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Lie groups and their forms.

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49finite group records41Lie group records38Lie algebras22specified maps

163 released records. Individual groups and parameterized families are counted separately.

Mathematical records
Object	Kind	Essential data	Coverage
$\mathrm{SU}(2)$ Special unitary group	Group Lie theory	dimℝ 3	executable 7 linked workbenchs
$\mathrm{SO}(3)$ Rotation group in three dimensions	Group Lie theory	dimℝ 3	executable 2 linked workbenchs
$\mathrm{SU}(3)$ Special unitary group	Group Lie theory	dimℝ 8	executable 8 linked workbenchs
$\mathrm{PSU}(3)$ Projective special unitary group	Group Lie theory	dimℝ 8	executable 1 linked workbench
$\mathrm{U}(1)$ Circle group	Group Lie theory	dimℝ 1	reference Sourced record
$\mathrm{Spin}(4)$ Euclidean spin group in four dimensions	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{SL}(2,\mathbb C)$ Lorentz spin group	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{A}_{r}$ Simple Lie type Aᵣ	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{B}_{r}$ Simple Lie type Bᵣ	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{C}_{r}$ Simple Lie type Cᵣ	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{D}_{r}$ Simple Lie type Dᵣ	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{E}_{6}$ Simple Lie type E₆	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{E}_{7}$ Simple Lie type E₇	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{E}_{8}$ Simple Lie type E₈	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{F}_{4}$ Simple Lie type F₄	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{G}_{2}$ Simple Lie type G₂	Family Lie theory	Parameterized family	classification Sourced record
$\mathfrak{sl}(2, \mathbb C)$ sl(2, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 3	reference Sourced record
$\mathfrak{sl}(3, \mathbb C)$ sl(3, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 8	reference Sourced record
$\mathfrak{sl}(4, \mathbb C)$ sl(4, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 15	reference Sourced record
$\mathfrak{sl}(5, \mathbb C)$ sl(5, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 24	reference Sourced record
$\mathfrak{so}(5, \mathbb C) \cong \mathfrak{sp}(4, \mathbb C)$ so(5, C) ≅ sp(4, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 10	reference Sourced record
$\mathfrak{sp}(6, \mathbb C)$ sp(6, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 21	reference Sourced record
$\mathfrak{so}(8, \mathbb C)$ so(8, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 28	reference Sourced record
$\mathfrak{so}(10, \mathbb C)$ so(10, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 45	reference Sourced record
$\mathfrak{e}_{6}(\mathbb C)$ e₆(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 78	reference Sourced record
$\mathfrak{e}_{7}(\mathbb C)$ e₇(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 133	reference Sourced record
$\mathfrak{e}_{8}(\mathbb C)$ e₈(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 248	reference Sourced record
$\mathfrak{f}_{4}(\mathbb C)$ f₄(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 52	reference Sourced record
$\mathfrak{g}_{2}(\mathbb C)$ g₂(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 14	reference Sourced record
$\mathfrak{sl}(2, \mathbb C) \oplus \mathfrak{sl}(2, \mathbb C)$ sl(2, C) ⊕ sl(2, C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 6	reference Sourced record
$(\mathbb C,[\, ,\,]=0)$ C (zero bracket) — complex Lie algebra	Lie algebra Lie theory	dimℂ 1	reference Sourced record
$\mathfrak{sl}(2, \mathbb C) \oplus \mathbb C$ sl(2, C) ⊕ C — complex Lie algebra	Lie algebra Lie theory	dimℂ 4	reference Sourced record
$\mathfrak{su}(2)$ su(2) — real Lie algebra	Lie algebra Lie theory	dimℝ 3	reference Sourced record
$\mathfrak{su}(3)$ su(3) — real Lie algebra	Lie algebra Lie theory	dimℝ 8	reference Sourced record
$\mathfrak{su}(4) \cong \mathfrak{so}(6)$ su(4) ≅ so(6) — real Lie algebra	Lie algebra Lie theory	dimℝ 15	reference Sourced record
$\mathfrak{su}(5)$ su(5) — real Lie algebra	Lie algebra Lie theory	dimℝ 24	reference Sourced record
$\mathfrak{sp}(2) \cong \mathfrak{so}(5)$ sp(2) ≅ so(5) — real Lie algebra	Lie algebra Lie theory	dimℝ 10	reference Sourced record
$\mathfrak{sp}(3)$ sp(3) — real Lie algebra	Lie algebra Lie theory	dimℝ 21	reference Sourced record
$\mathfrak{so}(8)$ so(8) — real Lie algebra	Lie algebra Lie theory	dimℝ 28	reference Sourced record
$\mathfrak{so}(10)$ so(10) — real Lie algebra	Lie algebra Lie theory	dimℝ 45	reference Sourced record
$\mathfrak{e}_{6}\;\text{compact}$ e₆ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 78	reference Sourced record
$\mathfrak{e}_{7}\;\text{compact}$ e₇ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 133	reference Sourced record
$\mathfrak{e}_{8}\;\text{compact}$ e₈ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 248	reference Sourced record
$\mathfrak{f}_{4}\;\text{compact}$ f₄ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 52	reference Sourced record
$\mathfrak{g}_{2}\;\text{compact}$ g₂ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 14	reference Sourced record
$\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ su(2) ⊕ su(2) — real Lie algebra	Lie algebra Lie theory	dimℝ 6	reference Sourced record
$\mathfrak{u}(1)$ u(1) — real Lie algebra	Lie algebra Lie theory	dimℝ 1	reference Sourced record
$\mathfrak{u}(2)$ u(2) — real Lie algebra	Lie algebra Lie theory	dimℝ 4	reference Sourced record
$\mathfrak{sl}(2, \mathbb R)$ sl(2, R) — real Lie algebra	Lie algebra Lie theory	dimℝ 3	reference Sourced record
$\mathfrak{sl}(3, \mathbb R)$ sl(3, R) — real Lie algebra	Lie algebra Lie theory	dimℝ 8	reference Sourced record
$\mathfrak{su}(2,2) \cong \mathfrak{so}(2,4)$ su(2,2) ≅ so(2,4) — real Lie algebra	Lie algebra Lie theory	dimℝ 15	reference Sourced record
$\mathfrak{sp}(4, \mathbb R) \cong \mathfrak{so}(2,3)$ sp(4, R) ≅ so(2,3) — real Lie algebra	Lie algebra Lie theory	dimℝ 10	reference Sourced record
$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$ sl(2, C) viewed over R — real Lie algebra	Lie algebra Lie theory	dimℝ 6	reference Sourced record
$\mathfrak{su}(2)$ su(2) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{su}(3)$ su(3) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{su}(4) \cong \mathfrak{so}(6)$ su(4) ≅ so(6) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{su}(5)$ su(5) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{sp}(2) \cong \mathfrak{so}(5)$ sp(2) ≅ so(5) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{sp}(3)$ sp(3) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{so}(8)$ so(8) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{so}(10)$ so(10) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{e}_{6}\;\text{compact}$ e₆ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{e}_{7}\;\text{compact}$ e₇ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{e}_{8}\;\text{compact}$ e₈ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{f}_{4}\;\text{compact}$ f₄ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{g}_{2}\;\text{compact}$ g₂ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ su(2) ⊕ su(2) real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{sl}(2, \mathbb R)$ sl(2, R) real form	Real form Lie theory	Noncompact	reference Sourced record
$\mathfrak{sl}(3, \mathbb R)$ sl(3, R) real form	Real form Lie theory	Noncompact	reference Sourced record
$\mathfrak{su}(2,2) \cong \mathfrak{so}(2,4)$ su(2,2) ≅ so(2,4) real form	Real form Lie theory	Noncompact	reference Sourced record
$\mathfrak{sp}(4, \mathbb R) \cong \mathfrak{so}(2,3)$ sp(4, R) ≅ so(2,3) real form	Real form Lie theory	Noncompact	reference Sourced record
$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$ sl(2, C) viewed over R real form	Real form Lie theory	Noncompact	reference Sourced record
$\mathrm{SU}(4)$ Special unitary group in four dimensions	Group Lie theory	dimℝ 15	executable 4 linked workbenchs
$\mathrm{PSU}(4)$ Adjoint compact group of type A₃	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{SU}(5)$ Special unitary group in five dimensions	Group Lie theory	dimℝ 24	executable 4 linked workbenchs
$\mathrm{U}(2)$ Unitary group in two dimensions	Group Lie theory	dimℝ 4	reference Sourced record
$\mathrm{SO}(4)$ Rotation group in four dimensions	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{SO}(5)$ Rotation group in five dimensions	Group Lie theory	dimℝ 10	reference Sourced record
$\mathrm{SO}(6)$ Rotation group in six dimensions	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{Sp}(2)$ Compact symplectic group of quaternionic rank two	Group Lie theory	dimℝ 10	executable 1 linked workbench
$\mathrm{Sp}(3)$ Compact symplectic group of quaternionic rank three	Group Lie theory	dimℝ 21	reference Sourced record
$\mathrm{Spin}(8)$ Simply connected compact spin group in eight dimensions	Group Lie theory	dimℝ 28	executable 1 linked workbench
$\mathrm{SO}(8)$ Rotation group in eight dimensions	Group Lie theory	dimℝ 28	reference Sourced record
$\mathrm{PSO}(8)$ Adjoint compact group of type D₄	Group Lie theory	dimℝ 28	reference Sourced record
$\mathrm{Spin}(10)$ Simply connected compact spin group in ten dimensions	Group Lie theory	dimℝ 45	executable 1 linked workbench
$\mathrm{SO}(10)$ Rotation group in ten dimensions	Group Lie theory	dimℝ 45	reference Sourced record
$\mathrm{G}_{2}\;\text{compact}$ Compact group of octonion automorphisms	Group Lie theory	dimℝ 14	executable 1 linked workbench
$\mathrm{F}_{4}\;\text{compact}$ Compact exceptional group of type F₄	Group Lie theory	dimℝ 52	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₆	Group Lie theory	dimℝ 78	executable 1 linked workbench
$\mathrm{E}_{7}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₇	Group Lie theory	dimℝ 133	executable 1 linked workbench
$\mathrm{E}_{8}\;\text{compact}$ Compact exceptional group of type E₈	Group Lie theory	dimℝ 248	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₆	Group Lie theory	dimℝ 78	reference Sourced record
$\mathrm{E}_{7}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₇	Group Lie theory	dimℝ 133	reference Sourced record
$\mathrm{SL}(2, \mathbb R)$ Real special linear group in two dimensions	Group Lie theory	dimℝ 3	reference Sourced record
$\mathrm{PSL}(2, \mathbb R)$ Adjoint real group of type A₁	Group Lie theory	dimℝ 3	reference Sourced record
$\widetilde{\mathrm{SL}(2,\mathbb R)}$ Simply connected group integrating sl(2,R)	Group Lie theory	dimℝ 3	reference Sourced record
$\mathrm{SL}(3, \mathbb R)$ Real special linear group in three dimensions	Group Lie theory	dimℝ 8	reference Sourced record
$\mathrm{SU}(2,2)$ Indefinite special unitary group of signature (2,2)	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{Sp}(4, \mathbb R)$ Real symplectic group on a four-dimensional space	Group Lie theory	dimℝ 10	reference Sourced record
$\mathrm{SO}_0(1,3)$ Proper orthochronous Lorentz group in four dimensions	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{SO}_0(2,4)$ Identity component of the orthogonal group of signature (2,4)	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{SO}_0(2,3)$ Identity component of the orthogonal group of signature (2,3)	Group Lie theory	dimℝ 10	reference Sourced record
$\mathrm{SU}(2) \longrightarrow \mathrm{SO}(3)$ Spin cover of three-dimensional rotations	Map Lie theory	SO(3)	executable 1 linked workbench
$\mathrm{SU}(3) \longrightarrow \mathrm{PSU}(3)$ Adjoint quotient of SU(3)	Map Lie theory	PSU(3)	executable 1 linked workbench
$\mathrm{SU}(2) \hookrightarrow \mathrm{SU}(3)$ Regular block inclusion in SU(3)	Map Lie theory	{diag(A,1) : A ∈ SU(2)} ≅ SU(2)	executable 2 linked workbenchs
$\mathrm{SU}(2) \longrightarrow \mathrm{SU}(3),\quad j=1$ Principal spin-one homomorphism into SU(3)	Map Lie theory	An SU(3)-conjugate of the real SO(3) subgroup	executable 2 linked workbenchs
$\mathrm{U}(1) \hookrightarrow \mathrm{SU}(2)$ Normalized maximal torus of SU(2)	Map Lie theory	{diag(z,z⁻¹) : \|z\|=1}	executable 1 linked workbench
$\mathrm{U}(1) \cong \mathrm{SO}(2) \hookrightarrow \mathrm{SO}(3)$ Normalized axial rotation subgroup of SO(3)	Map Lie theory	Rotations fixing a chosen oriented axis, isomorphic to SO(2)	executable 1 linked workbench
$\mathrm{Spin}(4) \longrightarrow \mathrm{SO}(4)$ Quaternionic spin cover in four Euclidean dimensions	Map Lie theory	SO(4)	reference Sourced record
$\mathrm{SU}(4) \longrightarrow \mathrm{SO}(6)$ Exterior-square realization of the Spin(6) cover	Map Lie theory	SO(6)	reference Sourced record
$\mathrm{SU}(4) \longrightarrow \mathrm{PSU}(4)$ Full central quotient of SU(4)	Map Lie theory	PSU(4)	reference Sourced record
$\mathrm{Sp}(2) \longrightarrow \mathrm{SO}(5)$ Quaternionic construction of the Spin(5) cover	Map Lie theory	SO(5)	reference Sourced record
$\mathrm{Spin}(8) \longrightarrow \mathrm{SO}(8)$ Vector spin cover in eight Euclidean dimensions	Map Lie theory	SO(8)	reference Sourced record
$\mathrm{Spin}(10) \longrightarrow \mathrm{SO}(10)$ Vector spin cover in ten Euclidean dimensions	Map Lie theory	SO(10)	reference Sourced record
$\mathrm{SL}(2, \mathbb C) \longrightarrow \mathrm{SO}_0(1,3)$ Hermitian-matrix Lorentz spin cover	Map Lie theory	SO₀(1,3)	reference Sourced record
$\mathrm{SL}(2, \mathbb R) \longrightarrow \mathrm{PSL}(2, \mathbb R)$ Adjoint central quotient of SL(2,R)	Map Lie theory	PSL(2,R) ≅ SO₀(2,1)	reference Sourced record
$\widetilde{\mathrm{SL}(2,\mathbb R)} \longrightarrow \mathrm{SL}(2,\mathbb R)$ Universal covering map of SL(2,R)	Map Lie theory	SL(2,R)	reference Sourced record
$\mathrm{SU}(2,2) \longrightarrow \mathrm{SO}_0(2,4)$ Exterior-square spin cover in signature (2,4)	Map Lie theory	SO₀(2,4)	reference Sourced record
$\mathrm{Sp}(4, \mathbb R) \longrightarrow \mathrm{SO}_0(2,3)$ Primitive-exterior-square spin cover in signature (2,3)	Map Lie theory	SO₀(2,3)	reference Sourced record
$\mathrm{SU}(2)/\mathrm{U}(1) \cong S^2$ Hopf sphere as SU(2)/U(1)	Space Lie theory	Reference	executable 1 linked workbench
$\mathrm{SO}(3)/\mathrm{SO}(2) \cong S^2$ Rotation sphere as SO(3)/SO(2)	Space Lie theory	Reference	executable 1 linked workbench
$A_{1}:\quad V_{(1)}$ Doublet of SU(2)	Representation Lie theory	dimℂ 2	executable 2 linked workbenchs
$A_{1}:\quad V_{(2)}$ Adjoint of SU(2)	Representation Lie theory	dimℂ 3	executable 2 linked workbenchs
$A_{2}:\quad V_{(1,0)}$ Fundamental of SU(3)	Representation Lie theory	dimℂ 3	executable 5 linked workbenchs
$A_{2}:\quad V_{(0,1)}$ Conjugate fundamental of SU(3)	Representation Lie theory	dimℂ 3	executable 5 linked workbenchs
$A_{2}:\quad V_{(1,1)}$ Adjoint of SU(3)	Representation Lie theory	dimℂ 8	executable 5 linked workbenchs
$A_{2}:\quad V_{(2,0)}$ Symmetric sextet of SU(3)	Representation Lie theory	dimℂ 6	executable 5 linked workbenchs
$A_{2}:\quad V_{(3,0)}$ Decuplet of SU(3)	Representation Lie theory	dimℂ 10	executable 5 linked workbenchs
$A_{2}:\quad V_{(2,2)}$ 27-dimensional irrep of SU(3)	Representation Lie theory	dimℂ 27	executable 5 linked workbenchs
$A_{3}:\quad V_{(1,0,0)}$ Fundamental of SU(4)	Representation Lie theory	dimℂ 4	executable 3 linked workbenchs
$A_{3}:\quad V_{(0,1,0)}$ Two-index antisymmetric of SU(4)	Representation Lie theory	dimℂ 6	executable 3 linked workbenchs
$A_{3}:\quad V_{(1,0,1)}$ Adjoint of SU(4)	Representation Lie theory	dimℂ 15	executable 3 linked workbenchs
$A_{4}:\quad V_{(1,0,0,0)}$ Fundamental of SU(5)	Representation Lie theory	dimℂ 5	executable 3 linked workbenchs
$A_{4}:\quad V_{(0,1,0,0)}$ Two-index antisymmetric of SU(5)	Representation Lie theory	dimℂ 10	executable 3 linked workbenchs
$A_{4}:\quad V_{(1,0,0,1)}$ Adjoint of SU(5)	Representation Lie theory	dimℂ 24	executable 3 linked workbenchs
$C_{2}:\quad V_{(1,0)}$ Defining representation of Sp(2) = USp(4)	Representation Lie theory	dimℂ 4	executable 1 linked workbench
$C_{2}:\quad V_{(0,1)}$ Five-dimensional irrep of Sp(2) = USp(4)	Representation Lie theory	dimℂ 5	executable 1 linked workbench
$D_{4}:\quad V_{(1,0,0,0)}$ Vector of Spin(8)	Representation Lie theory	dimℂ 8	executable 1 linked workbench
$D_{4}:\quad V_{(0,0,1,0)}$ Half-spin representation of Spin(8)	Representation Lie theory	dimℂ 8	executable 1 linked workbench
$D_{4}:\quad V_{(0,0,0,1)}$ Other half-spin representation of Spin(8)	Representation Lie theory	dimℂ 8	executable 1 linked workbench
$D_{5}:\quad V_{(1,0,0,0,0)}$ Vector of Spin(10)	Representation Lie theory	dimℂ 10	executable 1 linked workbench
$D_{5}:\quad V_{(0,0,0,0,1)}$ Chiral spinor of Spin(10)	Representation Lie theory	dimℂ 16	executable 1 linked workbench
$D_{5}:\quad V_{(0,0,0,1,0)}$ Conjugate chiral spinor of Spin(10)	Representation Lie theory	dimℂ 16	executable 1 linked workbench
$D_{5}:\quad V_{(0,1,0,0,0)}$ Adjoint of Spin(10)	Representation Lie theory	dimℂ 45	executable 1 linked workbench
$G_{2}:\quad V_{(1,0)}$ Fundamental of G2 (compact)	Representation Lie theory	dimℂ 7	executable 1 linked workbench
$G_{2}:\quad V_{(0,1)}$ Adjoint of G2 (compact)	Representation Lie theory	dimℂ 14	executable 1 linked workbench
$F_{4}:\quad V_{(0,0,0,1)}$ Minimal representation of F4 (compact)	Representation Lie theory	dimℂ 26	executable 1 linked workbench
$F_{4}:\quad V_{(1,0,0,0)}$ Adjoint of F4 (compact)	Representation Lie theory	dimℂ 52	executable 1 linked workbench
$E_{6}:\quad V_{(1,0,0,0,0,0)}$ 27-dimensional representation of E6 (compact simply connected)	Representation Lie theory	dimℂ 27	executable 1 linked workbench
$E_{6}:\quad V_{(0,1,0,0,0,0)}$ Adjoint of E6 (compact simply connected)	Representation Lie theory	dimℂ 78	executable 1 linked workbench
$E_{7}:\quad V_{(0,0,0,0,0,0,1)}$ 56-dimensional representation of E7 (compact simply connected)	Representation Lie theory	dimℂ 56	executable 1 linked workbench
$E_{7}:\quad V_{(1,0,0,0,0,0,0)}$ Adjoint of E7 (compact simply connected)	Representation Lie theory	dimℂ 133	executable 1 linked workbench
$E_{8}:\quad V_{(0,0,0,0,0,0,0,1)}$ Adjoint of E8 (compact simply connected)	Representation Lie theory	dimℂ 248	executable 1 linked workbench
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$ Compact Standard Model gauge algebra	Lie algebra Lie theory	dimℝ 12	reference Sourced record
$SU(3)\times SU(2)\times U(1)$ Standard Model compact covering product	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(2)}$ Standard Model global group with diagonal Z2 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(3)}$ Standard Model global group with diagonal Z3 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(6)}$ Standard Model global group with diagonal Z6 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(1)}\longrightarrow G_{\mathrm{SM}}^{(2)}$ Standard Model quotient map of order 2	Map Lie theory	G_SM^(2)	executable 1 linked workbench
$G_{\mathrm{SM}}^{(1)}\longrightarrow G_{\mathrm{SM}}^{(3)}$ Standard Model quotient map of order 3	Map Lie theory	G_SM^(3)	executable 1 linked workbench
$G_{\mathrm{SM}}^{(1)}\longrightarrow G_{\mathrm{SM}}^{(6)}$ Standard Model quotient map of order 6	Map Lie theory	G_SM^(6)	executable 1 linked workbench
$SU(3)\times SU(2)\times U(1)_X\longrightarrow SU(5)$ Color-first SU(5) block homomorphism	Map Lie theory	S(U(3) × U(2))	executable 1 linked workbench
$G_{\mathrm{SM}}^{(6)}\hookrightarrow SU(5)$ 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.

Finite-simple classification Lie classification