Definition 1. - We call Clifford algebra of \(Q\), denoted \(C(Q)\), the quotient of the tensor algebra \(T(E)\) of the module \(E\) by the two-sided ideal, denoted \(I(Q)\), generated by the elements of the form \(x \otimes x - Q(x) . 1 \quad (x \in E)\). 🇫🇷 Définition 1. - On appelle algèbre de Clifford de \(Q\) et on note \(C(Q)\) l'algèbre quotient de l'algèbre tensorielle \(T(E)\) du module \(E\) par l'idéal bilatère (noté \(I(Q)\)) engendré par les éléments de la forme \(x \otimes x - Q(x) . 1 \quad (x \in E)\).

We will denote by \(\rho _{\tiny Q}\) (or simply \(\rho \) when there is no risk of confusion) the mapping of \(E\) into \(C(Q)\) composed of the canonical mapping of \(E\) into \(T(E)\) and of the canonical mapping \(\sigma \) of \(T(E)\) onto \(C(Q)\); the mapping \(\rho _{\tiny Q}\) is said to be canonical. 🇫🇷 Nous noterons \(\rho _{e}\) (ou simplement \(\rho \) quand aucune confusion n'est à craindre) l'application de E dans \(C(Q)\) composée de l'application canonique de \(E\) dans \(T(E)\) et de l'application canonique \(\sigma \) de \(T(E)\) sur \(C(Q)\); l'application \(\rho _{\tiny Q}\) est dite canonique .

Note that \(C(Q)\) is generated by \(\rho _{\tiny Q}(E)\), and that, for \(x \in E\), we have \[\rho (x)^{2} = Q(x) . 1 ; \tag{1}\] hence, replacing \(x\) by \(x + y\) (\(x, y\) in \(E\)): \[\rho (x) \rho (y) + \rho (y) \rho (x) = \Phi (x, y) . 1 \tag{2}\] 🇫🇷 Remarquons que \(C(Q)\) est engendrée par \(\rho _{\tiny Q}(E)\), et que, pour \(x \in E\), on a \[\rho (x)^{2} = Q(x) . 1 ; \tag{1}\] d'où, en remplaçant \(x\) par \(x + y\) (\(x, y\) dans \(E\)): \[\rho (x) \rho (y) + \rho (y) \rho (x) = \Phi (x, y) . 1 \tag{2}\]

Example: If \(E\) admits a base composed of a single element \(e\), \(T(E)\) is isomorphic to the polynomial algebra \(A[X]\), and \(C(Q)\) is a quadratic extension of \(A\), based on \((1, u)\), where \(u\) is the element \(u = \rho (e)\) and satisfies \(u^{2} = Q(e)\). 🇫🇷 Exemple. Si \(E\) admet une base composée d'un seul élément \(e\), \(T(E)\) est isomorphe à l'algèbre de polynômes \(A[X]\), et \(C(Q)\) est une extension quadratique de \(A\), ayant pour base \((1, u)\), où \(u\) est l'élément \(u = \rho (e)\) et vérifie \(u^{2} = Q(e)\).

We denote \(T^{h}\) the \(h\)-th tensor power \(\bigotimes \limits ^{h} E\) in \(T(E)\), and \(T^{+}\) (resp. \(T^{-}\)) being the sum of the \(T^{h}\) for even (resp. odd) number of \(h\). 🇫🇷 Notons \(T^{h}\) la puissance tensorielle \(h\)-ème \(\bigotimes \limits ^{h} E\) dans \(T(E)\), et soit \(T^{+}\) (resp. \(T^{-}\)) la somme des \(T^{h}\) pour \(h\) pair (resp. impair).

Since \(T(E)\) is a direct sum of \(T^{+}\) and \(T^{-}\), and \(I(Q)\) is generated by elements of \(T^{+}\), \(I(Q)\) is a direct sum of \(T^{+} \cap I(Q)\) and \(T^{-} \cap I(Q)\), and \(C(Q)\) is a direct sum of the two submodules \(C^{+}(Q) = \sigma (T^{+})\) and \(C^{-}(Q) = \sigma (T^{-})\) (also denoted \(C^{+}\) and \(C^{-}\)). The elements of \(C^{+}\) are called even (resp. odd). 🇫🇷 Comme \(T(E)\) est somme directe de \(T^{+}\) et \(T^{-}\), et \(I(Q)\) est engendré par des éléments de \(T^{+}\), \(I(Q)\) est somme directe de \(T^{+} \cap I(Q)\) et \(T^{-} \cap I(Q)\), et \(C(Q)\) est somme directe des deux sous-modules \(C^{+}(Q) = \sigma (T^{+})\) et \(C^{-}(Q) = \sigma (T^{-})\) (que l'on note aussi \(C^{+}\) et \(C^{-}\)). Les éléments de \(C^{+}\) seront dits pairs (resp. impairs).

We have the relations \[C^{+} C^{+} \subset C^{+}, \quad C^{+} C^{-} \subset C^{-}, \quad C^{-} C^{+} \subset C^{-}, \quad C^{-} C^{-} \subset C^{+}. \tag{3}\] In particular \(C^{+}\) is a subalgebra of \(C(Q)\). 🇫🇷 On a les relations \[C^{+} C^{+} \subset C^{+}, \quad C^{+} C^{-} \subset C^{-}, \quad C^{-} C^{+} \subset C^{-}, \quad C^{-} C^{-} \subset C^{+}. \tag{3}\] En particulier \(C^{+}\) est une sous-algèbre de \(C(Q)\).

Proposition 1. - Let \(f\) be a linear map of \(E\) in an algebra \(D\) over \(A\) such that \(f(x)^{2} = Q(x) . 1\) for all \(x \in E\). There is one and only one homomorphism \(\bar {f}\) of \(C(Q)\) into \(D\) such that \(f = \bar {f} \circ \rho _{\tiny Q}\). 🇫🇷 Proposition 1. - Soit \(f\) une application linéaire de \(E\) dans une algèbre \(D\) sur \(A\) telle que \(f(x)^{2} = Q(x) . 1\) pour tout \(x \in E\). Il existe un homomorphisme \(\bar {f}\) et un seul de \(C(Q)\) dans \(D\) tel que \(f = \bar {f} \circ \rho _{\tiny Q}\).

Prop. 1 shows that \(C(Q)\) is a solution to a problem of universal (mapping) property ( Ens., chap. IV, § 3, \(n^{\circ }\) 1 ). 🇫🇷 La prop. 1 exprime que \(C(Q)\) est solution d'un problème d'application universelle ( Ens., chap. IV, § 3, \(n^{\circ }\) 1 ).

Let us take in particular for \(D\) the opposite algebra of \(C(Q)\) and for \(f\) the mapping \(\rho \); prop. 1 implies that there is one and only one anti-automorphism \(\beta \) of \(C(Q)\) whose restriction to \(\rho (E)\) is the identity; it is called the main anti-automorphism of \(C(Q)\). It is clear that \(\beta ^{2} = 1\). 🇫🇷 Prenons en particulier pour \(D\) l'algèbre opposée de \(C(Q)\) et pour \(f\) l'application \(\rho \); la prop. 1 entraine qu'il existe un antiautomorphisme \(\beta \) et un seul de \(C(Q)\) dont la restriction à \(\rho (E)\) soit l'identité ; on l'appelle l'antiautomorphisme principal de \(C(Q)\). Il est clair que \(\beta ^{2} = 1\).

On the other hand, let \(Q'\) be a quadratic form over an \(A\)-module \(E'\), and \(f\) a linear map of \(E\) into \(E'\) such that \(Q' \circ f = Q\). We have \(\rho _{\tiny Q'}(f(x))^{2} = Q'(f(x)) . 1 = Q(x) . 1\), and consequently there is one and only one homomorphism \(C(f)\) of \(C(Q)\) into \(C(Q')\) such that \(C(f) \circ \rho _{\tiny Q} = \rho _{\tiny Q'} \circ f\). 🇫🇷 D'autre part, soient \(Q'\) une forme quadratique sur un \(A\)-module \(E'\), et \(f\) une application linéaire de \(E\) dans \(E'\) telle que \(Q' \circ f = Q\). On a \(\rho _{\tiny Q'}(f(x))^{2} = Q'(f(x)) . 1 = Q(x) . 1\), et par suite il existe un homomorphisme \(C(f)\) et un seul de \(C(Q)\) dans \(C(Q')\) tel que \(C(f) \circ \rho _{\tiny Q} = \rho _{\tiny Q'} \circ f\).

If \(f\) is the identity, \(C(f)\) is the identity ; if \(Q'\) is a quadratic form over an \(A\)-module \(E'\), and \(g\) a linear map of \(E'\) into \(E''\) such that \(Q'' \circ g = Q'\), we have \(C(g \circ f) = C(g) \circ C(f)\). When \(E'\) is a submodule of \(E\) and \(f\) is the canonical injection of \(E'\) into \(E\) (so that \(Q'\) is the restriction of \(Q\) to \(E'\)), we say that \(C(f)\) is the canonical homomorphism of \(C(Q')\) into \(C(Q)\). 🇫🇷 Si \(f\) est l'identité, \(C(f)\) est l'identité ; si \(Q'\) est une forme quadratique sur un \(A\)-module \(E'\), et \(g\) une application linéaire de \(E'\) dans \(E''\) telle que \(Q'' \circ g = Q'\), on a \(C(g \circ f) = C(g) \circ C(f)\). Lorsque \(E'\) est un sous-module de \(E\) et \(f\) l'injection canonique de \(E'\) dans \(E\) (de sorte que \(Q'\) est la restriction de \(Q\) à \(E'\)), on dit que \(C(f)\) est l'homomorphisme canonique de \(C(Q')\) dans \(C(Q)\).

Let us take in particular \(Q' = Q\) and for \(f\) the mapping \(x \to -x\); we see that there is an automorphism \(\alpha \) and only one of \(C(Q)\) such that \(\alpha \circ \rho = -\rho \); it is called the main automorphism of \(C(Q)\). It is clear that \(\alpha ^{2} = 1\), and that the restriction of \(\alpha \) to \(C^{+}\) (resp. \(C^{-}\)) is the identity (resp. the mapping \(u \to -u\)). 🇫🇷 Prenons en particulier \(Q' = Q\) et pour \(f\) l'application \(x \to -x\); on voit qu'il existe un automorphisme \(\alpha \) et un seul de \(C(Q)\) tel que \(\alpha \circ \rho = -\rho \); on l'appelle l'automorphisme principal de \(C(Q)\). Il est clair que \(\alpha ^{2} = 1\), et que la restriction de \(\alpha \) à \(C^{+}\) (resp. \(C^{-}\)) est l'identité (resp. l'application \(u \to -u\)).

Proposition 2. - Let \(A'\) be a commutative ring, \(\varphi \) a homomorphism of \(A\) into \(A'\), \(Q'\) the quadratic form over \(E' = A' \otimes _{ A } E\) induced from \(Q\) by extension of scalars (§ 3, \(n^{\circ }\) 4, prop. 3 ). There is one and only one isomorphism \(j\) of the algebra \(A' \otimes _{ A } C ( Q )\) onto \(C(Q')\) such that \(j (1 \otimes \rho _{\tiny Q}(x) )=\rho _{\tiny Q'}(1 \otimes x)\) for all \(x \in E\). 🇫🇷 Proposition 2. - Soient \(A'\) un anneau commutatif, \(\varphi \) un homomorphisme de \(A\) dans \(A'\), \(Q'\) la forme quadratique sur \(E'= A' \otimes _{ A } E\) déduite de \(Q\) par extension des scalaires (§ 3, \(n^{\circ }\) 4, prop. 3 ). Il existe un isomorphisme \(j\) et un seul de l'algèbre \(A' \otimes _{ A } C ( Q )\) sur \(C(Q')\) tel que \(j (1 \otimes \rho _{\tiny Q}(x) )=\rho _{\tiny Q'}(1 \otimes x)\) pour tout \(x \in E\).

In this section we will designate by \(e_{x}\) (\(x \in E\)) the linear mapping \(u \rightarrow x \otimes u\) of the tensor algebra \(T( E )\) into itself. 🇫🇷 Dans ce \(n^{\circ }\) nous désignerons par \(e_{x}\) (\(x \in E\)) l'application linéaire \(u \rightarrow x \otimes u\) de l'algèbre tensorielle \(T( E )\) dans elle-même.

Lemma 1. - Let \(f\) be an element of the dual \(E^{*}\) of \(E\). There exists one linear mapping \(i_{f}\) and only one from \(T ( E )\) into itself such that \[i_{f}(1)=0 \tag{4}\] \[i_{f} \circ e_{x} + e_{x} \circ i_{f} = f(x) . I \text {\quad for all \,} x \in E \tag{5}\] (where \(I\) denotes the identical mapping). The mapping \(f \rightarrow i_{f}\) from \(E^{*}\) into \(\mathscr {L}(T(E))\) is linear. We have \(i_{f}\left (T^{n}\right ) \subset T^{n-1},\left (i_{f}\right )^{2}=0,\) and \(i_{f} \circ i_{g}+i_{g} \circ i_{f}=0\) for \(f, g\) in \(E ^{*}\). The mapping \(i_{f}\) is null on the subalgebra of \(T ( E )\) generated by the kernel of \(f\). The ideal \(I(Q)\) is stable by \(i_{f}\); through quotient \(i_{f}\), we define a linear mapping (again \(i_{f}\)) of \(C(Q)\) into itself. 🇫🇷 Lemme 1. - Soit \(f\) un élément du dual \(E^{*}\) de \(E\). Il existe une application linéaire \(i_{f}\) et une seule de \(T ( E )\) dans elle-même telle que \[i_{f}(1)=0 \tag{4}\] \[i_{f} \circ e_{x} + e_{x} \circ i_{f} = f(x) . I \text {\quad pour tout \,} x \in E \tag{5}\] (où \(I\) désigne l'application identique). L'application \(f \rightarrow i_{f}\) de \(E^{*}\) dans \(\mathscr {L}(T(E))\) est linéaire. On a \(i_{f}\left (T^{n}\right ) \subset T^{n-1},\left (i_{f}\right )^{2}=0,\) et \(i_{f} \circ i_{g}+i_{g} \circ i_{f}=0\) pour \(f, g\) dans \(E ^{*}\). L'application \(i_{f}\) est nulle sur la sous-algèbre de \(T ( E )\) engendrée par le noyau de \(f\). L'idéal \(I(Q)\) est stable par \(i_{f}\); par passage au quotient \(i_{f}\), définit donc une application linéaire (notée encore \(i_{f}\) ) de \(C(Q)\) dans elle-mème.