FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 11 definitions, 1 theorem, 4 propositions, 3 lemmas, 1 corollary, 6 examples, 4 constructions, 3 remarks and 1 convention.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
The spin group of a quadratic vector space \(( V , q )\) over a field \(k\) is defined by means of its Clifford algebra \({ \mathsf { C } } ( q )\) . As a vector space this is just the exterior algebra \(\Lambda ( V )\) and it contains \(V\) . The algebra structure comes from \(q\) . For us its importance lies in the action of the algebra conjugation. Firstly, conjugation with a non-isotropic vector \(u \in V\) preserves \(V\) and, secondly, in \(V\) this gives the hyperplane reflection \(\sigma _ { u }\) up to a sign. This property for \(u\) can be extended to a ”large” subgroup of \({ \mathsf { C } } ( q )\) . To do this properly, one makes use of a canonical splitting of \({ \mathsf { C } } ( q )\) into even and odd elements. The even elements form a subalgebra, the even Clifford algebra. The conjugation with \(u\) receives a sign according to whether \(u\) is odd or even. The Clifford group consists of the invertible elements \(u\) of the Clifford algebra which have the property that twisted conjugation with \(u\) preserves \(V\) .
Definition 13.1.1
Definition 13.1.1. The Clifford algebra \({ \mathsf { C } } ( q )\) is defined as the quotient of the tensor algebra \(T ( V )\) on \(V\) by the two-sided ideal \(I ( q )\) generated by \(x \otimes x - q ( x ) \cdot { \bf 1 }\) , \(x \in V\) , and with multiplication induced by the multiplication in \(T ( V )\) . We denote the multiplication in the Clifford algebra by a dot.
Definition / construction
The quotient construction of the Clifford algebra carries a canonical linear map \(\iota:V\to\mathsf C(q)\) with \(\iota(x)^2=q(x)1\); this map is injective, and \(V\) is hence identified with its image in \(\mathsf C(q)\).
Definition / construction
Equivalently, the Clifford algebra is characterized by the universal property that every linear map \(u:V\to A\) into a unital algebra with \(u(x)^2=q(x)1\) extends uniquely to an algebra homomorphism \(\mathsf C(q)\to A\).
Definition / construction
The Clifford algebra has a canonical \(\mathbf Z/2\mathbf Z\)-grading \(\mathsf C(q)=\mathsf C^0(q)\oplus\mathsf C^1(q)\). The grading involution \(\alpha\) is \(+1\) on the even Clifford algebra and \(-1\) on the odd part.
Construction / comparison
The tensor-degree filtration descends to \(\mathsf C(q)\) and its associated graded algebra is canonically isomorphic to the exterior algebra \(\Lambda V\).
Lemma 13.1.2
Lemma 13.1.2. There is a canonical vector space isomorphism \(\Lambda ( V ) { \stackrel { \cong } { \to } } \mathsf C ( q )\) . In particular, dim?? C(??) = 2dim ??.
Example / comparison
Hamiltonian quaternions revisited. Consider the standard diagonal form \(x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 }\) on \(\mathbb { R } ^ { 3 }\) . Its Clifford algebra has as a vector space basis \(\{ 1 , e _ { 1 } , e _ { 2 } , e _ { 3 } , i = e _ { 2 } e _ { 3 } , j = e _ { 3 } e _ { 1 } , k = e _ { 2 } e _ { 1 } , e _ { 1 } e _ { 2 } e _ { 3 } \}\) , where \(\{ e _ { 1 } , e _ { 2 } , e _ { 3 } \}\) is the standard basis of \(\mathbb { R } ^ { 3 }\) . Since for \(i , j = 1 , 2 , 3\) we have \(e _ { i } e _ { j } = - e _ { j } e _ { i }\) for \(i \neq j\) , and \(e _ { i } ^ { 2 } = 1\) , it follows that \(\pmb { i } \cdot \pmb { j } = e _ { 2 } e _ { 3 } e _ { 3 } e _ { 1 } = e _ { 2 } e _ { 1 } = \pmb { k }\) and similarly for the cyclic permutations of \(i , j , k\) . Moreover, \(\pmb { i } ^ { 2 } = \pmb { j } ^ { 2 } = \pmb { k } ^ { 2 } = - 1\) . So the vector space with basis \(\{ 1 , \pmb { i } , \pmb { j } , \pmb { k } \}\) generates the algebra of the Hamilton quaternions \(\mathbb { H }\) which we already encountered in \(\ S\) 5.4.A. This is the even Clifford algebra for the dot product form on \(\mathbb { R } ^ { 3 }\) . As a special case of the second example below we shall see that \(\mathbb { H }\) also figures as the full Clifford algebra of the form \(- x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 }\) on \(\mathbb { R } ^ { 2 }\) . Recall also that we explained in Section 5.4 how to associate a ternary quadratic form to any quaternion algebra \(\begin{array} { r } { D = \left( \frac { a , b } { k } \right) } \end{array}\) , namely the norm form restricted to the subspace \({ \cal D } ^ { ( 0 ) }\) of the quaternions of trace \(0\) . For the Hamilton quaternions this is the dot product form on \(\mathbb { R } ^ { 3 }\) where \(\mathbb { R } ^ { 3 }\) is identified with the real space with basis \(\{ \dot { \pmb { i } } , \dot { \pmb { j } } , \pmb { k } \}\) , the so-called pure quaternions. In general the even Clifford algebra of the norm form on \({ \cal D } ^ { ( 0 ) }\) is precisely \(D\) .
Example / comparison
Real Clifford algebras (cf. [68, 179]) By Sylvester’s law 8.1.3 nondegenerate real forms are classified by their signature \(( r , s )\) with representing diagonal quadratic forms \(\textstyle \sum _ { i = 1 } ^ { r } x _ { i } ^ { 2 } - \sum _ { i = r + 1 } ^ { n } x _ { i } ^ { 2 }\) , \(n = r + s\) . The corresponding Clifford algebra \({ \mathsf C } _ { r , s }\) is then generated by the standard basis \(\{ \pmb { e } _ { 1 } , \ldots , \pmb { e } _ { r + s } \}\) subject to the relations \[ \begin{array} { r } { e _ { i } ^ { 2 } = \left\{ \begin{array} { l l } { 1 } & { \mathrm { ~ i f ~ } i \leq r } \\ { - 1 } & { \mathrm { ~ i f ~ } i > r } \end{array} \right. \quad \mathrm { a n d } \quad e _ { i } \cdot e _ { j } + e _ { i } \cdot e _ { j } = 0 \mathrm { ~ i f ~ } i \neq j . } \end{array} \] For example, \({ \mathrm { C } } _ { 0 , 0 } = \mathbb { R }\) , \(\mathrm { C } _ { 1 , 0 } = \mathbb { R } \oplus \mathbb { R }\) , while \({ \mathsf C } _ { 0 , 1 } = \mathbb C\) . We shall now show that the isomorphism class of the Clifford algebra of a real quadratic form does not determine the signature. To do so we make basic use of quadratic forms over \(\mathbb { F } _ { 2 }\) as suggested by the signs that appear in the above relations. We start by considering the group ring \(\mathbb { R } [ G ]\) , \(G = \Pi ^ { n } C _ { 2 }\) . where \(C _ { 2 }\) is the cyclic group of order 2. Viewing \(C _ { 2 }\) as the field \(\mathbb { F } _ { 2 }\) and switching to additive notation and identifying groupelements of \(G\) with vectors \(\pmb { x } \in \mathbb { F } _ { 2 } ^ { n }\) , this group ring becomes the real vector space \(V : = \oplus _ { x \in G } \mathbb { R } \cdot e _ { x }\) with algebra structure induced by the additive group structure, i.e., the product \(^ *\) is given by \(e _ { x } * e _ { y } = e _ { x + y }\) . As vector spaces \(V\) and \({ \mathsf C } _ { r , s }\) are isomorphic: identify \(e _ { x } \in \ V\) with the element \(\pmb { e } _ { i _ { 1 } } \cdot \cdot \cdot \pmb { e } _ { i _ { k } } \in \mathbb { C } _ { r , s }\) where \(_ x\) has \(i\) -th coordinate 1 precisely if \(i \in \{ i _ { 1 } , \ldots , i _ { k } \}\) . In particular, if \(\{ e _ { 1 } , \ldots , e _ { n } \}\) is the standard basis of \(\mathbb { F } _ { 2 }\) , \[ e _ { e _ { k } } { \mathrm { ~ c o r r e s p o n d s ~ t o ~ } } e _ { k } . \] The possibly non-commutative Clifford algebra \({ \mathsf C } _ { r , s }\) comes from the twisted group action \[ e _ { x } \cdot e _ { y } = ( - 1 ) ^ { \beta ^ { r , s } ( x , y ) } e _ { x } * e _ { y } , \] where \(\beta ^ { r , s }\) is the \(\mathbb { F } _ { 2 }\) -valued (in general non-symmetric) bilinear form on \(\mathbb { F } _ { 2 } ^ { n }\) , which in coordinates with respect to the standard basis \(\{ e _ { 1 } , \ldots , e _ { n } \}\) is given by \(x A ^ { r , s } { \bf y } ^ { \scriptscriptstyle 1 }\) , where \(A\) is the upper-triangular matrix \[ A ^ { r , s } = \left( \begin{array} { l l l l } { 0 } & { 1 } & { \cdots } & { 1 } \\ { \vdots } & { 0 } & { \ddots } & { \vdots } \\ { \vdots } & { \ldots } & { \ddots } & { 1 } \\ { 0 } & { \cdots } & { \ldots } & { 0 } \end{array} \right) + \left( \begin{array} { l } { 0 _ { r } } \\ { \phantom { - } } \\ { 0 _ { s \times r } } \end{array} \right| \left( \begin{array} { l } { 0 _ { r \times s } } \\ { \phantom { - } } \\ { \phantom { - } } \\ { 0 _ { s \times r } } \end{array} \right) \] By the correspondence (13.4) the relations \(( - 1 ) ^ { A _ { i j } ^ { r s } } = - 1\) for \(j > i\) and \(( - 1 ) ^ { A _ { i j } ^ { r s } } = 1\) for \(j < i\) imply that the desired anti-commutation relations hold, while \(( - 1 ) ^ { A _ { i i } ^ { r s } } = \mathbf { 1 }\) for \(i \leq r\) and \(( - 1 ) ^ { A _ { i i } ^ { r s } } = - { \bf 1 }\) for \(i > r\) , proving the remaining two relations. Observe that as in equation (1.1) in Chapter 1, a bilinear form defines a unique quadratic form on \(\mathbb { F } _ { 2 } ^ { n }\) . Conversely, given a basis for a \(\mathbb { F } _ { 2 }\) -vector space, a quadratic form \(q\) determines an upper-triangular bilinear form \(\beta\) which can be used to define a twisted product (13.5). The resulting associative algebra is denoted \(C _ { \mathbb { R } } [ \mathbb { F } _ { 2 } ^ { n } , q ]\) . This algebra has a special structure: either the generators \(\mathbf { \boldsymbol { e } } _ { i }\) and \(e _ { j }\) for \(i \neq j\) anticommute (namely if \(\beta _ { i j } = 1\) for \(i < j\) since then \(\pmb { e } _ { i } \cdot \pmb { e } _ { j } = - \pmb { e } _ { i } * \pmb { e } _ { j }\) and \(\pmb { e } _ { j } \cdot \pmb { e } _ { i } = \pmb { e } _ { i } * \pmb { e } _ { j }\) ), or they commute (if \(\beta _ { i j } = 0\) for \(i < j\) , since then \(\pmb { e } _ { i } \cdot \pmb { e } _ { j } = \pmb { e } _ { j } \cdot \pmb { e } _ { i } = \pmb { e } _ { i } * \pmb { e } _ { j }\) ). It follows that isometric forms \(q\) give isomorphic algebras. So the classification of the real Clifford algebras follows from the classification of the quadratic forms on the vector spaces \(\mathbb { F } _ { 2 } ^ { n }\) as given in Example 8.2.7. The nondegenerate forms live on even-dimensional spaces and come in two types depending on the Arf invariant. The ones with zero Arf invariant are the form \(q _ { 0 } = \mathbb { Q } ^ { m } U\) , \(n = 2 m\) , the ones with Arf invariant 1 are of the form \(q _ { 1 } = \Phi ^ { m - 1 } U \odot \Phi ^ { 2 } \langle 1 \rangle\) , \(n = 2 m\) . On odd dimensional spaces, say of dimension \(n = 2 m + 1\) , one has three types with \(\mathrm { d i m } \operatorname { r a d } ( q ) \leq 1\) , namely \(q _ { 0 } ^ { \prime } = \Phi ^ { m } U \circledast 0\) , \(q _ { 1 } ^ { \prime } = \oplus ^ { m - 1 } U \oplus \oplus ^ { 2 } \langle 1 \rangle \oplus 0\) and \(q _ { 2 } = \Phi ^ { m } U \odot\) [1]. We shall show that these types indeed give all the real Clifford algebras \({ \mathsf C } _ { r , s }\) . Note however, to identify \(\mathsf { C } _ { \mathbb { R } } [ \mathbb { F } _ { 2 } ^ { n } , q ]\) with some \({ \mathsf C } _ { r , s }\) , one has to transform each of the 5 types of quadratic forms \(q ( { \pmb x } )\) into the desired shape \(\pmb { x } A ^ { r , s } \pmb { x } ^ { \scriptscriptstyle 1 }\) , where \(A ^ { r , s }\) is as in formula (13.6). This poses no problem for \(\mathsf { C } _ { \mathbb { R } } \left[ \mathbb { F } _ { 2 } ^ { 2 } , q _ { i } \right]\) , \(i \in \{ 0 , 1 \}\) : these are the 4-dimensional Clifford algebras \(\mathrm { C _ { 2 , 0 } , C _ { 0 , 2 } }\) with generators \(\mathbf { \boldsymbol { e } } _ { 1 } , \mathbf { \boldsymbol { e } } _ { 2 }\) subject to \[ \begin{array} { r l } { \pmb { e } _ { 1 } ^ { 2 } = \pmb { e } _ { 2 } ^ { 2 } = 1 , \quad } & { \pmb { e } _ { 1 } \cdot \pmb { e } _ { 2 } = - \pmb { e } _ { 2 } \cdot \pmb { e } _ { 1 } \mathrm { ~ f o r ~ } q _ { 0 } } \\ { \pmb { e } _ { 1 } ^ { 2 } = \pmb { e } _ { 2 } ^ { 2 } = - 1 , \quad } & { \pmb { e } _ { 1 } \cdot \pmb { e } _ { 2 } = - \pmb { e } _ { 2 } \cdot \pmb { e } _ { 1 } \mathrm { ~ f o r ~ } q _ { 1 } . } \end{array} \] Next, since \(x _ { 1 } x _ { 2 } + x _ { 2 } ^ { 2 }\) has Arf invariant \(0\) it is isometric to \(x _ { 1 } x _ { 2 }\) (in fact \(( x _ { 1 } , x _ { 2 } ) \mapsto\) \(\left( x _ { 1 } + x _ { 2 } , x _ { 2 } \right)\) is an isometry). The first corresponds to \(\mathrm { C _ { 1 , 1 } }\) and the second to \(\mathrm { C _ { 2 , 0 } }\) which indeed shows that the Clifford algebra does not determine the signature. To continue, one checks that \(\mathsf C _ { 2 m , 0 } = \mathsf C _ { \mathbb R } [ \mathbb F _ { 2 } ^ { 2 m } , q _ { 0 } ]\) since \(A ^ { 2 m , 0 }\) corresponds to \(q _ { 0 }\) . Likewise \({ \mathsf C } _ { 0 , 2 m } = { \mathsf C } _ { \mathbb R } [ \oplus ^ { 2 m } \mathbb { F } _ { 2 } , q _ { 1 } ]\) . The forms \(q _ { 0 } ^ { \prime } , q _ { 1 } ^ { \prime }\) correspond to \(\mathrm { C } _ { 2 m + 1 , 0 }\) , respectively \(\mathsf { C } _ { 0 , 2 m + 1 }\) and \(q _ { 2 }\) corresponds to \(\mathsf { C } _ { m , m + 1 }\) . The resulting 2 types of \(2 ^ { 2 m }\) - dimensional real Clifford algebras and 3 types of \(2 ^ { 2 m + 1 }\) -dimensional real Clifford algebras fit in a remarkable periodic table of period 8 displayed as [137, Table II in §I.4]. This table makes use of an identification of these Clifford algebras with familiar matrix algebras as follows. The relations (13.7) show that the assignments \(\pmb { e } _ { 1 } \mapsto \left( \begin{array} { c c } { { 1 } } & { { 0 } } \\ { { 0 } } & { { - 1 } } \end{array} \right)\) −1 , ??2 ↦→ \(\pmb { e } _ { 2 } \mapsto \left( \begin{array} { c c } { { 0 } } & { { 1 } } \\ { { 1 } } & { { 0 } } \end{array} \right)\) give an algebra-isomorphism of \(C _ { 2 , 0 }\) with \(\mathsf { M a t } ( 2 , \mathbb { R } )\) . Similarly, the map induced by \(\pmb { e } _ { 1 } \mapsto \pmb { i }\) , \(\pmb { e } _ { 2 } \mapsto \pmb { j }\) (13.8) identifies the Clifford algebra \(\mathsf { C } _ { 0 , 2 }\) with the Hamiltonian quaternions. It follows that \(\mathsf C _ { 2 m , 0 } = \mathsf C _ { \mathbb R } [ \mathbb F _ { 2 } ^ { 2 m } , q _ { 0 } ]\) leads to \(\otimes ^ { m } \mathsf { M a t } ( 2 , \mathbb { R } ) = \mathsf { M a t } ( 2 ^ { m } , \mathbb { R } )\) and \(\mathsf { C } _ { 0 , 2 m } = \mathsf { C } _ { \mathbb { R } } \left[ \oplus ^ { 2 m } \mathbb { F } _ { 2 } , q _ { 1 } \right]\) to \(\mathsf { M a t } ( 2 , \mathbb { R } ) \otimes \mathbb { H } \simeq\) \({ \mathsf { M a t } } ( 2 ^ { m - 1 } , \mathbb { H } )\) . Finally, the Clifford algebras \(\mathsf { C } _ { 2 m + 1 , 0 }\) , \(\mathrm { C } _ { 0 , 2 m + 1 }\) and \(\mathsf { C } _ { m , m + 1 }\) are isomorphic to \(\mathsf { M a t } ( 2 ^ { m } , \mathbb { R } ) + \mathsf { M a t } ( 2 ^ { m } , \mathbb { R } )\) , \(\mathsf { M a t } ( 2 ^ { m - 1 } , \mathbb { H } ) + \mathsf { M a t } ( 2 ^ { m - 1 } , \mathbb { H } )\) , respectively \({ \mathsf { M a t } } ( 2 ^ { m } , \mathbb { C } )\) .
Proposition 13.1.5
Proposition 13.1.5. Suppose that \(\operatorname { c h a r } ( k ) = 2\) . Let \(q\) be a non-degenerate quadratic form, and let \(\{ e _ { 1 } , \ldots , e _ { 2 n } \}\) be a symplectic basis such that \(\begin{array} { r } { q = \sum _ { j = 1 } ^ { n } x _ { i } x _ { n + i } { + } \sum _ { j = 1 } ^ { 2 n } a _ { j } x _ { j } ^ { 2 } } \end{array}\) . Setting \(\pmb { z } = e _ { 1 } e _ { n + 1 } + \cdot \cdot \cdot + e _ { n } e _ { 2 n } \in \mathsf { C } ^ { 0 } ( q )\) , the center of \({ \mathsf { C } } ( q )\) is \(k\) and the center of \({ \mathsf { C } } ^ { 0 } ( q )\) equals \(k + k z\) . Moreover, \(z\) satisfies the relation \({ \pmb z } ^ { 2 } + { \pmb z } \equiv { \pmb a } ( q )\) mod \(\wp ( k )\) , where we recall that \({ \mathfrak { a } } ( q )\) is the Arf invariant of \(q\) .
Definition / construction
The Clifford group consists of the invertible Clifford-algebra elements whose twisted adjoint action preserves \(V\).
Example / comparison
3.2). The involution \(\mathbf { \Delta } \mathbf { x } \mapsto \mathbf { \Delta } \mathbf { x } ^ { * }\) sending \(\pmb { x } = \pmb { x } _ { 0 } +\) \(x _ { 1 } \pmb { i } \pmb { x } _ { 2 } \pmb { j } \pmb { + } x _ { 3 } \pmb { k }\) to \(\pmb { x } ^ { * } = x _ { 0 } - x _ { 1 } \pmb { i } - x _ { 2 } \pmb { j } - x _ { 3 } \pmb { k }\) (the usual extension of complex conjugation to quaternions) can be used to define the norm \(\mathsf { N m } ( \pmb { x } ) = \pmb { x } \pmb { x } ^ { * } = \pmb { x } _ { 0 } ^ { 2 } + \pmb { x } _ { 1 } ^ { 1 } + \pmb { x } _ { 2 } ^ { 2 } + \pmb { x } _ { 3 } ^ { 2 } \in \mathbb { R }\) of a quaternion \(_ x\) . A non-zero element \(\pmb { x } \in \mathbb { H }\) has \(\mathsf { N m } ( \pmb { x } ) ^ { - 1 } \pmb { x }\) as its inverse. It is easy to verify that conjugating by \(_ x\) preserves the span of \(e _ { 1 } , e _ { 2 } , e _ { 3 }\) . So \(\mathbb { H } - \{ 0 \}\) is the special Clifford group. Now identify the pure quaternions, i.e. those with \(x _ { 0 } = 0\) , with (another copy of) \(\mathbb { R } ^ { 3 }\) . These are preserved under conjugation with a non-zero quaternion, as one easily verifies. This gives a representation of the special Clifford group on \(\mathbb { R } ^ { 3 }\) by means of orthogonal transformations. In fact, one can show that it gives a surjection onto the rotation group SO(3) with kernel \(\mathbb { R } ^ { \times }\) , whence a finite surjective homomorphism \[ \mathrm { S p i n ( 3 ) } = \{ { \pmb x } \in \mathbb { H } \mid \mathsf { N m } ( { \pmb x } ) = 1 \} \mathrm { S O ( 3 ) } . \] The left-hand side is the classical spin group. The map turns out to have degree 2 and gives the universal cover of SO(3). See also Remark 13.3.6.3.
Example / comparison
Let us consider the Lorentzian type inner product space \(\{ e _ { 1 } , e _ { 2 } , e _ { 3 } \}\) and quadratic form given by \(q ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - x _ { 3 } ^ { 2 }\) \(\mathbb { R } ^ { 3 }\) with standard basis . T \({ \mathsf { C } } ^ { 0 } ( q )\) is spanned by \(\boldsymbol { j } = e _ { 1 } e _ { 3 }\) \(\pmb { k } = e _ { 2 } e _ { 1 }\) , but now \(\pmb { i } ^ { 2 } = \pmb { j } ^ { 2 } = 1\) \(\pmb { k } ^ { 2 } = - 1\) , \({ \pmb i } { \pmb j } = - { \pmb j } { \pmb i } = - { \pmb k }\) and similarly for cyclic permutations of \(i , j , k\) . This algebra has a representation as a matrix algebra, where \(x + i y + j z + k w\) corresponds to the matrix \(\begin{array} { r l } { \left( x + y \right. } & { { } z - u } \\ { \left. z + u \right. } & { { } x - y } \end{array}\) The twisted adjoint is not a faithful representation. Its kernel is going to play a central role in Section 13.3.
Proposition 13.2.2
Proposition 13.2.2. The kernel of the homomorphism \(\operatorname { C l i f } ( q ) \to \operatorname { A u t } ( V )\) given by \(u \mapsto \widetilde { \mathrm { A d } } _ { u } \vert V\) is equal to \(k ^ { \times } \cdot { \bf 1 }\) .
Lemma 13.2.3
Lemma 13.2.3. The two involutions \(\alpha\) and \(^ *\) preserve the Clifford group \(\operatorname { C l i f } ( q )\)
Example / comparison
Example 13.2.4. For Examples 13.2.1 we find that \(1 ^ { * } = 1\) and \(\mathbf { i } ^ { * } = - \mathbf { i }\) , \(\pmb { j } ^ { * } = - \pmb { j }\) and \(k ^ { * } = - k\) . This shows that the notation is consistent with the notation for the classical Hamiltonians. For the latter we have seen that \({ \pmb x } \cdot { \pmb x } ^ { * } \in \mathbb { R }\) . This is also true for the second example: we find that \(\pmb { x } \cdot \pmb { x } ^ { * } = x ^ { 2 } - y ^ { 2 } - z ^ { 2 } + w ^ { 2 }\) . However, for mixed degree elements in a Clifford algebra this need not be the case as is shown by the example \({ \pmb x } = e _ { 1 } + e _ { 1 } e _ { 2 }\) , where \({ \pmb x } \cdot { \pmb x } ^ { * } = ( e _ { 1 } + e _ { 1 } e _ { 2 } ) ( e _ { 1 } + e _ { 2 } e _ { 1 } ) = 2 - 2 e _ { 2 }\) .
Proposition 13.3.1
Proposition 13.3.1. The spinor norm is a homomorphism from \(\operatorname { C l i f } ( q )\) to \(k ^ { \times }\) .
Remark 13.3.2
Remark 13.3.2. For non-isotropic vectors \(v \in V\) , \(\mathrm { N m } _ { \mathrm { s p i n } } ( v ) = - q ( v )\) .
Corollary 13.3.3
Corollary 13.3.3. 1. For \(u \in \operatorname { C l i f } ( q )\) the induced map \(\widetilde { \mathrm { A d } _ { u } }\) on \(V\) is an orthogonal transformation, and if \(u \in \mathrm { C l i f } ^ { 0 } ( q )\) , it is a rotation. 2. \(u \in \operatorname { C l i f } ( q )\) can be written as a product \(u = v _ { 1 } \cdot \cdot \cdot v _ { r }\) with non-isotropic \(v _ { j } \in V\) , \(j = 1 , \dotsc , r\) . Consequently \[ \mathsf { N m } _ { \mathrm { s p i n } } ( u ) = u \cdot \alpha ( u ^ { * } ) = ( - 1 ) ^ { r } q ( v _ { 1 } ) \cdot \cdot \cdot q ( v _ { r } ) \in k ^ { \times } . \] If \(u \in \mathrm { C l i f } ^ { 0 } ( q )\) , then \(r\) can be taken even. 3. Ad induces isomorphisms \(\mathsf { C l i f } ( q ) / k ^ { \times } \overset { \simeq } { \to } \mathsf { O } ( q )\) and \(\mathrm { C l i f } ^ { 0 } ( q ) / k ^ { \times } \overset { \simeq } { \to } \mathrm { S O } ( q ) .\)
Definition 13.3.4
Definition 13.3.4. The (s)pin group is the group \[ \begin{array} { r c l } { { \mathsf { P i n } ( q ) } } & { { = } } & { { \mathrm { k e r } \left( \mathsf { N m } _ { \mathsf { s p i n } } : \mathsf { C l i f } ( q ) \to k ^ { \times } \right) , \mathrm { ~ r e s p e c t i v e l y , } } } \\ { { \mathsf { S p i n } ( q ) } } & { { = } } & { { \mathrm { k e r } \left( \mathsf { N m } _ { \mathsf { s p i n } } : \mathsf { C l i f } ^ { 0 } ( q ) \to k ^ { \times } \right) . } } \end{array} \] that is, the subgroup of those \(u \in \mathrm { C l i f } ( q )\) , respectively \(u \in \mathsf { C l i f } ^ { 0 } ( q )\) , for which \(u \alpha ( u ^ { * } ) = 1\) .
Definition / construction
The spinor norm of an orthogonal transformation written as a product of reflections \(\varphi=\sigma_{v_1}\cdots\sigma_{v_r}\) is the square class \(q(v_1)\cdots q(v_r)\in k^\times/(k^\times)^2\); this is independent of the chosen reflection factorization.
Construction / comparison
The scalar elements of the pin or spin group are exactly \(\{\pm1\}\).
Theorem 13.3.5
Theorem 13.3.5. The pin and spin groups map two-to-one onto normal subgroups \(O^+(q)\triangleleft O(q)\) and \(SO^+(q)\triangleleft SO(q)\), the reduced orthogonal groups. Equivalently, \(SO^+(q)\) consists of products of an even number of reflections whose represented values have square product.
Definition / construction / notation
The image of the spinor norm on the rotation group is a subgroup of \(\mathsf { D } ( k )\) which might be a proper subgroup as shown in Remark 13.3.6.1 below. We introduce a shorthand notation for this group: \[ \mathsf { S } _ { V } = \mathrm { I m } ( \mathsf { N m } _ { \mathrm { s p i n } } : \mathsf { S O } ( q ) \to \mathsf { D } ( k ) ) . \]
Remark 13.3.6
Remark 13.3.6. 1. For \(k = \mathbb { R }\) the group \(\mathsf { D } ( k )\) is the cyclic group \(C _ { 2 }\) represented by the real numbers \(\{ 1 , - 1 \}\) . Often we use these representatives instead of their classes. Note that if \(q\) is positive definite, the spinor norm is always 1 and so \(\mathsf { N m } _ { \mathrm { s p i n } }\) is not surjective in this case. In particular, \({ \mathsf { S O } } ^ { + } ( q ) = { \mathsf { S O } } ( q )\) and \(0 ^ { + } ( q ) = 0 ( q )\) . 2. Suppose \(k\) is a topological field. Then the orthogonal groups \({ \mathrm { O } } ( q )\) and SO(??) are in a natural way topological groups. The preceding constructions give the Clifford group \(\operatorname { C l i f } ( q )\) and the spin group \(\mathrm { S p i n } ( q )\) the structure of a topological group. The adjoint representation as well as the spinor norm are continuous. So all the maps in the preceding diagram are continuous. This holds for instance for the fields \(\mathbb { R , C }\) with the usual topology, or for \(\mathbb { Q } _ { p }\) endowed with the \(p\) -adic topology.
Convention / notation
For a real quadratic form of signature \((r,s)\) the orthogonal, special orthogonal, and spin groups are denoted \(O(r,s)\), \(SO(r,s)\), and \(Spin(r,s)\); in the positive-definite case one writes \(O(n)\) and \(SO(n)\).
Proposition 13.3.7
Proposition 13.3.7 (Topological structure of orthogonal groups). 1. SO(??), \(n \geq 2\) . is connected, but not simply connected: • SO(2) is the circle group and thus \(\pi _ { 1 } ( { \mathsf { S O } } ( 2 ) ) = \mathbb { Z }\) ; • for \(n \geq 3\) , one has \(\pi _ { 1 } ( \operatorname { S O } ( n ) ) = \mathbb { Z } / 2 \mathbb { Z }\) and the universal cover of \({ \mathsf { S O } } ( n )\) is the group \(\mathrm { S p i n } ( n )\) . 2. The groups \(\mathsf { S O } ( r , s )\) have two connected components in case \(r , s ~ \geq ~ 1\) . The component of the identity is \({ \mathsf { S O } } ^ { + } ( r , s )\) . • \({ \mathsf { S O } } ( 1 , s )\) , \(s ~ \geq ~ 1\) has fundamental group \(\mathbb { Z } / 2 \mathbb { Z }\) and the universal cover is \(\mathsf { S p i n } ( r , s )\) ; • \(\mathsf { S O } ( r , s )\) , \(r , s \ge 2\) has fundamental group \(\mathbb { Z } / 2 \mathbb { Z } \times \mathbb { Z } / 2 \mathbb { Z }\) and the universal cover is an unramified double cover of \(\mathsf { S p i n } ( r , s )\) .
Definition / construction / notation
Analogous to Definition 13.3.4 we set \[ \begin{array} { r } { \mathsf { O } ^ { + } ( L ) = \ker ( \mathsf { N m } _ { \mathsf { s p i n } } : \mathsf { O } ( L , b ) \to \mathsf { D } ( Q ( R ) ) ) } \\ { \mathsf { S O } ^ { + } ( L ) = \ker ( \mathsf { N m } _ { \mathsf { s p i n } } : \mathsf { S O } ( L ) \to \mathsf { D } ( Q ( R ) ) ) . } \end{array} \]
Example / comparison
Example 13.4.1. For \(p\)-adic lattices, the examples identify common direct-summand configurations whose orthogonal or special-orthogonal groups have spinor norm containing all \(p\)-adic unit square classes.
Construction / comparison
An isometry of ambient quadratic spaces carrying a lattice \(L\) to \(L\prime\) induces by conjugation a rotation-preserving isomorphism \(O(L)\simeq O(L\prime)\), and corresponding isometries have the same spinor norm.
Lemma 13.4.2
Lemma 13.4.2. \(\mathrm { S } _ { L }\) , the image of \(\mathsf { N m } _ { \mathrm { s p i n } } : \mathsf { S O } ( L ) \to \mathsf { D } ( Q ( R ) )\) , is an isometry invariant.
Remark 13.4.3
Remark 13.4.3. Since for integral lattices the spinor norm takes values in \(\mathsf { D } ( \mathbb { Q } ) =\) \(\mathbb { Q } ^ { \times } / ( \mathbb { Q } ^ { \times } ) ^ { 2 }\) , this spinor norm is a more refined invariant than the real spinor norm which, as we have explained above, only takes values 1 or \(^ { - 1 }\) .
Definition / construction / notation
To extend the definition of the \(\mathrm { ( s ) p i n }\) group itself to the setting of lattices, we define \[ \begin{array} { r l } & { \mathrm { \ } \mathsf { C } ( L ) : = \mathrm { s u b r i n g ~ o f ~ } \mathsf { C } ( q ) \mathrm { ~ g e n e r a t e d ~ b y ~ } 1 \mathrm { ~ a n d ~ t h e ~ e l e m e n t s ~ } x \in L , } \\ & { \mathrm { \ } \mathsf { C } ^ { 0 } ( L ) = \mathsf { C } ( L ) \cap \mathsf { C } ^ { 0 } ( q ) , } \\ & { \mathrm { \ } \mathsf { P i n } ( L ) = \mathsf { C } ( L ) \cap \mathsf { P i n } ( q ) , } \\ & { \mathrm { \ } \mathsf { S p i n } ( L ) = \mathsf { C } ( L ) \cap \mathsf { S p i n } ( q ) . } \end{array} \]
Construction / comparison
For an integral lattice, the adjoint map \(Spin(L)\to SO^+(L)\), and likewise the twisted-adjoint map from \(Pin(L)\), need not be surjective.