Spin group - Knowlpedia

Let $n\ge 2$ , let $V=\mathbb R^n$ with its standard inner product $\langle\cdot,\cdot\rangle$ , and let $\mathrm{Cl}(V)$ be the real Clifford algebra generated by $V$ subject to the relations

v\cdot v=-\langle v,v\rangle\,1 \quad (v\in V).

The spin group $\mathrm{Spin}(n)$ is the subgroup of the even Clifford algebra $\mathrm{Cl}^0(V)$ generated by products of an even number of unit vectors:

\mathrm{Spin}(n)=\langle v_1v_2\cdots v_{2k}\mid v_i\in V,\ \langle v_i,v_i\rangle=1\rangle \subset \mathrm{Cl}^0(V)^\times.

Covering map to

SO(n)

There is a canonical group homomorphism

\rho:\mathrm{Spin}(n)\to SO(n)

defined by the conjugation action on $V\subset \mathrm{Cl}(V)$ :

\rho(s)(v)=s\,v\,s^{-1}.

One checks that $\rho(s)$ preserves $\langle\cdot,\cdot\rangle$ and has determinant $1$ , hence lands in the special orthogonal group $SO(n)$ . Moreover,

\ker(\rho)=\{\pm 1\},

so $\rho$ is a $2$ -fold covering map. For $n\ge 3$ , $\mathrm{Spin}(n)$ is connected and simply connected, and $\rho$ identifies it as the universal covering group of $SO(n)$ .

Lie algebra and context

The differential $d\rho_e$ is an isomorphism of Lie algebras, so the Lie algebra of $\mathrm{Spin}(n)$ is canonically identified with the orthogonal Lie algebra $\mathfrak{so}(n)$ . This makes $\mathrm{Spin}(n)$ fundamental in topology and representation theory: “spin representations” are representations of $\mathrm{Spin}(n)$ that do not descend to $SO(n)$ , reflecting the nontriviality of the covering.

Positive-definite scope

This knowl uses a positive-definite inner product, which has signature $(0,n)$ in this collection's negative-first ordering, and the Clifford convention $v^2=-\langle v,v\rangle$ . For a quadratic form of mixed signature, the related groups $\operatorname{Pin}(p,q)$ , $\operatorname{Spin}(p,q)$ , and their identity components require additional component bookkeeping; see indefinite spin group. In the Lorentzian case,

\operatorname{Spin}^+(1,3)\cong SL(2,\mathbb C)_{\mathbb R}.

This link does not redefine the compact group $\operatorname{Spin}(n)$ treated above.

References

H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter I, §§2–5. Publisher record.
Thomas Friedrich, Dirac Operators in Riemannian Geometry, AMS, 2000, §1.3. Publisher record.