# nLab spin group

Contents

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

see also

string geometry

## Ninebrane geometry

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

#### Higher Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

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 $SO(n)$.

For special cases in low dimensions see at: Spin(2), Spin(3), Spin(4), Spin(5), Spin(6), Spin(7), Spin(8)

## Definition

###### Definition

A quadratic vector space $(V, \langle -,-\rangle)$ is a vector space $V$ over finite dimension over a field $k$ of characteristic 0, and equipped with a symmetric bilinear form $\langle -,-\rangle \colon V \otimes V \to k$.

Conventions as in (Varadarajan 04, section 5.3).

We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding quadratic form.

###### Definition

The Clifford algebra $CL(V,q)$ of a quadratic vector space, def. , is the associative algebra over $k$ which is the quotient

$Cl(V,q) \coloneqq T(V)/I(V,q)$

of the tensor algebra of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.

Since the tensor algebra $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.

Let $(\mathbb{R}^n, q = {\vert -\vert})$ be the $n$-dimensional Cartesian space with its canonical scalar product. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the complexification of its Clifford algebra.

###### Proposition

There exists a unique complex representation

$Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)$

of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest dimension

$dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]} \,.$
###### Definition

The Pin group $Pin(V;q)$ of a quadratic vector space, def. , is the subgroup of the group of units in the Clifford algebra $Cl(V,q)$

$Pin(V,q) \hookrightarrow GL_1(Cl(V,q))$

on those elements which are multiples $v_1 \cdots v_{n}$ of elements $v_i \in V$ with $q(v_i) = 1$.

The Spin group $Spin(V,q)$ is the further subgroup of $Pin(V;q)$ on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(v_i) = 1$.

Specifically, “the” Spin group is

$Spin(n) \coloneqq Spin(\mathbb{R}^n) \,.$

A spin representation is a linear representation of the spin group, def. .

## Properties

### General

By definition the spin group sits in a short exact sequence of groups

$\mathbb{Z}_2 \to Spin \to SO \,.$

### Relation to Whitehead tower of orthogonal group

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

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

The homotopy groups of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$

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

By co-killing these groups step by step one gets

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

Via the J-homomorphism this is related to the stable homotopy groups of spheres:

$n$012345678910111213141516
Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\mathbb{Z}_{240}$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_6$$\mathbb{Z}_{504}$0$\mathbb{Z}_3$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\mathbb{Z}_2$

### Exceptional isomorphisms

In low dimensions the spin groups happens to be isomorphic to various other classical Lie groups. One speaks of exceptional isomorphisms or sporadic isomorphisms.

See for instance (Garrett 13). See also division algebra and supersymmetry.

In the following $Sp(n)$ denotes the quaternionic unitary group in quaternionic dimension $n$.

We have

• in Euclidean signature

• $Spin(1) \simeq O(1)$

• Spin(2)$\simeq U(1) \simeq SO(2) \simeq S^1$ (SO(2), the circle group, see also at real Hopf fibration)

the projection $Spin(2)\to SO(2)$ corresponds to $S^1\stackrel{\cdot 2}{\longrightarrow} S^1$, see also at Theta characteristic

• Spin(3)$\simeq Sp(1) \simeq SU(2) \simeq S^3$ (the special unitary group SU(2)

the inclusion $Spin(2) \hookrightarrow Spin(3)$ corresponds to the canonical $S^1 \hookrightarrow S^3$ (see e.g. Gorbunov-Ray 92)

• Spin(4)$\simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3$

this is given by identifying $\mathbb{R}^4 \simeq \mathbb{H}$ with the quaternions and $SU(2) \simeq S^3$ with the group of unit quternions. Then left and right quaternion multiplication gives a homomorphism

$SU(2) \times SU(2) \longrightarrow SO(4)$
$(g,h) \mapsto ( x \mapsto \; g^{-1} x h )$

which is a double cover and hence exhibits the isomorphism.

In particular therefore the inclusion $Spin(3) \hookrightarrow Spin(4)$ corresponds to the diagonal $S^3 \hookrightarrow S^3 \times S^3$.

At the level of Lie algebras $\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4$ and the $\pm 1$-eigenspaces of the Hodge star operator $\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4$ gives the direct sum decomposition $\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)$

• Spin(5)$\simeq Sp(2)$ (an indirect consequence of triality, see e.g. Čadek-Vanžura 97)

• Spin(6)$\simeq SU(4)$ (the special unitary group SU(4))

• in Lorentzian signature

• $Spin(1,1) \simeq GL(1,\mathbb{R})$

• $Spin(2,1) \simeq SL(2, \mathbb{R})$ – 2d special linear group of real numbers

• $Spin(3,1) \simeq SL(2,\mathbb{C})$ – 2d special linear group of complex numbers

• $Spin(4,1) \simeq Sp(1,1)$

• $Spin(5,1) \simeq SL(2,\mathbb{H})$ – 2d special linear group of quaternions

• $Spin(9,1) \simeq_{in\;some\;sense} SL(2, \mathbb{O})$ – 2d special linear group of octonions (see SL(2,O) for more on this would-be isomorphism)

• in anti de Sitter signature

• $Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$

• $Spin(3,2) \simeq Sp(4,\mathbb{R})$

• $Spin(4,2) \simeq SU(2,2)$

• in mixed signature

• $Spin(3,3) \simeq SL(4,\mathbb{R})$ (Garrett 13 (2.12))

Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq$ SL(2,H)$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$Spin(9,1) ${\simeq}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

## Examples

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

see also

## Applications

### 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.

The Whitehead tower of the orthogonal group looks like

$\cdots \to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}(n)$Pin group$Pin(n)$Tring group$Tring(n)$
special orthogonal group$SO(n)$Spin group$Spin(n)$String group$String(n)$
Lorentz group$\mathrm{O}(n,1)$$\,$$Spin(n,1)$$\,$$\,$
anti de Sitter group$\mathrm{O}(n,2)$$\,$$Spin(n,2)$$\,$$\,$
conformal group$\mathrm{O}(n+1,t+1)$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$Poincaré spin group$\widehat {ISO}(n,1)$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$
superconformal group

## References

Textbook accounts:

See also

Examples of sporadic (exceptional) spin group isomorphisms incarnated as isogenies onto orthogonal groups are discussed in

The exceptional isomorphism Spin(5) $\simeq$ Sp(2) is discussed via triality in

• Martin Čadek, Jiří Vanžura, On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

Discussion of the cohomology of the classifying space $B Spin$ includes

• E. Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.

• Harsh Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

Last revised on October 31, 2023 at 17:00:19. See the history of this page for a list of all contributions to it.