Pin group

**spin geometry**, **string geometry**, **fivebrane geometry** …

The simply connected cover of the orthogonal group. Its restriction along the inclusion of the special orthogonal group is the Spin group. Hence the $Pin$-group is “like the Spin group, but including reflections”.

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.

The *Clifford algebra* $CL(V,q)$ of a quadratic vector space, def. 1, 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.

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

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

The Pin group $Pin(V;q)$ of a quadratic vector space, def. 1, 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 $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(V) = 1$.

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 $q(V) = 1$.

Specifically, “the” Spin group is

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

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 | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |

Narain group | $O(n,n)$ | ||||

Poincaré group | $ISO(n,1)$ | Poincaré spin group | $\widehat {ISO}(n,1)$ | $\,$ | $\,$ |

super Poincaré group | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |

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)

Revised on February 4, 2015 18:35:10
by Urs Schreiber
(78.102.228.246)