Let $$g$$ be an bilinear symmetric form on a real vector space $$V$$.

Definition: The complex Clifford algebra $$\mathbb{C}l(V)$$ is the complex unital algebra generated by elements $$v \in V$$ with product defined by

$u \cdot v + v \cdot u = 2 g(u,v).$

When $$V = \mathbb{R}^n$$ and $$g$$ is the usual euclidean metric, we denote the Clifford algebra by $$\mathbb{C}l(V)$$.

By choosing a basis $$e_1,\dots, e_n$$ for $$V$$, we can see that $$e_{i_1}\cdots e_{i_k}$$ where $$1 \leq i_1 \le \cdots \le i_k \leq n$$ form a basis and the complex dimension of $$\mathbb{C}l(V)$$ is $$2^n$$. This post aims to show some of the structures that the Clifford algebras has.

The many structures enjoyed by $$\mathbb{C}l$$

The Clifford algebra comes naturally with a $$\mathbb{Z}$$-grading given by $$\mathbb{C}l(n)_k = \langle e_{i_1}\cdots e_{i_k}\rangle_{1 \leq i_1 \le \cdots \le i_k \leq n}$$.

We have a natural Hilbert space structure in $$\mathbb{C}l(V)$$ given by

$\langle e_{i_1} \dots e_{i_k}, e_{j_1} \dots e_{j_l}\rangle = 0$

if $$k \not= s$$ or if $$i_1 < \cdots < i_k$$ is not the same sequence as $$j_1 < \cdots < j_s$$, and declaring that the elements $$e_{i_1} \dots e_{i_k}$$ are normal.

We also have a $$^*$$-algebra structure given by

$(e_{i_1} \dots e_{i_k})^* = e_{i_k} \dots e_{i_1}$

and a trace $$\tau$$, such that $$\tau(1) = 1$$ given by $$\tau(e_{i_1} \dots e_{i_k}) = 0$$ whenever $$k \geq 1$$.

Proposition: We have that the inner product is given by

$\langle a, b \rangle = \tau(a^*b).$

The inner product coincides with $$g$$ in $$V$$ and we have $$\lVert a^* a \rVert = \lVert a \rVert^2$$, $$\mathbb{C}l(V)$$ becomes a $$C^*$$-algebra.

Let $$n= 2m, 2m+1$$ and let $$\gamma = (-1)^m e_1\cdots e_n$$ be the top element in the Clifford algebra, the so-called chirality element, we have that $$\gamma$$ does not depend on the choice of orthogonal basis and

$\gamma^* = \gamma,\quad \gamma^* \gamma = 1.$

Let $$\chi : \mathbb{C}l(V) \to \mathbb{C}l(V)$$ given by $$\chi(a) = \gamma\, a\, \gamma^*$$, we have that $$\chi(v) = (-1)^{n + 1} v$$ for all $$v \in V$$. If the dimension of $$V$$ is even, the endomorphism $$\chi$$ induces a $$\mathbb{Z}_2$$-grading given by

$\mathbb{C}l(V)^{\pm} = \{ a \in \mathbb{C}l(V) \mid \chi(a) = \pm a \}.$

otherwise $$\chi = \text{Id}$$ and $$\gamma$$ is in the center of the Clifford algebra.

Lemma: The center of the Clifford algebra is $$\mathbb{C}$$ if $$V$$ is even-dimensional or $$\mathbb{C} \oplus \gamma \mathbb{C}$$ if $$V$$ is odd-dimensional.

The Spin group

Definition: We say that $$u \in \mathbb{C}l(V)$$ is unitary if $$u^*u = uu^*= 1$$ and the operators $$a \mapsto ua$$ are unitary for any unitary element $$u$$.

The Spin group $$Spin(V)$$ is the set of unitary elements $$u$$ such that

$u = \sum_K a_K e_K$

where $$K = i_1 < \cdots < i_{2s}$$ is a multi-index of even length and $$a_K$$ is real for all $$K$$. $$Spin(V)$$ has a natural multiplication and inversion and $$1 \in \mathbb{C}l(V)$$ is in $$Spin(V)$$ and corresponds to its identity.

Proposition: $$Spin(V)$$ is a connected Lie group, simply connected for $$n > 2$$. It is the double cover of $$SO(V)$$.

Proof: $$Spin(V)$$ is given by algebraic equations and therefore is an algebraic group.

Let $$\phi: Spin(V) \to \text{End}(V)$$ given by $$\phi(u)x = uxu^*$$, then since

$g(\phi(u) x ,\phi(u) x ) = \tau( (uxu^*)^* uxu^*) = \tau(x^*x) = g(x,x), \quad x \in V.$

So $$\phi(u)$$ is actually in $$O(V)$$ and $$\det(\phi(u)) = \pm 1$$. We extend the definition of $$\phi$$ for all invertible $$a \in \mathbb{C}l(V)$$ as $$\phi(a)x = \chi(a)xa^{-1}$$, we note that for a unitary vector $$v\in V$$

$\phi(v)x = -vxv = (xv - g(x,v))v = x - g(x,v)v$

which is the reflection with respect of $$v$$, since all elements of $$SO(V)$$ are generated by an even number of these we see that $$\phi$$ is onto $$SO(V)$$.

Suppose that $$u$$ is in the kernel of $$\phi$$ then $$u$$ commutes with all $$x \in V$$ and $$u$$ is in the center of the Clifford algebra. By considering by separated the cases of odd and even dimension we see that $$u \in \mathbb{C}$$ is real and unitary, hence $$u = \pm 1$$.

By the exact sequence

$1 \longrightarrow \mathbb{Z}_2 \longrightarrow Spin(V) \longrightarrow SO(V) \longrightarrow 1$

and the long exact sequence of homotopy groups we have that $$\pi_0(Spin(V)) = 1$$ and we also have the exact sequence $$1 \longrightarrow \pi_1(SO(V)) \longrightarrow \pi_1(Spin(V))) \longrightarrow 1 .$$

Finishing the proof. $$\square$$