Clifford algebra

2020 Mathematics Subject Classification: Primary: 15A66 [MSN][ZBL]

The Clifford algebra of a quadratic form is a finite-dimensional associative algebra over a commutative ring; it was first investigated by W. Clifford in 1876. Let $K$ be a commutative ring with an identity, let $E$ be a free $K$-module and let $Q$ be a quadratic form on $E$. By the Clifford algebra of the quadratic form $Q$ (or of the pair $(E,Q)$) one means the quotient algebra $C(Q)$ of the tensor algebra $T(E)$ of the $K$-module $E$ by the two-sided ideal generated by the elements of the form $x\otimes x-Q(x)\cdot 1$, where $x\in E$. Elements of $E$ are identified with their corresponding cosets in $C(Q)$. For any $x,y\in E$ one has $xy+yx=\Phi(x,y)$, where $\Phi(E\times E)\to K$ is the symmetric bilinear form associated with $Q$.

For the case of the null quadratic form $Q$, $C(Q)$ is the same as the exterior algebra $\Lambda(E)$ of $E$. If $K=\R$, the field of real numbers, and $Q$ is a non-degenerate quadratic form on the $n$-dimensional vector space $E$ over $\R$, then $C(G)$ is the algebra ${}^lA_{n+1}$ of alternions, where $l$ is the number of positive squares in the canonical form of $Q$ (cf. Alternion).

Let $e_1,\dots,e_n$ be a basis of the $K$-module $E$. Then the elements $1, e_{i_1}\cdots e_{i_k}\; (i_1<\cdots < i_k)$ form a basis of the $K$-module $C(Q)$. In particular, $C(Q)$ is a free $K$-module of rank $2^n$. If in addition the $e_1,\dots,e_n$ are orthogonal with respect to $Q$, then $C(Q)$ can be presented as a $K$-algebra with generators $1,e_1,\dots,e_n$ and relations $e_i e_j = -e_je_i\; (i\ne j)$ and $e_i^2 = Q(e_i)$. The submodule of $C(Q)$ generated by products of an even number of elements of $E$ forms a subalgebra of $C(Q)$, denoted by $C^+(Q)$.

Suppose that $K$ is a field and that the quadratic form $Q$ is non-degenerate. For even $n$, $C(Q)$ is a central simple algebra over $K$ of dimension $2^n$, the subalgebra $C^+(Q)$ is separable, and its centre $Z$ has dimension 2 over $K$. If $K$ is algebraically closed, then when $n$ is even $C(Q)$ is a matrix algebra and $C^+(Q)$ is a product of two matrix algebras. (If, on the other hand, $n$ is odd, then $C^+(Q)$ is a matrix algebra and $C(Q)$ is a product of two matrix algebras.)

The invertible elements $s$ of $C(Q)$ (or of $C^+(Q)$) for which $sEs^{-1} = E$ form the Clifford group $G(Q)$ (or the special Clifford group $G^+(Q)$) of the quadratic form $Q$. The restriction of the transformation

$$x\mapsto sxs^{-1}\quad (x\in G(Q))$$ to the subspace $E$ defines a homomorphism $\def\phi{\varphi}\phi : G(Q)\to \def\O{ {\rm O}}\O(Q)$, where $\O(Q)$ is the orthogonal group of the quadratic form $Q$. The kernel $\def\Ker{ {\rm Ker}\;}\Ker \phi$ consists of the invertible elements of the algebra $Z$ and $(\Ker \phi)\cap G^+(Q) = k^*$. If $n$ is even, then $\phi(G(Q))=\O(G)$ and $\phi(G^+(Q))=\O^+(G)$ is a subgroup of index 2 in $\O(Q)$, which in the case when $K$ is not of characteristic 2, is the same as the special orthogonal group $\def\SO{ {\rm SO}}\SO(Q)$. If $n$ is odd, then

$$\phi(G(Q)) = \phi(G^+(Q)) = \SO(Q).$$ Let $\beta : C(Q) \to C(Q)$ be the anti-automorphism of $C(Q)$ induced by the anti-automorphism

$$x_1\otimes\cdots \otimes x_n \mapsto x_n\otimes\cdots \otimes x_1$$ of the tensor algebra $T(E)$. The group

$$\def\Spin{ {\rm Spin}}\Spin(Q) = \{s\in G^+(Q) : \beta(s) = s^{-1} \}$$ is called the spinor group of the quadratic form $Q$ (or of the Clifford algebra $C(Q)$).

The homomorphism $\phi: \Spin(Q) \to \O^+(Q) $ has kernel $\{\pm1\}$. If $K=\C$ or $K=\R$ and $Q$ is positive definite, then ${\rm Im}\;\phi : \O^+(Q) = \SO(Q)$ and $\Spin(Q)$ coincides with the classical spinor group.

Comments

The algebra $C^+(Q)$ generated by products of an even number of elements of the free $K$-module $E$ is also called the even Clifford algebra of the quadratic form $Q$. See also the articles Exterior algebra (or Grassmann algebra), and Cartan method of exterior forms for more details in the case $Q=0$.

References

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[Ki]	A.A. Kirillov, "Elements of the theory of representations", Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
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