nLab pin group

Idea

A $Pin$-group is a double cover of an orthogonal group. Its restriction along the inclusion of the special orthogonal group is a Spin group. Hence a $Pin$-group is “like the corresponding Spin group, but including reflections”.

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

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

where we understand the standard quadratic form on $\mathbb{R}^n$ for either global sign

$$\array{ \mathbb{R}^n &\overset{q_{\pm}}{\longrightarrow}& \mathbb{R} \\ \vec x &\mapsto& \pm \underset{i}{\sum} (x^i)^2 }$$

The corresponding two $Pin$-groups are denoted

$$Pin_\pm(n) \;\coloneqq\; Pin\big( \mathbb{R}^n, q_\pm\big)$$

Examples

• Pin(2)
• Pin(5)

Related concepts

group	symbol	universal cover	symbol	higher cover	symbol
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

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

• pseudoscalar representation, pseudovector representation

References

A standard textbook account is

• H. Blaine Lawson, Marie-Louise Michelsohn, chapter I, section 2 of Spin geometry, Princeton University Press (1989)

See also

• Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

The following article discusses which of the Pin groups are in fact compatible with general relativity

• Bas Janssens, The Pin Groups in General Relativity (arXiv:1709.02742)