Curated collection

Spacetime groups and spin covers

Keep dimension, signature, connected component, and covering group visible when discussing fields and states.

23 released records · classifications and individual objects are labeled separately.

Spacetime groups and spin covers
Object	Kind	Essential data	Coverage
$\mathrm{SU}(2)$ Special unitary group	Group Lie theory	dimℝ 3	executable 7 linked workbenchs
$\mathrm{Spin}(4)$ 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
$\mathrm{Spin}(8)$ Simply connected compact spin group in eight dimensions	Group Lie theory	dimℝ 28	executable 1 linked workbench
$\mathrm{Spin}(10)$ Simply connected compact spin group in ten dimensions	Group Lie theory	dimℝ 45	executable 1 linked workbench
$\mathrm{SO}_0(1,3)$ Proper orthochronous Lorentz group in four dimensions	Group Lie theory	dimℝ 6	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
$\mathrm{SU}(2) \longrightarrow \mathrm{SU}(3),\quad j=1$ Principal spin-one homomorphism into SU(3)	Map Lie theory	An SU(3)-conjugate of the real SO(3) subgroup	executable 2 linked workbenchs
$\mathrm{Spin}(4) \longrightarrow \mathrm{SO}(4)$ 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
$\mathrm{Sp}(2) \longrightarrow \mathrm{SO}(5)$ 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
$\mathrm{SU}(2,2) \longrightarrow \mathrm{SO}_0(2,4)$ 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
$D_{4}:\quad V_{(1,0,0,0)}$ 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