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. \(\square\)