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Pin group

Contents

Idea

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

Definition

Definition

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

Conventions as in (Varadarajan 04, section 5.3).

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

Definition

The Clifford algebra CL(V,q)CL(V,q) of a quadratic vector space, def. 1, is the associative algebra over kk which is the quotient

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

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

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

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

Proposition

There exists a unique complex representation

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

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

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

The Pin group Pin(V;q)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)Cl(V,q)

Pin(V,q)GL 1(Cl(V,q)) Pin(V,q) \hookrightarrow GL_1(Cl(V,q))

on those elements which are multiples v 1v 2kv_1 \cdots v_{2k} of elements v iVv_i \in V with q(V)=1q(V) = 1.

The Spin group Spin(V,q)Spin(V,q) is the further subgroup of Pin(V;q)Pin(V;q) on those elements which are even number multiples v 1v 2kv_1 \cdots v_{2k} of elements v iVv_i \in V with q(V)=1q(V) = 1.

Specifically, “the” Spin group is

Spin(n)Spin( n). Spin(n) \coloneqq Spin(\mathbb{R}^n) \,.
groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group

References

A standard textbook account is

See also

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