | Representation Lie theory | dimℂ 2 | executable 2 linked workbenchs |
| A1 · (2) Adjoint of SU(2) | Representation Lie theory | dimℂ 3 | executable 2 linked workbenchs |
| A2 · (1,0) Fundamental of SU(3) | Representation Lie theory | dimℂ 3 | executable 5 linked workbenchs |
| A2 · (0,1) Conjugate fundamental of SU(3) | Representation Lie theory | dimℂ 3 | executable 5 linked workbenchs |
| A2 · (1,1) Adjoint of SU(3) | Representation Lie theory | dimℂ 8 | executable 5 linked workbenchs |
| A2 · (2,0) Symmetric sextet of SU(3) | Representation Lie theory | dimℂ 6 | executable 5 linked workbenchs |
| A2 · (3,0) Decuplet of SU(3) | Representation Lie theory | dimℂ 10 | executable 5 linked workbenchs |
| A2 · (2,2) 27-dimensional irrep of SU(3) | Representation Lie theory | dimℂ 27 | executable 5 linked workbenchs |
| A3 · (1,0,0) Fundamental of SU(4) | Representation Lie theory | dimℂ 4 | executable 3 linked workbenchs |
| A3 · (0,1,0) Two-index antisymmetric of SU(4) | Representation Lie theory | dimℂ 6 | executable 3 linked workbenchs |
| A3 · (1,0,1) Adjoint of SU(4) | Representation Lie theory | dimℂ 15 | executable 3 linked workbenchs |
| A4 · (1,0,0,0) Fundamental of SU(5) | Representation Lie theory | dimℂ 5 | executable 3 linked workbenchs |
| A4 · (0,1,0,0) Two-index antisymmetric of SU(5) | Representation Lie theory | dimℂ 10 | executable 3 linked workbenchs |
| A4 · (1,0,0,1) Adjoint of SU(5) | Representation Lie theory | dimℂ 24 | executable 3 linked workbenchs |
| C2 · (1,0) Defining representation of Sp(2) = USp(4) | Representation Lie theory | dimℂ 4 | executable 1 linked workbench |
| C2 · (0,1) Five-dimensional irrep of Sp(2) = USp(4) | Representation Lie theory | dimℂ 5 | executable 1 linked workbench |
| D4 · (1,0,0,0) Vector of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D4 · (0,0,1,0) Half-spin representation of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D4 · (0,0,0,1) Other half-spin representation of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D5 · (1,0,0,0,0) Vector of Spin(10) | Representation Lie theory | dimℂ 10 | executable 1 linked workbench |
| D5 · (0,0,0,0,1) Chiral spinor of Spin(10) | Representation Lie theory | dimℂ 16 | executable 1 linked workbench |
| D5 · (0,0,0,1,0) Conjugate chiral spinor of Spin(10) | Representation Lie theory | dimℂ 16 | executable 1 linked workbench |
| D5 · (0,1,0,0,0) Adjoint of Spin(10) | Representation Lie theory | dimℂ 45 | executable 1 linked workbench |
| G2 · (1,0) Fundamental of G2 (compact) | Representation Lie theory | dimℂ 7 | executable 1 linked workbench |
| G2 · (0,1) Adjoint of G2 (compact) | Representation Lie theory | dimℂ 14 | executable 1 linked workbench |
| F4 · (0,0,0,1) Minimal representation of F4 (compact) | Representation Lie theory | dimℂ 26 | executable 1 linked workbench |
| F4 · (1,0,0,0) Adjoint of F4 (compact) | Representation Lie theory | dimℂ 52 | executable 1 linked workbench |
| E6 · (1,0,0,0,0,0) 27-dimensional representation of E6 (compact simply connected) | Representation Lie theory | dimℂ 27 | executable 1 linked workbench |
| E6 · (0,1,0,0,0,0) Adjoint of E6 (compact simply connected) | Representation Lie theory | dimℂ 78 | executable 1 linked workbench |
| E7 · (0,0,0,0,0,0,1) 56-dimensional representation of E7 (compact simply connected) | Representation Lie theory | dimℂ 56 | executable 1 linked workbench |
| E7 · (1,0,0,0,0,0,0) Adjoint of E7 (compact simply connected) | Representation Lie theory | dimℂ 133 | executable 1 linked workbench |
| E8 · (0,0,0,0,0,0,0,1) Adjoint of E8 (compact simply connected) | Representation Lie theory | dimℂ 248 | executable 1 linked workbench |
| su(3) ⊕ su(2) ⊕ u(1) Compact Standard Model gauge algebra | Lie algebra Lie theory | dimℝ 12 | reference Sourced record |
| SU(3) × SU(2) × U(1) Standard Model compact covering product | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(2) Standard Model global group with diagonal Z2 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(3) Standard Model global group with diagonal Z3 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(6) Standard Model global group with diagonal Z6 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(1) → G_SM^(2) Standard Model quotient map of order 2 | Map Lie theory | G_SM^(2) | executable 1 linked workbench |
| G_SM^(1) → G_SM^(3) Standard Model quotient map of order 3 | Map Lie theory | G_SM^(3) | executable 1 linked workbench |
| G_SM^(1) → G_SM^(6) Standard Model quotient map of order 6 | Map Lie theory | G_SM^(6) | executable 1 linked workbench |
| SU(3) × SU(2) × U(1)_X → SU(5) Color-first SU(5) block homomorphism | Map Lie theory | S(U(3) × U(2)) | executable 1 linked workbench |
| G_SM^(6) ↪ SU(5) The diagonal-Z6 Standard Model group inside SU(5) | Map Lie theory | S(U(3) × U(2)) | reference Sourced record |
Coverage is specific. Classification identifies a mathematical object or family. Reference supplies stated properties. Executable links to a bounded calculation. Read the coverage policy.