Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.

In general the map from the Pin group to the orthogonal group is not surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.

The non-trivial element of the kernel is denoted $-1,$ which should not be confused with the orthogonal transform of reflection through the origin, generally denoted $-I.$

General definition

Let $V$ be a vector space with a non-degenerate quadratic form $Q$ . The pin group $\operatorname {Pin} (V,Q)$ is the subset of the Clifford algebra $\operatorname {Cl} (V,Q)$ consisting of elements of the form $v_{1}v_{2}\cdots v_{k}$ , where the $v_{i}$ are vectors such that $Q(v_{i})=\pm 1$ . The spin group $\operatorname {Spin} (V,Q)$ is defined similarly, but with $k$ restricted to be even; it is a subgroup of the pin group.^[1]

In this article, $V$ is always a real vector space. When $V$ has basis vectors $e_{1},...,e_{p+q}$ satisfying $e_{1}^{2},...,e_{p}^{2}=1$ and $e_{p+1}^{2},...,e_{p+q}^{2}=-1,$ the pin group is denoted Pin(p, q).

Geometrically, for vectors $v$ with $Q(v)\neq 0$ , $-vwv^{-1}$ is the reflection of a vector $w$ across the hyperplane orthogonal to $v$ . More generally, an element $v_{1}v_{2}\cdots v_{k}$ of the pin group acts on vectors by transforming $w$ to $(-1)^{k}v_{1}v_{2}\cdots v_{k}w(v_{1}v_{2}\cdots v_{k})^{-1}$ , which is the composition of k reflections. Since every orthogonal transformation can be expressed as a composition of reflections (the Cartan–Dieudonné theorem), it follows that this representation of the pin group is a homomorphism from the pin group onto the orthogonal group. This is often called the twisted adjoint representation. The elements ±1 of the pin group are the elements which map to the identity $I\in O(p,q)$ , and every element of $O(p,q)$ corresponds to exactly two elements of $\operatorname {Pin} (p,q)$ .^[2]

Definite form

The pin group of a definite form maps onto the orthogonal group, and each component is simply connected (in dimension 3 and higher): it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are.^{[note 1]}

In terms of the standard forms, O(n, 0) = O(0, n), but Pin(n, 0) and Pin(0, n) are in general not isomorphic. Using the "+" sign convention for Clifford algebras (where $v^{2}=Q(v)\in \mathrm {Cl} (V,Q)$ ), one writes

{\mbox{Pin}}_{+}(n):={\mbox{Pin}}(n,0)\qquad {\mbox{Pin}}_{-}(n):={\mbox{Pin}}(0,n)

and these both map onto O(n) = O(n, 0) = O(0, n).

By contrast, we have the natural isomorphism^{[note 2]} Spin(n, 0) ≅ Spin(0, n) and they are both the (unique) non-trivial double cover of the special orthogonal group SO(n), which is the (unique) universal cover for n ≥ 3.

Indefinite form

There are as many as eight different double covers of O(p, q), for p, q ≠ 0, which correspond to the extensions of the center (which is either C₂ × C₂ or C₄) by C₂. Only two of them are pin groups—those that admit the Clifford algebra as a representation. They are called Pin(p, q) and Pin(q, p) respectively.

As topological group

Every connected topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group. For a disconnected topological group, there is a unique universal cover of the identity component of the group, and one can take the same cover as topological spaces on the other components (which are principal homogeneous spaces for the identity component) but the group structure on other components is not uniquely determined in general.

The Pin and Spin groups are particular topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.

In 2001, Andrzej Trautman^{[note 3]} found the set of all 32 inequivalent double covers of O(p) x O(q), the maximal compact subgroup of O(p, q) and an explicit construction of 8 double covers of the same group O(p, q).

Construction

The two pin groups correspond to the two central extensions

1\to \{\pm 1\}\to {\mbox{Pin}}_{\pm }(V)\to \mathrm {O} (V)\to 1.

The group structure on Spin(V) (the connected component of determinant 1) is already determined; the group structure on the other component is determined up to the center, and thus has a ±1 ambiguity.

The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in O(V), r² = 1, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of Spin_±(V) → SO(V), so ${\tilde {r}}^{2}=\pm 1$ , and either choice determines a pin group (since all reflections are conjugate by an element of SO(V), which is connected, all reflections must square to the same value).

Concretely, in Pin₊, ${\tilde {r}}$ has order 2, and the preimage of a subgroup {1, r} is C₂ × C₂: if one repeats the same reflection twice, one gets the identity.

In Pin₋, ${\tilde {r}}$ has order 4, and the preimage of a subgroup {1, r} is C₄: if one repeats the same reflection twice, one gets "a rotation by 2π"—the non-trivial element of Spin(V) → SO(V) can be interpreted as "rotation by 2π" (every axis yields the same element).

Low dimensions

In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups:

{\begin{aligned}{\mbox{Pin}}_{+}(1)&\cong \mathrm {C} _{2}\times \mathrm {C} _{2}={\mbox{Dih}}_{1}\\{\mbox{Pin}}_{-}(1)&\cong \mathrm {C} _{4}={\mbox{Dic}}_{1}.\end{aligned}}

In 2 dimensions, the distinction between Pin₊ and Pin₋ mirrors the distinction between the dihedral group of a 2n-gon and the dicyclic group of the cyclic group C_2n.

In Pin₊, the preimage of the dihedral group of an n-gon, considered as a subgroup Dih_n < O(2), is the dihedral group of a 2n-gon, Dih_2n < Pin₊(2), while in Pin₋, the preimage of the dihedral group is the dicyclic group Dic_n < Pin₋(2).

The resulting commutative square of subgroups for Spin(2), Pin₊(2), SO(2), O(2) – namely C_2n, Dih_2n, C_n, Dih_n – is also obtained using the projective orthogonal group (going down from O by a 2-fold quotient, instead of up by a 2-fold cover) in the square SO(2), O(2), PSO(2), PO(2), though in this case it is also realized geometrically, as "the projectivization of a 2n-gon in the circle is an n-gon in the projective line".

In 3 dimensions the situation is as follows. The Clifford algebra generated by 3 anticommuting square roots of +1 is the algebra of 2×2 complex matrices, and Pin₊(3) is isomorphic to $\{A\in U(2):\det A=\pm 1\}$ .^[3] The Clifford algebra generated by 3 anticommuting square roots of -1 is the algebra $\mathbb {H} \oplus \mathbb {H}$ , and Pin₋(3) is isomorphic to SU(2) × C₂. These groups are nonisomorphic because the center of Pin₊(3) is C₄ while the center of Pin₋(3) is C₂ × C₂.

Center

Suppose $n>0$ . The center of $\mathrm {Pin} _{+}(n)$ is $\mathrm {C} _{2}\times \mathrm {C} _{2}$ when $n=0,1\ \mathrm {mod} \ 4$ , and $\mathrm {C} _{4}$ when $n=2,3\ \mathrm {mod} \ 4$ . The center of $\mathrm {Pin} _{-}(n)$ is $\mathrm {C} _{2}\times \mathrm {C} _{2}$ when $n=0,3\ \mathrm {mod} \ 4$ , and $\mathrm {C} _{4}$ when $n=2,1\ \mathrm {mod} \ 4$ .

Name

The name was introduced in (Atiyah, Bott & Shapiro 1964, page 3, line 17), where they state "This joke is due to J-P. Serre". It is a back-formation from Spin: "Pin is to O(n) as Spin is to SO(n)", hence dropping the "S" from "Spin" yields "Pin".

Notes

^ In fact, they are equal as subsets of GL(V), not just isomorphic as abstract groups: an operator preserves a form if and only if it preserves the negative form.
^ They are included into different algebras $\mathrm {Cl} (n,0)\not \cong \mathrm {Cl} (0,n)$ , but they are equal as subsets of the vector spaces $\mathrm {Cl} (n,0)=\mathrm {Cl} (0,n)=\Lambda ^{*}\mathbf {R} ^{n}$ , and carry the same product structure, hence they are naturally identified.
^ A. Trautman (2001). "Double Covers of Pseudo-orthogonal Groups". In F. Brackx; J. S. R. Chisholm; V. Souček (eds.). Clifford Analysis and Its Applications. NATO Science Series. Vol. 25. pp. 377–388. doi:10.1007/978-94-010-0862-4_32. ISBN 978-0-7923-7045-1.

References

^ Lawson & Michelsohn 1989, p. 18
^ Lawson & Michelsohn 1989, pp. 16–20
^ Harvey 1990, p. 302

Atiyah, M.F.; Bott, R.; Shapiro, A. (1964), "Clifford modules", Topology, 3, suppl. 1: 3–38, doi:10.1016/0040-9383(64)90003-5
M. Karoubi (1968). "Algèbres de Clifford et K-théorie". Ann. Sci. Éc. Norm. Supér. 1 (2): 161–270. doi:10.24033/asens.1163.
Dabrowski, L. (1988), Group Actions on Spinors, Bibliopolis, ISBN 88-7088-205-5
Carlip, S.; DeWitt-Morette, C. (1988), "Where the sign of the metric makes a difference", Phys. Rev. Lett., 60 (16): 1599–1601, Bibcode:1988PhRvL..60.1599C, doi:10.1103/physrevlett.60.1599, PMID 10038088
Chamblin, A. (1994), "On the obstructions to non-Cliffordian pin structures", Comm. Math. Phys., 164 (1): 65–85, arXiv:gr-qc/9509039, Bibcode:1994CMaPh.164...65C, doi:10.1007/bf02108806, S2CID 18305909
Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4
Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
Karoubi, Max (2008). K-Theory. Springer. pp. 212–214. ISBN 978-3-540-79889-7.

[3] In fact, they are equal as subsets of GL(V), not just isomorphic as abstract groups: an operator preserves a form if and only if it preserves the negative form.

[4] They are included into different algebras $\mathrm {Cl} (n,0)\not \cong \mathrm {Cl} (0,n)$ , but they are equal as subsets of the vector spaces $\mathrm {Cl} (n,0)=\mathrm {Cl} (0,n)=\Lambda ^{*}\mathbf {R} ^{n}$ , and carry the same product structure, hence they are naturally identified.

[5] A. Trautman (2001). "Double Covers of Pseudo-orthogonal Groups". In F. Brackx; J. S. R. Chisholm; V. Souček (eds.). Clifford Analysis and Its Applications. NATO Science Series. Vol. 25. pp. 377–388. doi:10.1007/978-94-010-0862-4_32. ISBN 978-0-7923-7045-1.

[1] Lawson & Michelsohn 1989, p. 18

[2] Lawson & Michelsohn 1989, pp. 16–20

[6] Harvey 1990, p. 302

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