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Chapter 19

The Clifford Algebra in the Theory of Algebras, Quadratic Forms, and Classical Groups

Alexander Hahn1

ABSTRACT This article is an expanded version of my plenary lecture for the conference. It was the aim of the lecture to introduce the participants of the conference – their diverse realms of expertise ranged from , to computer science, to pure – to the algebraic matters of the title above. The basic texts listed in the references – note that this listing is by no means complete – serve as illustration of the rich and persistent interest in these topics and provide a reader with an opportunity to explore them in detail.

Keywords: Clifford algebra, , classical , involutions, Clif- ford modules

1 Basic quadratic forms

We will set the stage in very general terms. Let R be a commutative (and associative) with 1. A quadratic form in n variables over R is a homo- geneous polynomial of degree 2

p(X1,...,Xn)= X ai,j XiXj 1≤i,j≤n with coefficients ai,j in R. Two quadratic forms are equivalent if there is an invertible linear substitution of variables that transforms the one into the

1Many thanks go to Rafa l Ab lamowicz and John Ryan for their wonderful organiza- tional effort in bringing together in stimulating conversation scholars from all disciplines of the multifaceted reality that is the Clifford algebra. I wish to dedicate this article to the memory of one of them: Many of us will miss the dynamic mathematical and personal presence of Pertti Lounesto. AMS∗ Subject Classification: 11E81, 11E88, 16D70, 20G15. 308 Alexander Hahn other. 2 2 Example 1. If 1+1=2 is an invertible element in R, then X1 − X2 and Y1Y2 are equivalent. What is the linear change of variables and why does 2 have to be invertible?

Let M be a free left R- with {x1,...,xn} and let p be a quadratic form in n variables. Define q : M → R by setting q(r1x1 + ···+ rnxn)=p(r1,...,rn). With the properties of this example in mind, we now turn to a more general concept of quadratic form. Let M be any module over R. A quadratic form on M is a map q : M → R such that q(rx)=r2q(x) for all r ∈ R and x ∈ M, and b : M × M → R given by b(x, y)=q(x + y) − q(x) − q(y) is bilinear. The composite M = (M,q)isaquadratic module over R. If M is finitely generated projective and q non-singular – this means that the map M → M ∗ provided by b is an – then M is a quadratic space. A quadratic module is isotropic if q(x) = 0 for some non-zero x in M. Let S be a containing R. The quadratic form

p(X1,...,Xn)= X ai,j XiXj 1≤i,j≤n over R is also a quadratic form over S. In terms of quadratic modules this is captured by going from (M,q) to the product (M ⊗R S, qS). Two quadratic modules (M,q) and (M 0,q0) over R are equivalent or isometric if there is an invertible ϕ : M → M 0 such that q0(ϕx)=q(x) for all x in M. If this is so, we will write M =∼ M 0. The notation ∼ M = X ai,j XiXj 1≤i,j≤n means that the quadratic module M is isometric to a quadratic module con- structed from the quadratic form P ≤ ≤ ai,j XiXj in the way described ∼ 1 i,j n above. Check that if M = X1X2, then M is a two-dimensional isotropic quadratic space. The following “diagonalization theorem” is a very basic result in the theory of quadratic forms. Theorem 1. If R is a field of not 2 and M is a finite di- mensional quadratic module over R, then ∼ 2 2 2 M = a1X1 + a2X2 + ···+ anXn, where n = dim M and ai ∈ R. The Clifford Algebra in the Theory of Algebras 309

For an introduction to the basic concepts and facts discussed above, see [6, Chapter IV], [7, Chapter I], [8, Chapter IV], [9, Chapter 1], [14, Chapter 1], or [16, Chapter I].

2 Basic Clifford algebras

Let M =(M,q) with associated b be a quadratic module over R. A Clifford algebra of M is a pair C(M)=(C(M),γ) such that (a) C(M) is an R-algebra and

2 (b) γ : M → C(M) is an R-module map satisfying γ(x) = q(x)1C and γ(x)γ(y)+γ(y)γ(x)=b(x, y)1C for all x and y in M, and (c) (C(M),γ) is “minimal” (in the sense of a universal mapping property) with respect to (a) and (b). The following very basic facts introduce concepts that will be relevant in the discussions that follow. Theorem 2. Any (M,q) has a unique Clifford algebra. If M is finitely gen- erated projective, then γ is injective. There is a unique anti- on C(M) taking γx to γx for all x. This anti-automorphism is an involu- tion, as two successive applications of it give the identity map on C(M).

Theorem 3. If {xi}i∈I , where I is an index set, spans M as R-module, then 1C together with all the elements of the form

γxi1 γxi2 ···γxik ,i1 < ···

Taking the submodules of C(M) spanned by 1C and all the elements above with k even, and then in turn the submodule spanned by all the elements above with k odd, splits C(M) into an even and odd part C(M)= C0(M) ⊕ C1(M), and provides C(M) with a Z2-grading, in other words, a two-component grading. Note in particular that C0(M) is a of C(M) and C1(M)isa C0(M)-module.

Theorem 4. If M is free with finite basis {x1,...,xn}, then C(M) is a free R-module with basis

{1C,γxi1 γxi2 ···γxik }, where 1 ≤ k ≤ n and i1 < ···

Example 2. If q =0, then C(M) is an .IfR is a field, dim M =2, and q is non-singular, then C(M) is either the algebra Mat2(R) or a over R.

The algebra A(M) is the centralizer of C0(M)inC(M). More precisely,

A(M)={c ∈ C(M) | cd = dc, ∀ d ∈ C0(M)}.

The next theorem illustrates the important role that A(M) plays in the structure theory of C(M) and C0(M). Theorem 5. Let R be a field and M a quadratic space over R. Then

(1) A = A(M) =∼ R[X]/(X2 − aX − b) with a2 +4b =06 .

(2) Suppose dimRM is even. Then there is a division algebra D over R such that ∼ C(M) = Matk(D)

with dimRD and k both powers of 2. So k =2m for some m.

(2a) Suppose X2 − aX − b has a root in R. Then A =∼ R ⊕ R, and ∼ C0(M) = Matm(D) ⊕ Matm(D).

(2b) Suppose X2 − aX − b does not have a root in R, but does have a root in D. Then A is a subfield of D, the centralizer D0 of the root in D is a central division algebra over A, and ∼ C0(M) = Matm(D0).

2 (2c) Suppose X − aX − b does not have a root in D. Then D ⊗R A is a central division algebra over A and ∼ C0(M) = Matm(D ⊗R A).

(3) Suppose dimRM is odd. Then there is a division algebra D over R ∼ such that C0(M) = Matk(D), with dimRD and k both powers of 2. Statements analogous to those for C0(M) above hold for the algebra C(M).

Corollary 1. If n is even, then C(M) is a over R. If n is odd, then C0(M) is a central simple algebra over R. See [6, Chapter V], [9, Chapter 5], [14, Chapter 9], [16, Chapter IV], [18, Chapters 5, 9, and 11], and [22, Chapter 3] for the details. The Clifford Algebra in the Theory of Algebras 311 3 CSA’s with

We have seen that the existence of an involution and – over a field – the property of being central and simple are basic features of the Clifford alge- bra. Could it possibly be that any finite dimensional central simple algebra that comes equipped with an involution is isomorphic to a Clifford algebra? If the requirement “isomorphic” is replaced by the weaker “is similar to,” that is to say, “is Brauer equivalent to,” then the answer is yes! Theorem A. Let R be a field and A a finite–dimensional central simple algebra over R with an involution. Then there is a quadratic space M of even rank (and discriminant 1) such that A is similar to C(M). At the core of the proof lies a deep theorem of Merkurjev asserting that the map from a certain “K-group” to the 2 part of the is surjective. If R is specialized to be a global field, then Theorem A can be sharpened. Recall that a global field is a finite extension field of either the rational numbers or of the quotient field of a in one variable over a finite field. A global field is an algebraic number field in the first case and an algebraic function field in the second. Theorem B. Let R be a global field and let A be a finite–dimensional central simple algebra over R with an involution. Then there is a quadratic space M of dimension 2 such that A is similar to C(M). This follows from the fact that the only division algebras with involution over such an R are the quaternion division algebras (and R itself). This fact also provides the stronger isomorphism result for global fields. Theorem C. Let R be a global field and let A be a finite–dimensional central simple algebra over R with an involution. Then there is a quadratic space M of even dimension such that A is isomorphic to C(M). The question arises as to whether these field theoretic results have ring theoretic analogues, once “finitely generated (as R-module), central sep- arable” is inserted in place of “finite–dimensional central simple.” In the situation of Theorem B, there is a theorem that says “yes,” if R is an arithmetic Dedekind domain. There are a number of equivalent definitions of Dedekind domain. According to one of them, an R is Dedekind if every of R is the product of prime ideals. A Dedekind domain is arithmetic if its field of fractions is a global field. Theorem B0. Let R be an arithmetic Dedekind domain and let A be any central separable algebra over R with an involution. Then there is a quadratic space M over R of rank 4 (and discriminant 1) such that A is similar to C(M). Do there exist ring theoretic analogues of Theorems A and C? The article 312 Alexander Hahn

R. Parimala and R. Sridharan, Non-surjectivity of the Clifford invariant map, Proc. Indian Acad. Sci. 104 (1994), 49–56, shows that if R is the affine algebra of the smooth projective hyperelliptic curve over the 3-adic numbers defined by

y2 =(t2 − 3)(t4 + t3 + t2 + t +1), then there are central separable algebras over R (finitely generated as R- modules) that are not similar to the Clifford algebra of any quadratic space. (The concepts just used can be found in R. Hartshorne, Algebraic , Springer-Verlag, Berlin Heidelberg New York, 1977). In particular, there appears to be no obvious analogue for Theorem A. My guess is that this is also true for Theorem C. I have given the matter no thought, but there ought to be versions of Theorems A–C that take the graded structure of the Clifford algebra into account. A graded Brauer group – see [9, Chapter 4] and [16, Chapter III (§6)] – is available for the purpose. See [11, Chapters 7 and 8], [14, Chapter 2 (§14)] or [15, Sections 1.5 and 2.3] for introductions to the theorem of Merkurjev and [18, Chapters 11 and 14] for the classification results described above. Refer to [14, Chap- ter 8], [21, Chapter I], and [22, Chapter 6] for related concerns.

4 Quadratic forms over fields

We all know that a large part of the agenda of mathematics is the matter of classification. We just saw an example of this: Clifford algebras were identified as structures with a number of interesting properties, and the question was as to whether any structure with these properties is equivalent or similar to a Clifford algebra. This section discusses this question for quadratic forms with coefficients in a field. Here too, the Clifford algebra has a role to play. Let F be a field of characteristic not 2 and let M (with q and b)bea quadratic space over F. Since b(x, x)=2q(x),band q determine each other, so that the theories of quadratic and symmetric bilinear forms are the same. We call M hyperbolic if dim M is even and there is a basis {x1,...,x2k} of M such that there is the equality

OI (b(x ,x )) =   i j IO of 2k × 2k matrices, where O and I are the k × k zero and identity matrices respectively. Notice that hyperbolic spaces have lots of isotropy (indeed, they have maximal isotropy). Two quadratic spaces M and M 0 are Witt equivalent if there are hyperbolic spaces H and H0 such that

M ⊥ H =∼ M 0 ⊥ H0. The Clifford Algebra in the Theory of Algebras 313

The equivalence classes form an additive monoid. This monoid can be ex- panded to a Grothendieck group (this is like going from the additive set of non-zero positive to the group Z). With the of modules this additive group becomes the Witt ring W (F )ofF. The equiv- alence classes of even–dimensional spaces determine the fundamental ideal I(F )ofW (F ). Theorem 6. Let M and M 0 be quadratic spaces. Then M =∼ M 0 if and only if dim M = dim M 0 and [M]=[M 0] in W (F ). Consider the special cases F = C and F = R. The diagonalization theo- rem of Section 1 plus the fact that everything in C is a square, tells us that the dimension alone suffices to characterize complex quadratic spaces. By similar considerations, any quadratic space M in the real case satisfies ∼ 2 2 2 2 M = X1 + ···+ Xs − Xs+1 −···−Xs+r, where s + r = dim M. The fact that s and r are uniquely determined by M (this is a theorem of Sylvester), provides the signature s(M)= s − r ∈ Z of M. Observe that the dimension together with the signature characterize real quadratic spaces. The assignment sig[M] 7→ s(M) defines an isomorphism sig : W (R) → Z. Return to an arbitrary F. In addition to the surjective group homomor- phism dim : W (F ) → Z2 given by dimension mod 2, there are the surjective group homomorphisms disc : I(F ) → Qu(F ), where Qu(F ) is a group of equivalence classes of quadratic algebras and disc[M]=[A(M)] with A(M) the discriminant algebra of M, and 2 Cliff : I (F ) → Br2(F ), where Br2(F ) are the elements of 2 (and 1) in the Brauer group of F and Cliff[M]=[C(M)]. The kernels of these three maps are, respectively, the fundamental ideal I(F ) and its powers I2(F ) and I3(F ). There are also graded analogues of these homomorphisms defined on all of W (F ). If F is a local field (one that is equipped with a discrete and complete valuation with respect to which the field is locally compact), then dim , disc, and Cliff suffice to classify quadratic spaces. The same is true if F is an algebraic function field. However, if F is an algebraic number field, then one more invariant is needed. Let M be a quadratic space over an algebraic number field F. Suppose that F has k real . For each of these, there is a signature via M ⊗F R. Putting all these signatures together, we get the total signature sig : W (F ) → Zk. 314 Alexander Hahn

The fact is that dim , disc, Cliff, and sig classify quadratic spaces over F. ∼ 2 2 2 ∼ 2 Example 3. The quadratic spaces M1 = −X1 +3X2 +5X3 and M2 = X1 + 2 2 ∼ 2 2 2 7X2 − 105X3 over Q are isometric, but the spaces M3 = X1 − 6X2 +15X3 ∼ 2 2 2 and M4 = 3X1 − 10X2 +3X3 over Q are not isometric. All four quadratic spaces of the example are three–dimensional with signa- ture 1. In view of the earlier discussion, they are all isometric over R. Refer to [6, Chapter VI], [9, Chapters 2, 4, and 6] and [14, Chapters 2, 5, and 6] for the details and much more. The suggestion that the three surjective homomorphisms described above might be the beginning of an infinite pattern of invariants that would clas- sify quadratic forms over any field was made by Milnor in 1970. The proof of Milnor’s conjecture became a holy grail of quadratic form theory. More precisely, the question was whether for any field F of characteristic not 2, n there exist for each n, a suitable group H (F, Z2) and a surjective homomorphism

n n en : I (F ) → H (F, Z2) with In+1(F ). If the answer were yes, then these invariants would classify quadratic spaces over F as follows: For two quadratic spaces M1 and M2, let [M1] − [M2]=[M] ∈ W (F ). If en[M]=0, then [M] is in In+1(F ), so if all invariants of the difference [M] are zero, then [M]=0, and [M1] and [M2] are equal in W (F ). Voevodsky announced a proof of Milnor’s conjecture a few years ago and he provided complete details in the summer of 2001. For this impressive accomplishment, he was awarded the Fields Medal at the International Congress of Mathematicians in Beijing in 2002. For a glimpse of how Vo- evodsky did it and for the broader implications of what he did, refer to the surveys A. Pfister, On the Milnor Conjectures: History, Influence, and Applications, Jahresbericht d. Dt. Math.-Verein. 102 (2000), 15–41, and W. Scharlau, On the History of the Algebraic Theory of Quadratic Forms, Contemporary Mathematics 272 (2000), 229–259.

5 The

We turn next to a connection between the Clifford algebra and an interest- ing slice of group theory. We continue to let F be a field of characteristic not 2 and M a quadratic space over F. Recall that γ : M → C(M) is injective and that there is a unique invo- lution on C(M) taking γx to γx for all x. Consider M to be a subset of C(M) via γ, and define the group Spin(M)by

× −1 Spin(M)={c ∈ C0(M) | cMc = M, cc¯ =1C}, The Clifford Algebra in the Theory of Algebras 315

× where C0(M) is the group of invertible elements of the ring C0(M). The isometries from M onto M constitute the O(M) and SO(M) is the subgroup of elements of 1. For c in Spin(M), define πc : M → M by πc(x)=cxc−1. This provides a homomorphism

π : Spin(M) → SO(M).

By a theorem of Cartan and Dieudonn´e,any element σ in O(M) is a prod- uct σ = τy1 ···τyk of hyperplane reflections τyi . The assignment Θ(σ)= × 2 q(y1) ···q(yk)(F ) defines the norm homomorphism

Θ: SO(M) → F ×/(F ×)2.

Theorem 7. The sequence

π Θ × × 2 1 →{±1C} ,→ Spin(M) −→ SO(M) −→ F /(F ) is exact. So π induces an isomorphism

∼ 0 Spin(M)/{±1C} = O (M) where O0(M)=kerΘ. Let Ω(M) be the subgroup of O(M). If dim M ≥ 5, then

Ω(M)/Center Ω(M) is a simple group if F is a finite field, a local, or a global field, and for any field F if M is isotropic. It turns out that in all the situations just singled out, Ω(M)=O0(M). In particular, ∼ Spin(M)/{±1C} = Ω(M),

Spin(M) is a central extension of Ω(M), and the quotient

SO(M)/Ω(M) is an elementary Abelian 2-group. Our discussion has connected the Clif- ford algebra to some interesting group theory for the orthogonal group: generators and relations and questions of length, analysis of central exten- sions, characterizations of the normal subgroups, as well as the theory of finite simple groups (there are also generalizations to some commutative rings). The Spin groups have an impact on another issue that has received much attention over the years. Let M be a finitely generated projective module over a ring R. Suppose that M is equipped with a non-singular quadratic, 316 Alexander Hahn alternating, or hermitian form, or the form that is identically zero. The group I(M) of isometries of M is an orthogonal, symplectic, or unitary group, or a general linear group (in the case of the zero form). These groups, certain of their subgroups, and their quotients by scalar transformations are the classical groups. Consider two such modules M1 and M2 over the rings R1 and R2 and let I(M1) and I(M2) be the respective groups of isometries. A semi-isometry α from M1 to M2 is an invertible map that is linear up to a ring isomorphism from R1 to R2 and that preserves the forms up to multiplication by an invertible ring element. Such a map induces a group isomorphism

Φα : I(M1) −→ I(M2) in a natural way. This construction raises the following question. Pick two classical groups at “random” but suppose that they are isomorphic. Could it be that the isomorphism comes (via restriction and/or factoring scalars) from an isomorphism Φα as just defined. In particular, are the underlying modules semi-isometric? The short and generic answer is “yes.”

Theorem 8. Let G1 and G2 be two isomorphic classical groups over inte- gral domains (possibly non-commutative). If the dimensions of the under- lying spaces are large enough (say at least 6, and not both 8 in the case of two orthogonal groups), isotropic enough (say at least three independent or- thogonal isotropic vectors), then the underlying modules are semi-isometric and the isomorphism is induced by a semi-isometry. For instance, a unitary group cannot be isomorphic to a linear, sym- plectic, or orthogonal group, and it can be isomorphic to another unitary group only if the underlying (hermitian) are semi-isometric. The case of two 8-dimensional orthogonal groups needs to be excluded due to existence of the exceptional triality . What about the low–dimensional barrier in the theorem? There are ex- ceptions that come from Spin groups! Suppose dim M =6. If M is hyper- bolic, then ∼ Spin(M) = SL4(F ) and if M is “almost” hyperbolic (there exist two independent orthogonal isotropic vectors rather than three), then ∼ Spin(M) = SU4(F ).

If dim M = 5 and M is almost hyperbolic, then ∼ Spin(M) = Sp4(F ).

By factoring centers we get isomorphic classical groups in these low– dimensional situations that have different underlying geometries. So these examples go “counter” to the theorem. The Clifford Algebra in the Theory of Algebras 317

See [4, Chapters II and IV], [6, Chapters IV, V, IX (§95), and X (§101)], [15], [16, Chapter V], and [21, Chapter VI (§27)] for lots of information about the concerns of this section. The recent article A. Hahn, The Zassen- haus Decomposition for the Orthogonal Groups: Properties and Applica- tions, Documenta Mathematica (2001), 165–181, accessible at http://www.mathematik.uni-bielefeld.de/documenta/ also falls into this realm.

6 Quadratic forms over arithmetic domains

This section provides an overview of that part of the arithmetic theory of quadratic forms that attempts to come to grips with the classification of quadratic forms over arithmetic Dedekind domains. The Clifford algebra is connected to this enterprise via the spinor norm. Let R be a Dedekind domain with global field of fractions F. We let M =(M,q) and M 0 =(M 0,q0) be quadratic spaces over R and turn to the fundamental question: When is (M,q) =∼ (M 0,q0)? Suppose that M =∼ M 0. Since they are equivalent over R, they are equivalent after any extension of scalars. In particular, ∼ 0 0 (M ⊗R F, qF ) = (M ⊗R F, qF ). This allows for the following reformulation of our question: Assume that both M and M 0 sit inside and span the same quadratic space V over F. Does there exist a σ ∈ O(V ) such that σM = M 0? Defining

class M = {σM | σ ∈ O(V )}, the original question becomes: When is M 0 in the class of M? Let p be a non-trivial valuation on F. Let Fp be the completion of F at p. There are three possibilities: ∼ ∼ Fp = R,Fp = C, or Fp is a local field (with p discrete).

For p discrete, let Rp be the in Fp. The fact is that there is a set S of discrete valuations such that

R = \ Rp. p∈S Our Dedekind domain is a Hasse domain if every valuation of F – except for a finite number of them – is equivalent to one of the valuations in S. This is what we will now assume. If F is an algebraic number field and R its ring of algebraic integers, then R is a Hasse domain. The integers Z in Q and the Gaussian integers Z[i]inQ(i) are standard examples. 318 Alexander Hahn

Consider for any p the extensions of scalars, or “localizations,”

0 0 Mp = M ⊗R Rp,Mp = M ⊗R Rp, and Vp = V ⊗F Fp.

We say that the spaces M and M 0 are in the same genus if for any p ∈ S 0 there is a σp ∈ O(Vp) so that σp(Mp)=Mp. This concept defines the set genus M. Notice that class M ⊆ genus M. There are invariants that let us decide whether M 0 is in genus M. But the question of when M 0 is in class M is much more subtle. There is an “intermediate” concept that splits it into two parts. We say that M 0 is in the spinor genus of M if there exist a σ ∈ O(V ) and for every p ∈ S an 0 element Σp ∈ Op(V ), such that

0 Mp = σpΣpMp.

The corresponding equivalence class gives rise to the set spingenus M. No- tice that class M ⊆ spingenus M ⊆ genus M. The set spingenus M is partitioned into classes and genus M is partitioned into both classes and spinor genera. The basic facts are these: the number of classes in any genus M is fi- nite, and the number of spinor genera in any genus M is a power of 2. By varying M within V, any 2-power can be obtained. If the completion of V is isotropic for at least one p ∈ S – this is the so-called indefinite case – then spingenus M = class M for any M. Among the results that additional analysis provides is the following Theorem 9. Suppose M is a quadratic space over Z of rank n. Assume that M ⊗Z R is isotropic (this is the indefinite condition), b(M,M)=Z, and discM = ±1. Then

∼ 2 2 2 M = a1X1 + a2X2 + ···+ anXn with all ai = ±1. Refer for these matters and much more to [1, Chapters IV, V, and VI], [6, Chapters VIII, IX, and X], [12], [17, Chapters 5, 6, and 7], and [20], as well as to the recent survey, J. Hsia, Arithmetic of Indefinite Quadratic Forms, Contemp. Math. Vol. 249 (1999), 1–15.

7 Generalized even Clifford algebras

There is a generalization of the Clifford algebra – more precisely of the even subalgebra – that plays an important role in the analysis of involutions on The Clifford Algebra in the Theory of Algebras 319 central simple algebras as well as the classification of algebraic groups over arbitrary fields. Let F be a field, let A be a finite–dimensional central simple algebra over F, and let σ be an involution on A. The restriction of σ to F is an involution on F. If this involution is the identity on F, then σ is of the first kind and if not, σ is of the second kind. This section will concentrate exclusively on involutions of the first kind. A large class of examples of such involutions can be constructed as follows. Let V be a finite–dimensional over F and consider the central simple algebra EndF (V ). Take a non- singular bilinear from b on V (as given by an appropriate for instance) that is either symmetric or alternating (i.e., symmetric up 0 to a − sign). For any t ∈ EndF (V ) there is a unique t ∈ EndF (V ) such that b(x, ty)=b(t0x, y) for all x, y ∈ V. The assignment t 7→ t0 defines an involution of the first kind on EndF (V ). This is the adjoint involution corresponding to b. The fact is that every involution of the first kind on EndF (V ) arises in this way. Let us return to our general central simple algebra A with its involution σ of the first kind. It is a classical fact that there is a finite separable extension K of F such that ∼ A ⊗F K = Matk(K).

Because σ ⊗ idK defines an involution of the first kind on Matk(K), it arises from a non-singular quadratic or alternating form. The involution σ is orthogonal or symplectic accordingly. Incidentally, the isomorphism above also supplies the reduced trace TrdA(a) for any a ∈ A as the matrix trace of the of a ⊗ 1. The definition of the generalized even Clifford algebra has as starting point the concept of a quadratic pair. Recall first that the dimension of A over F has the form m2 for a positive m known as the degree of A over F. Consider the sets Sym(A, σ)={a ∈ A | σa = a} and Skew(A, σ)={a ∈ A | σa = −a} and observe that Sym(A, σ) ⊇ F. A quadratic pair (σ, f )onA consists of an involution σ on A and a linear map f : Sym(A, σ) → F such that 1 (i) Sym(A, σ) has dimension 2 m(m + 1) over F,

(ii) TrdA(Skew(A, σ)) = {0}, and

(iii) f is F -linear and f(a + σa)=TrdA(a) for all a ∈ A. A quadratic space (V,q)overF gives rise to a quadratic pair as follows. Let b be the associated non-singular bilinear form of V. Let σb be the corresponding adjoint involution on the algebra EndF (V ). Fix v ∈ V and note that the element in EndF (V ) defined by x 7→ vb(v, x) is fixed by σb. There is a unique linear map

fq : Sym(EndF (V ),σb) −→ F 320 Alexander Hahn such that fq(x 7→ vb(v, x)) = q(v). It turns out that (σb,fb) is a quadratic pair on the central simple algebra EndF (V ). Return to an arbitrary quadratic pair (σ, f ) on a central simple algebra A. If char F =26 , then the defining conditions imply that σ is orthogonal 1 and that f is determined by f(s)= 2 TrdA(s). If char F =2, then σ is symplectic and m is even. In this case, there are a number of quadratic pairs with the same σ. In any characteristic there exists an element l ∈ A such that f(s)=TrdA(ls) for all s ∈ Sym(A, σ). Focus next on the F -vector space structure of A. Start with the T (A)ofA. Let I be the ideal of T (A) generated by the elements of the form s − f(s)1 with s ∈ Sym(A, σ). Consider the isomorphism

ϕ A ⊗F A −→ EndF (A) that satisfies ϕ(a ⊗ b)x = axb for all a, b, and x in A. Let J be the ideal of T (A) generated by all u − (ϕu)1 for all u ∈ A ⊗F A such that (ϕu)(x)= (ϕu)(σx) for all x ∈ A. The Clifford algebra C(A, σ, f) is the F -algebra defined by T (A) C(A, σ, f)= . I + J

The involution on T (A) defined by a1 ⊗···⊗ar 7−→ σar ⊗···⊗σa1 induces an involution σ on C(A, σ, f). Let (V,q) be a quadratic space over F and consider the quadratic pair (σb,fq) on EndF (V ) that it defines. The corresponding Clifford algebra C((EndF (V ),σb,fq) turns out to be isomorphic to the even Clifford algebra C0(V ) as follows. The standard identification V ⊗ V → EndF (V ) given by ∼ (v, w) 7→ (x 7→ vb(w, x)) defines an isomorphism T (V ⊗ V ) = T (EndF (V )) which in turn induces an isomorphism C0(V ) → C((EndF (V ),σb,fq) that satisfies vw 7→ (x 7→ vb(w, x)) + (I + J). The involution σb corresponds to the restriction to C0(V ) of the involution defined in Section 2. To define the generalized Spin group, start with the submodule T+(A) of T (A) given by ⊗k T+(A)=M A . k≥1

The multiplication of T (A) provides T+(A) with the structure of a right T (A)-module. Letting T (A) act on T+(A) on the left by a permuted version ∗ of this multiplication, gives T+(A) the structure of a T (A)-bimodule. By going to the induced structure, the quotient T (A) B(A, σ, f)= + [I ∗ T+(A)] + [T+(A) · I] becomes a C(A, σ, f)-bimodule. In the example just discussed, B(A, σ, f) corresponds to the C0(V )-bimodule V ⊗C1(V ) where both actions are given by the multiplication of C(V ). From a 7→ a ∈ T+(A), we get an injective The Clifford Algebra in the Theory of Algebras 321

F -linear map A → B(A, σ, f). The image of A is written Ab. The Spin group of (A, σ, f) is defined by

Spin(A, σ, f)={c ∈ C(A, σ, f)× | c−1 ∗ Ab · c ⊆ Ab and σc = c−1}. Theorem 10. Suppose that G is a simply connected semisimple group of type Dn over F. If n =46 , then G is isomorphic – as group scheme over F –toSpin(A, σ, f) for some central simple F -algebra A of degree 2n with quadratic pair (σ, f ). When n = 4 the groups Spin(A, σ, f) are still simply connected semisimple groups of type Dn, but the additional symmetry (triality) of the Dynkin diagram gives rise to more such groups. Refer to [21, Chapters I (§5), II (§8, 9), III (§13), and VI] for the complete exposition of the theory outlined above. The study of these generalized even Clifford algebras goes back to N. Jacobson, Clifford algebras for algebras with involution of type D, J. Algebra, 1 (1964), 288–300, and J. Tits, Formes quadratiques, groupes orthogonaux et alg`ebresde Clifford, Inventiones Math. 5 (1968), 19–41. Two additional developments deserve attention. The first is related to the theme of Section 3. Theorem 11. Let F be a field of characteristic not 2. Suppose that the ideal I(F )3 of the Witt ring of F is zero. Local fields and algebraic number fields with no real embeddings satisfy this condition. Let A be a central simple algebra over F. Then all symplectic involutions on A are isomorphic, and two orthogonal involutions on A are isomorphic if and only if they have isomorphic Clifford algebras. This theorem is proved in D. Lewis and J.-P. Tignol, Classification the- orems for central simple algebras with involution, Manuscripta Math. 100 (1999), 259–276. A different generalization of the Clifford algebra over a field F of charac- teristic not 2 is developed in K. Hannabuss, Bilinear forms, Clifford alge- bras, q-commutation relations, and quantum groups, J. Algebra 228 (2000), 227–256. It is defined for any vector space V over F equipped with a non- B. If B is symmetric, it specializes to the classical Clifford algebra of Section 2. Like the classical Clifford algebra, this generalized Clifford algebra is a quotient of the tensor algebra of V. This generalization is related to the quantized function algebras studied in Y. Manin, Topics in non-commutative Geometry, Princeton, 1991.

8 Clifford modules and vector fields on spheres

Clifford modules, especially over the real numbers R, are a point of de- parture for some important interconnections and applications. One of the 322 Alexander Hahn most spectacular of these is Adams’s solution of the vector field problem for the sphere. We will briefly recall the essence of his achievement on the occasion of its 40th anniversary. Let V be a quadratic space of dimension n over R. Recall from Section 4 that ∼ 2 2 2 2 2 V = X1 + X2 + ···+ Xs − Xs+1 ···−Xs+r, where s+r = n. Because the signature is well defined, s and r are uniquely determined by V. It has become custom to denote the Clifford algebra C(V ) by C`r,s. An R-representation of C`r,s is a homomorphism of R-algebras

ρ : C`r,s → EndR(W ), where W is a finite–dimensional vector space over R. It turns out that every representation of C`r,s is a direct sum of irreducible representations.

Theorem 12. Let wr,s be the number of inequivalent irreducible R-repre- sentations of C`r,s. Then

(1, if r − s is even; wr,s = 2, if r − s is odd.

We now turn to vector fields on the sphere. Let N be a positive integer, consider RN+1, and let

SN = {x ∈ RN+1 |kxk2 =1} be the N-sphere. A tangent vector field on SN is a continuous function v:SN → RN+1 such that v(x) is tangent to SN at x for all x. Example 4. Let N be odd. Then v : SN → RN+1 defined by

v(x)=(x1, −x0,x3, −x2,...,xN−2,xN−1) is a tangent vector field on SN that is nowhere 0. For N even, on the other hand, it turns out that every tangent vector field on SN vanishes somewhere. N A set of tangent vector fields v1,...,vn on S is linearly independent, N+1 if {v1(x),...,vn(x)} is an independent set of vectors in R for every x in SN . The existence of linearly independent tangent vector fields on SN is connected to the representation theory of Clifford algebras as follows. Turn to the special case s = 0 and r = n, and denote C`r,s by C`n. Let ρ : C`n → EndR(W )beanR-representation. Defining cw =(ρc)w for c ∈ C`n and w ∈ W, makes W a module over the algebra C`n. Conversely, any C`n-module arises from such a representation. It is not very hard to prove the following The Clifford Algebra in the Theory of Algebras 323

N+1 Theorem 13. Suppose that R is a C`n-module. Then there exist n linearly independent vector fields on SN .

4a+b Let m be a positive integer and write m =2 m0 with m0 odd and 0 ≤ b ≤ 3. The number ρ(m)=8a +2b is the Radon–Hurwitz number of m. Set k =4a + b. One can check that

2k +1 if k ≡ 0 (mod 4);  2k if k ≡ 1 (mod 4); ρ(m)= 2k if k ≡ 2 (mod 4);  2k +2 if k ≡ 3 (mod 4).

Theorem 14. The integer ρ(N +1)− 1 is the largest integer n such that N+1 R is a C`n-module. Check that this conclusion is consistent with the earlier example, and that Theorem 15. There exist ρ(N +1)− 1 linearly independent vector fields on SN . is an obvious consequence of the two theorems above. The proofs of the results so far are largely routine. But now comes the important and very difficult question that the celebrated theorem of Adams answers: What is the largest number of linearly independent tangent vector fields that SN can support? Theorem 16. The number ρ(N +1)− 1 is in fact the largest number of linearly independent vector fields that can exist on SN . The proof analyzes the group of stable fiber homotopy classes on real pro- jective space and applies topological K-theory in the process. Refer to D. Husemoller, Fibre Bundles, third edition, Springer-Verlag, Berlin Heidel- berg, New York, 1994, for the complete details, and to [22, Chapters 0 and 12] for background information and other connections.

Acknowledgments

I wish to thank Richard Elman, Max-Albert Knus, Carl Riehm, and espe- cially Jean-Pierre Tignol and Adrian Wadsworth, for the insights, sugges- tions, and corrections that they provided.

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Alexander J. Hahn Department of Mathematics The Clifford Algebra in the Theory of Algebras 325

University of Notre Dame Notre Dame, Indiana 46556 E-mail: [email protected] Submitted: September 17, 2002.