nLab
spin group

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Integration and differentiation

Cohomology

∞-Connections

∞-Chern-Weil theory

Examples

∞-Lie groupoids

∞-Lie algebroids

Edit this sidebar

The spin group Spin(n) is the universal covering space of the special orthogonal group SO(n). By the usual arguments it inherits a group structure for which the operations are smooth and so is a Lie group like SO(n).

The spin group is one element in the Whitehead tower of O(n), which starts out like

⋯→Fivebrane(n)→String(n)→Spin(n)→SO(n)→O(n).\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

Fivebrane group to string group to spin group to special orthogonal group to orthogonal group.

The homotopy groups of O(n) are for k∈ℕ and for sufficiently large n

π 8k+0(O) =ℤ 2 π 8k+1(O) =ℤ 2 π 8k+2(O) =0 π 8k+3(O) =ℤ π 8k+4(O) =0 π 8k+5(O) =0 π 8k+6(O) =0 π 8k+7(O) =ℤ.\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

By co-killing these groups step by step one gets

cokillthis toget π 0(O) =ℤ 2 SO π 1(O) =ℤ 2 Spin π 2(O) =0 π 3(O) =ℤ String π 4(O) =0 π 5(O) =0 π 6(O) =0 π 7(O) =ℤ Fivebrane.\array{ cokill this &&&& to get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.

Spin group in physics

The name arises due to the requirement that the structure group of the tangent bundle of spacetime lifts to Spin(n) so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)

See spin structure.

Revised on December 16, 2009 17:41:31 by Urs Schreiber (193.198.162.13)
Edit | Back in time (8 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Lie groupoid, Lie theory for stacky Lie groupoids, Lie theory, Lie's three theorems, Lie algebroid, Lie integration, differentiable stack, Lie algebra, L-infinity-algebra, Chevalley-Eilenberg algebra, Weil algebra, Lie infinity-groupoid, Lie infinity-algebroid, Courant algebroid, generalized complex geometry, differential graded Lie algebra, cohomology, Lie infinity-algebroid representation, path groupoid, Atiyah Lie groupoid, orbifold, Lie group, path n-groupoid, path infinity-groupoid, groupoid of Lie-algebra valued forms, symplectic manifold, Lie 2-algebra, tangent Lie algebroid, invariant polynomial, String Lie 2-algebra, supergravity Lie 3-algebra, Whitehead tower, string 2-group, spin group, string group, Fivebrane group, special orthogonal group, orthogonal group, Lie algebra cohomology, fivebrane 6-group, Maurer-Cartan form, n-symplectic manifold, Poisson Lie algebroid, nonabelian Lie algebra cohomology, strict Lie 2-algebra, infinity-Lie theory - contents, Atiyah Lie algebroid, curvature, Lie operad, differential 2-crossed module, 2-groupoid of Lie 2-algebra valued forms, infinity-Lie algebra cohomology, rational homotopy theory in an (infinity,1)-topos, The Dirac Electron, Lie 2-group, Lie 3-algebra, Chern-Simons form, curvature characteristic form, infinity-Lie theory, Chern-Weil homomorphism, groupal model for universal principal infinity-bundles, automorphism infinity-Lie algebra, Bianchi identity, 3-groupoid of Lie 3-algebra valued forms, Lie 3-group, infinity-Lie group, Jacobi identity