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Chapter 19 the Clifford Algebra in the Theory of Algebras, Quadratic

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Chapter 19 the Clifford Algebra in the Theory of Algebras, Quadratic This is page 307 Printer: Opaque this 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 theoretical physics, to computer science, to pure mathematics – 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, quadratic form, classical group, involutions, Clif- ford modules 1 Basic quadratic forms We will set the stage in very general terms. Let R be a commutative (and associative) ring 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-module with basis {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 isomorphism – 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 commutative ring 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 tensor 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 linear map 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 characteristic 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 bilinear form 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-automorphism 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 < ···<ik span C(M) as R-module. 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 subalgebra 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 < ··· <ik. So C(M) is a free R-module of n dimension 2 . It follows that C0(M) and C1(M) are free R-modules as well and that both have dimension 2n−1. 310 Alexander Hahn Example 2. If q =0, then C(M) is an exterior algebra.IfR is a field, dim M =2, and q is non-singular, then C(M) is either the matrix algebra Mat2(R) or a quaternion division algebra over R. The discriminant 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 central simple algebra 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 involution 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 Brauer group 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 polynomial ring in one variable over a finite field.
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