Spin group

Definition (via the Clifford algebra)

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

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:

Covering map to $SO(n)$

There is a canonical group homomorphism

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

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

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.