Chapter 22 Tensor Algebras, Symmetric Algebras and Exterior

Home , Basis (linear algebra), Einstein notation, Exterior algebra, Interior product, Linear map, Logarithm of a matrix

Chapter 22

Tensor Algebras, Symmetric Algebras and Exterior Algebras

22.1 Tensors Products

We begin by defining tensor products of vector spaces over a field and then we investigate some basic properties of these tensors, in particular the existence of bases and duality. After this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Tensor products of modules over a commutative ring with identity will be discussed very briefly. They show up naturally when we consider the space of sections of a tensor product of vector bundles.

Given a linear map, f : E F ,weknowthatifwehaveabasis,(ui)i I ,forE,thenf → ∈ is completely determined by its values, f(ui), on the basis vectors. For a multilinear map, f : En F ,wedon’tknowifthereissuchanicepropertybutitwouldcertainlybevery useful. → In many respects, tensor products allow us to define multilinear maps in terms of their action on a suitable basis. The crucial idea is to linearize,thatis,tocreateanewvectorspace, n n n E⊗ , such that the multilinear map, f : E F ,isturnedintoalinear map, f : E⊗ F , ⊗ which is equivalent to f in a strong sense. If→ in addition, f is symmetric, then we can define→ asymmetrictensorpower,Symn(E), and every symmetric multilinear map, f : En F ,is turned into a linear map, f :Symn(E) F ,whichisequivalenttof in a strong→ sense. ⊙ Similarly, if f is alternating, then we can define→ a skew-symmetric tensor power, n(E), and every alternating multilinear map is turned into a linear map, f : n(E) F ,whichis ∧ equivalent to f in a strong sense. → Tensor products can be defined in various ways, some more abstract than others. We tried to stay down to earth, without excess!

Let K be a given ﬁeld, and let E1,...,En be n 2givenvectorspaces.Foranyvector space, F ,recallthatamap,f : E E F≥,ismultilinear iﬀit is linear in each of 1 ×···× n → 585 586 CHAPTER 22. TENSOR ALGEBRAS its argument, that is,

f(u1,...ui1 ,v+ w, ui+1,...,un)=f(u1,...ui1 ,v,ui+1,...,un)

+ f(u1,...ui1 ,w,ui+1,...,un)

f(u1,...ui1 ,λv,ui+1,...,un)=λf(u1,...ui1 ,v,ui+1,...,un), for all u E (j = i), all v, w E and all λ K,fori =1...,n. j ∈ j ∈ i ∈ The set of multilinear maps as above forms a vector space denoted L(E1,...,En; F )or Hom(E1,...,En; F ). When n = 1, we have the vector space of linear maps, L(E,F)or Hom(E,F). (To be very precise, we write HomK (E1,...,En; F ) and HomK (E,F).) As usual, the dual space, E∗, of E is deﬁned by E∗ = Hom(E,K). Before proceeding any further, we recall a basic fact about pairings. We will use this fact to deal with dual spaces of tensors.

Definition 22.1 Given two vector spaces, E and F ,amap,( , ): E F K,isa nondegenerate pairing iffit is bilinear and iff(u, v)=0forall−v −F implies× →u =0and (u, v)=0forallu E implies v = 0. A nondegenerate pairing induces∈ two linear maps, ∈ ϕ: E F ∗ and ψ : F E∗,definedby → → ϕ(u)(y)=(u, y) ψ(v)(x)=(x, v), for all u, x E and all v, y F . ∈ ∈ Proposition 22.1 For every nondegenerate pairing, ( , ): E F K, the induced maps − − × → ϕ: E F ∗ and ψ : F E∗ are linear and injective. Furthermore, if E and F are finite → → dimensional, then ϕ: E F ∗ and ψ : F E∗ are bijective. → →

Proof .Themapsϕ: E F ∗ and ψ : F E∗ are linear because u, v (u, v)isbilinear. Assume that ϕ(u)=0.Thismeansthat→ →ϕ(u)(y)=(u, y)=0forall→y F and as our pairing is nondegenerate, we must have u =0.Similarly,ψ is injective. If E∈and F are finite dimensional, then dim(E)=dim(E∗)anddim(F ) = dim(F ∗). However, the injectivity of ϕ and ψ implies that that dim(E) dim(F ∗)anddim(F ) dim(E∗). Consequently dim(E) ≤ ≤ ≤ dim(F )anddim(F ) dim(E), so dim(E)=dim(F ). Therefore, dim(E)=dim(F ∗)andϕ ≤ is bijective (and similarly dim(F )=dim(E∗)andψ is bijective). Proposition 22.1 shows that when E and F are finite dimensional, a nondegenerate pairing induces canonical isomorphims ϕ: E F ∗ and ψ : F E∗,thatis,isomorphismsthatdo not depend on the choice of bases. An→ important special→ case is the case where E = F and we have an inner product (a symmetric, positive definite bilinear form) on E.

Remark: When we use the term “canonical isomorphism” we mean that such an isomorphism is deﬁned independently of any choice of bases. For example, if E is a ﬁnite dimensional 22.1. TENSORS PRODUCTS 587

vector space and (e1,...,en) is any basis of E,wehavethedualbasis,(e1∗,...,en∗ ), of E∗ (where, ei∗(ej)=δij)andthus,themapei ei∗ is an isomorphism between E and E∗.This isomorphism is not canonical. → On the other hand, if , is an inner product on E,thenProposition22.1showsthat − − the nondegenerate pairing, , ,inducesacanonicalisomorphismbetweenE and E∗. − − This isomorphism is often denoted : E E∗ and we usually write u for (u), with u E. → ∈ Given any basis, (e1,...,en), of E (not necessarily orthonormal), if we let gij =(ei,ej), then for every u = n u e ,sinceu(v)= u, v ,forallv V ,weget i=1 i i ∈ n n u = ωiei∗, with ωi = gijuj. i=1 j=1 If we use the convention that coordinates of vectors are written using superscripts n i (u = i=1 u ei)andcoordinatesofone-forms(covectors)arewrittenusingsubscripts n (ω = i=1 ωiei∗), then the map, ,hastheeﬀectoflowering(ﬂattening!)indices.The n inverse of is denoted : E∗ E.Ifwewriteω E∗ as ω = i=1 ωiei∗ and ω E as n j → ∈ ∈ ω = (ω ) ej,since j=1 n ω = ω(e )= ω,e = (ω)jg , 1 i n, i i i ij ≤ ≤ j=1 we get n i ij (ω ) = g ωj, j=1 where (gij)istheinverseofthematrix(g ). The inner product, ( , ), on E induces an ij − − inner product on E∗ also denoted ( , )andgivenby − − (ω1,ω2)=(ω1,ω2),

for all ω ,ω E∗.Then,itisobviousthat 1 2 ∈ (u, v)=(u,v), for all u, v E. ∈

If (e1,...,en)isabasisofE and gij =(ei,ej), as

n ik (ei∗) = g ek, k=1 an easy computation shows that

ij (ei∗,ej∗)=((ei∗) , (ej∗) )=g , 588 CHAPTER 22. TENSOR ALGEBRAS

ij that is, in the basis (e1∗,...,en∗ ), the inner product on E∗ is represented by the matrix (g ), the inverse of the matrix (gij). The inner product on a finite vector space also yields a natural isomorphism between the space, Hom(E,E; K), of bilinear forms on E and the space, Hom(E,E), of linear maps from E to itself. Using this isomorphism, we can define the trace of a bilinear form in an intrinsic manner. This technique is used in differential geometry, for example, to define the divergence of a differential one-form.

Proposition 22.2 If , is an inner product on a finite vector space, E, (over a field, K), then for every bilinear− − form, f : E E K, there is a unique linear map, f : E E, such that × → → f(u, v)= f (u),v , for all u, v E. ∈ The map, f f , is a linear isomorphism between Hom(E,E; K) and Hom(E,E). → Proof .Foreveryg Hom(E,E), the map given by ∈ f(u, v)= g(u),v ,u,vE, ∈ is clearly bilinear. It is also clear that the above defines a linear map from Hom(E,E)to Hom(E,E; K). This map is injective because if f(u, v)=0forallu, v E,as , is an inner product, we get g(u)=0forallu E. Furthermore, both spaces∈ Hom(E,E− )and− Hom(E,E; K)havethesamedimension,soourlinearmapisanisomorphism.∈

If (e1,...,en)isanorthonormalbasisofE, then we check immediately that the trace of alinearmap,g,(whichisindependentofthechoiceofabasis)isgivenby

n tr(g)= g(e ),e , i i i=1 where n =dim(E). We deﬁne the trace of the bilinear form, f,by

tr(f)=tr(f ).

From Proposition 22.2, tr(f)isgivenby

tr(f)= f(ei,ei), i=1

for any orthonormal basis, (e1,...,en), of E.Wecanalsocheckdirectlythattheabove expression is independent of the choice of an orthonormal basis. We will also need the following Proposition to show that various families are linearly independent. 22.1. TENSORS PRODUCTS 589

Proposition 22.3 Let E and F be two nontrivial vector spaces and let (ui)i I be any family ∈ of vectors ui E. The family, (ui)i I , is linearly independent iﬀfor every family, (vi)i I , of ∈ ∈ vectors v F∈, there is some linear map, f : E F , so that f(u )=v , for all i I. i ∈ → i i ∈ Proof .Leftasanexercise. First, we deﬁne tensor products, and then we prove their existence and uniqueness up to isomorphism.

Deﬁnition 22.2 A tensor product of n 2vectorspacesE1,...,En,isavectorspaceT , together with a multilinear map ϕ: E ≥ E T , such that, for every vector space F 1 ×···× n → and for every multilinear map f : E1 En F ,thereisauniquelinearmapf : T F , ⊗ with ×···× → → f(u1,...,un)=f (ϕ(u1,...,un)), ⊗ for all u E ,...,u E ,orforshort 1 ∈ 1 n ∈ n f = f ϕ. ⊗ ◦ Equivalently, there is a unique linear map f such that the following diagram commutes: ⊗

ϕ E E T 1 ×···× n f ⊗ f F

First, we show that any two tensor products (T1,ϕ1)and(T2,ϕ2)forE1,...,En,are isomorphic.

Proposition 22.4 Given any two tensor products (T1,ϕ1) and (T2,ϕ2) for E1,...,En, there is an isomorphism h: T T such that 1 → 2 ϕ = h ϕ . 2 ◦ 1 Proof .Focusingon(T ,ϕ ), we have a multilinear map ϕ : E E T ,andthus, 1 1 2 1 ×···× n → 2 there is a unique linear map (ϕ2) : T1 T2,with ⊗ →

ϕ2 =(ϕ2) ϕ1. ⊗ ◦ Similarly, focusing now on on (T ,ϕ ), we have a multilinear map ϕ : E E T , 2 2 1 1 ×···× n → 1 and thus, there is a unique linear map (ϕ1) : T2 T1,with ⊗ →

ϕ1 =(ϕ1) ϕ2. ⊗ ◦ But then, we get ϕ1 =(ϕ1) (ϕ2) ϕ1, ⊗ ◦ ⊗ ◦ 590 CHAPTER 22. TENSOR ALGEBRAS and ϕ2 =(ϕ2) (ϕ1) ϕ2. ⊗ ◦ ⊗ ◦ On the other hand, focusing on (T1,ϕ1), we have a multilinear map ϕ1 : E1 En T1, but the unique linear map h: T T ,with ×···× → 1 → 1 ϕ = h ϕ 1 ◦ 1 is h =id,andsince(ϕ1) (ϕ2) is linear, as a composition of linear maps, we must have ⊗ ◦ ⊗

(ϕ1) (ϕ2) =id. ⊗ ◦ ⊗ Similarly, we must have (ϕ2) (ϕ1) =id. ⊗ ◦ ⊗ This shows that (ϕ1) and (ϕ2) are inverse linear maps, and thus, (ϕ2) : T1 T2 is an ⊗ ⊗ ⊗ → isomorphism between T1 and T2. Now that we have shown that tensor products are unique up to isomorphism, we give a construction that produces one.

Theorem 22.5 Given n 2 vector spaces E1,...,En, a tensor product (E1 En,ϕ) for E ,...,E can be constructed.≥ Furthermore, denoting ϕ(u ,...,u ) as u⊗···⊗ u , 1 n 1 n 1 ⊗···⊗ n the tensor product E1 En is generated by the vectors u1 un, where u1 E ,...,u E , and for⊗··· every⊗ multilinear map f : E E ⊗···F ,⊗ the unique linear∈ 1 n ∈ n 1 ×···× n → map f : E1 En F such that f = f ϕ, is deﬁned by ⊗ ⊗···⊗ → ⊗ ◦

f (u1 un)=f(u1,...,un), ⊗ ⊗···⊗ on the generators u u of E E . 1 ⊗···⊗ n 1 ⊗···⊗ n Proof .Givenanyset,I,viewedasanindexset,letK(I) be the set of all functions, f : I K, → such that f(i) =0onlyforfinitelymanyi I. As usual, denote such a function by (fi)i I , ∈ it is a family of finite support. We make K∈(I) into a vector space by defining addition and scalar multiplication by

(fi)+(gi)=(fi + gi)

λ(fi)=(λfi).

The family, (ei)i I ,isdeﬁnedsuchthat(ei)j =0ifj = i and (ei)i =1.Itisabasisof ∈ the vector space K(I),sothateveryw K(I) can be uniquely written as a ﬁnite linear ∈ (I) combination of the ei.Thereisalsoaninjection,ι: I K ,suchthatι(i)=ei for every i I. Furthermore, it is easy to show that for any vector→ space, F ,andforanyfunction, f ∈: I F ,thereisauniquelinearmap,f : K(I) F ,suchthat → → f = f ι, ◦ 22.1. TENSORS PRODUCTS 591

as in the following diagram: ι I K(I)

f f F This shows that K(I) is the free vector space generated by I. Now, apply this construction to the cartesian product, I = E E ,obtainingthefreevectorspaceM = K(I) on 1 ×···× n I = E1 En.Sinceevery,ei,isuniquelyassociatedwithsomen-tuple i =(u1,...,un) E ×···E×,wewilldenotee by (u ,...,u ). ∈ 1 ×···× n i 1 n Next, let N be the subspace of M generated by the vectors of the following type:

(u ,...,u + v ,...,u ) (u ,...,u,...,u ) (u ,...,v,...,u ), 1 i i n − 1 i n − 1 i n (u ,...,λu,...,u ) λ(u ,...,u,...,u ). 1 i n − 1 i n

We let E1 En be the quotient M/N of the free vector space M by N, π : M M/N be the quotient⊗··· map⊗ and set → ϕ = π ι. ◦ By construction, ϕ is multilinear, and since π is surjective and the ι(i)=ei generate M, since i is of the form i =(u ,...,u ) E E ,theϕ(u ,...,u )generateM/N.Thus, 1 n ∈ 1 ×···× n 1 n if we denote ϕ(u1,...,un)asu1 un,thetensorproductE1 En is generated by the vectors u u ,where⊗u···⊗E ,...,u E . ⊗···⊗ 1 ⊗···⊗ n 1 ∈ 1 n ∈ n For every multilinear map f : E1 En F ,ifalinearmapf : E1 En F ×···× → ⊗ ⊗···⊗ → exists such that f = f ϕ,sincethevectorsu1 un generate E1 En,themap ⊗ f is uniquely deﬁned by◦ ⊗···⊗ ⊗···⊗ ⊗

f (u1 un)=f(u1,...,un). ⊗ ⊗···⊗ (E1 En) On the other hand, because M = K ×···× is free on I = E1 En, there is a unique (E1 En) ×···× linear map f : K ×···× F ,suchthat → f = f ι, ◦ as in the diagram below:

ι (E1 En) E E K ×···× 1 ×···× n f f F Because f is multilinear, note that we must have f(w)=0,foreveryw N.Butthen, f : M F induces a linear map h: M/N F , such that ∈ → → f = h π ι, ◦ ◦ 592 CHAPTER 22. TENSOR ALGEBRAS by defining h([z]) = f(z), for every z M,where[z]denotestheequivalenceclassinM/N of z M: ∈ ∈ π ι (E1 En) E E ◦ K ×···× /N 1 ×···× n h f F Indeed, the fact that f vanishes on N insures that h is well defined on M/N,anditisclearly linear by definition. However, we showed that such a linear map h is unique, and thus it agrees with the linear map f defined by ⊗

f (u1 un)=f(u1,...,un) ⊗ ⊗···⊗ on the generators of E E . 1 ⊗···⊗ n What is important about Theorem 22.5 is not so much the construction itself but the fact that it produces a tensor product with the universal mapping property with respect to multilinear maps. Indeed, Theorem 22.5 yields a canonical isomorphism,

L(E E ,F) = L(E ,...,E ; F ), 1 ⊗···⊗ n ∼ 1 n between the vector space of linear maps, L(E1 En,F), and the vector space of multilinear maps, L(E ,...,E ; F ), via the linear⊗ map···⊗ ϕ deﬁned by 1 n −◦ h h ϕ, → ◦ where h L(E E ,F). Indeed, h ϕ is clearly multilinear, and since by Theorem ∈ 1 ⊗···⊗ n ◦ 22.5, for every multilinear map, f L(E1,...,En; F ), there is a unique linear map f ∈ ⊗ ∈ L(E1 En,F)suchthatf = f ϕ,themap ϕ is bijective. As a matter of fact, ⊗ its inverse⊗··· is⊗ the map ◦ −◦ f f . → ⊗ Using the “Hom” notation, the above canonical isomorphism is written

Hom(E E ,F) = Hom(E ,...,E ; F ). 1 ⊗···⊗ n ∼ 1 n

Remarks:

(1) To be very precise, since the tensor product depends on the ﬁeld, K,weshouldsubscript the symbol with K and write ⊗ E E . 1 ⊗K ···⊗K n However, we often omit the subscript K unless confusion may arise. 22.2. BASES OF TENSOR PRODUCTS 593

(2) For F = K,thebaseﬁeld,weobtainacanonicalisomorphismbetweenthevector space L(E E ,K), and the vector space of multilinear forms L(E ,...,E ; K). 1 ⊗···⊗ n 1 n However, L(E E ,K)isthedualspace,(E E )∗,andthus,thevector 1 ⊗···⊗ n 1 ⊗···⊗ n space of multilinear forms L(E1,...,En; K)iscanonicallyisomorphicto(E1 En)∗. We write ⊗···⊗ L(E ,...,E ; K) = (E E )∗. 1 n ∼ 1 ⊗···⊗ n

The fact that the map ϕ: E1 En E1 En is multilinear, can also be expressed as follows: ×···× → ⊗···⊗

u (u + v ) u =(u u u ) 1 ⊗···⊗ i i ⊗···⊗ n 1 ⊗···⊗ i ⊗···⊗ n +(u v u ), 1 ⊗···⊗ i ⊗···⊗ n u (λu ) u = λ(u u u ). 1 ⊗···⊗ i ⊗···⊗ n 1 ⊗···⊗ i ⊗···⊗ n Of course, this is just what we wanted! Tensors in E E are also called n-tensors, 1 ⊗···⊗ n and tensors of the form u1 un,whereui Ei,arecalledsimple (or indecomposable) n-tensors.Thosen-tensors⊗ that···⊗ are not simple are∈ often called compound n-tensors. Not only do tensor products act on spaces, but they also act on linear maps (they are functors). Given two linear maps f : E E and g : F F ,wecandeﬁneh: E F → → × → E F by ⊗ h(u, v)=f(u) g(v). ⊗ It is immediately veriﬁed that h is bilinear, and thus, it induces a unique linear map

f g : E F E F , ⊗ ⊗ → ⊗ such that (f g)(u v)=f(u) g(u). ⊗ ⊗ ⊗

If we also have linear maps f : E E and g : F F ,wecaneasilyverifythat → → the linear maps (f f) (g g)and(f g) (f g)agreeonallvectorsoftheform u v E F .Sincethesevectorsgenerate◦ ⊗ ◦ ⊗E ◦F ,weconcludethat⊗ ⊗ ∈ ⊗ ⊗ (f f) (g g)=(f g) (f g). ◦ ⊗ ◦ ⊗ ◦ ⊗

The generalization to the tensor product f1 fn of n 3linearmapsfi : Ei Fi is immediate, and left to the reader. ⊗···⊗ ≥ →

22.2 Bases of Tensor Products

We showed that E1 En is generated by the vectors of the form u1 un. However, there vectors are not⊗··· linearly⊗ independent. This situation can be ﬁxed⊗··· when⊗ considering bases, which is the object of the next proposition. 594 CHAPTER 22. TENSOR ALGEBRAS

k Proposition 22.6 Given n 2 vector spaces E1,...,En, if (ui )i Ik is a basis for Ek, ∈ 1 k n, then the family of≥ vectors ≤ ≤ 1 n (ui ui )(i1,...,in) I1 ... In 1 ⊗···⊗ n ∈ × × is a basis of the tensor product E E . 1 ⊗···⊗ n

Proof .Foreachk,1 k n,everyvk E can be written uniquely as ≤ ≤ ∈ k k k k v = vj uj , j I ∈ k k for some family of scalars (vj )j Ik .LetF be any nontrivial vector space. We show that for ∈ every family

(wi1,...,in )(i1,...,in) I1 ... In , ∈ × × of vectors in F ,thereissomelinearmaph: E E F , such that 1 ⊗···⊗ n → h(u1 un )=w . i1 ⊗···⊗ in i1,...,in Then, by Proposition 22.3, it follows that

1 n (ui ui )(i1,...,in) I1 ... In 1 ⊗···⊗ n ∈ × × k 1 n is linearly independent. However, since (ui )i Ik is a basis for Ek,theui ui also ∈ 1 n generate E E ,andthus,theyformabasisofE E . ⊗···⊗ 1 ⊗···⊗ n 1 ⊗···⊗ n We deﬁne the function f : E E F as follows: 1 ×···× n → f( v1 u1 ,..., vn un )= v1 vn w . j1 j1 jn jn j1 ··· jn j1,...,jn j1 I1 jn In j1 I1,...,jn In ∈ ∈ ∈ ∈ It is immediately veriﬁed that f is multilinear. By the universal mapping property of the tensor product, the linear map f : E1 En F such that f = f ϕ,isthedesired ⊗ ⊗ map h. ⊗···⊗ → ◦

In particular, when each Ik is ﬁnite and of size mk =dim(Ek), we see that the dimension k of the tensor product E1 En is m1 mn. As a corollary of Proposition 22.6, if (ui )i Ik ⊗···⊗ ··· ∈ is a basis for Ek,1 k n,theneverytensorz E1 En can be written in a unique way as ≤ ≤ ∈ ⊗···⊗ z = λ u1 un , i1,...,in i1 ⊗···⊗ in (i1,...,in) I1 ... In ∈ × × for some unique family of scalars λ K,allzeroexceptforaﬁnitenumber. i1,...,in ∈ 22.3. SOME USEFUL ISOMORPHISMS FOR TENSOR PRODUCTS 595 22.3 Some Useful Isomorphisms for Tensor Products

Proposition 22.7 Given 3 vector spaces E,F,G, there exists unique canonical isomorphisms

(1) E F F E ⊗ ⊗ (2) (E F ) G E (F G) E F G ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ (3) (E F ) G (E G) (F G) ⊕ ⊗ ⊗ ⊕ ⊗ (4) K E E ⊗ such that respectively

(a) u v v u ⊗ → ⊗ (b) (u v) w u (v w) u v w ⊗ ⊗ → ⊗ ⊗ → ⊗ ⊗ (c) (u, v) w (u w, v w) ⊗ → ⊗ ⊗ (d) λ u λu. ⊗ →

Proof . These isomorphisms are proved using the universal mapping property of tensor products. We illustrate the proof method on (2). Fix some w G.Themap ∈ (u, v) u v w → ⊗ ⊗ from E F to E F G is bilinear, and thus, there is a linear map fw : E F E F G, such that× f (u ⊗v)=⊗u v w. ⊗ → ⊗ ⊗ w ⊗ ⊗ ⊗ Next, consider the map (z, w) f (z), → w from (E F ) G into E F G.Itiseasilyseentobebilinear,andthus,itinducesa linear map⊗ × ⊗ ⊗ f :(E F ) G E F G, ⊗ ⊗ → ⊗ ⊗ such that f((u v) w)=u v w. ⊗ ⊗ ⊗ ⊗ Also consider the map (u, v, w) (u v) w → ⊗ ⊗ from E F G to (E F ) G.Itistrilinear,andthus,thereisalinearmap × × ⊗ ⊗ g : E F G (E F ) G, ⊗ ⊗ → ⊗ ⊗ such that g(u v w)=(u v) w.Clearly,f g and g f are identity maps, and thus, f and g are isomorphisms.⊗ ⊗ The⊗ other⊗ cases are similar.◦ ◦ 596 CHAPTER 22. TENSOR ALGEBRAS

Given any three vector spaces, E,F,G, we have the canonical isomorphism

Hom(E,F; G) ∼= Hom(E,Hom(F, G)). Indeed, any bilinear map, f : E F G,givesthelinearmap,ϕ(f) Hom(E,Hom(F, G)), where ϕ(f)(u) is the linear map× in Hom(→ F, G)givenby ∈

ϕ(f)(u)(v)=f(u, v).

Conversely, given a linear map, g Hom(E,Hom(F, G)), we get the bilinear map, ψ(g), given by ∈ ψ(g)(u, v)=g(u)(v), and it is clear that ϕ and ψ and mutual inverses. Consequently, we have the important corollary:

Proposition 22.8 For any three vector spaces, E,F,G, we have the canonical isomorphism,

Hom(E F, G) = Hom(E,Hom(F, G)), ⊗ ∼ 22.4 Duality for Tensor Products

In this section, all vector spaces are assumed to have finite dimension. Let us now see how tensor products behave under duality. For this, we define a pairing between E∗ E∗ and 1 ⊗···⊗ n E1 En as follows: For any fixed (v1∗,...,vn∗) E1∗ En∗,wehavethemultilinear map,⊗···⊗ ∈ ×···× lv ,...,v :(u1,...,un) v∗(u1) v∗(un), 1∗ n∗ → 1 ··· n from E1 En to K.Themaplv ,...,v extends uniquely to a linear map, ×···× 1∗ n∗ Lv ,...,v : E1 En K. We also have the multilinear map, 1∗ n∗ ⊗···⊗ −→

(v∗,...,v∗) Lv ,...,v , 1 n → 1∗ n∗ from E∗ E∗ to Hom(E E ,K), which extends to a linear map, L,from 1 ×···× n 1 ⊗···⊗ n E∗ E∗ to Hom(E E ,K). However, in view of the isomorphism, 1 ⊗···⊗ n 1 ⊗···⊗ n Hom(U V,W) = Hom(U, Hom(V,W)), ⊗ ∼ we can view L as a linear map,

L:(E∗ E∗) (E E ) K, 1 ⊗···⊗ n ⊗ 1 ⊗···⊗ n → which corresponds to a bilinear map,

(E∗ E∗) (E E ) K, 1 ⊗···⊗ n × 1 ⊗···⊗ n −→ 22.4. DUALITY FOR TENSOR PRODUCTS 597

via the isomorphism (U V )∗ ∼= L(U, V ; K). It is easy to check that this bilinear map is nondegenerate and thus,⊗ by Proposition 22.1, we have a canonical isomorphism,

(E E )∗ = E∗ E∗. 1 ⊗···⊗ n ∼ 1 ⊗···⊗ n

This, together with the isomorphism, L(E1,...,En; K) ∼= (E1 En)∗,yieldsacanonical isomorphism ⊗···⊗ L(E ,...,E ; K) = E∗ E∗. 1 n ∼ 1 ⊗···⊗ n We prove another useful canonical isomorphism that allows us to treat linear maps as tensors.

Let E and F be two vector spaces and let α: E∗ F Hom(E,F)bethemapdeﬁned such that × → α(u∗,f)(x)=u∗(x)f,

for all u∗ E∗, f F ,andx E. This map is clearly bilinear and thus, it induces a linear map, ∈ ∈ ∈ α : E∗ F Hom(E,F), ⊗ ⊗ → such that α (u∗ f)(x)=u∗(x)f. ⊗ ⊗ Proposition 22.9 If E and F are vector spaces with E of ﬁnite dimension, then the linear map, α : E∗ F Hom(E,F), is a canonical isomorphism. ⊗ ⊗ →

Proof .Let(ej)1 j n be a basis of E and, as usual, let ej∗ E∗ be the linear form deﬁned by ≤ ≤ ∈

ej∗(ek)=δj,k,

where δj,k =1iﬀj = k and 0 otherwise. We know that (ej∗)1 j n is a basis of E∗ (this is ≤ ≤ where we use the ﬁnite dimension of E). Now, for any linear map, f Hom(E,F), for every x = x e + + x e E,wehave ∈ 1 1 ··· n n ∈

f(x)=f(x e + + x e )=x f(e )+ + x f(e )=e∗(x)f(e )+ + e∗ (x)f(e ). 1 1 ··· n n 1 1 ··· n n 1 1 ··· n n Consequently, every linear map, f Hom(E,F), can be expressed as ∈

f(x)=e∗(x)f + + e∗ (x)f , 1 1 ··· n n

for some fi F .Furthermore,ifweapplyf to ei,wegetf(ei)=fi,sothefi are unique. Observe that∈

n n

(α (e1∗ f1 + + en∗ fn))(x)= (α (ei∗ fi))(x)= ei∗(x)fi. ⊗ ⊗ ··· ⊗ ⊗ ⊗ i=1 i=1 598 CHAPTER 22. TENSOR ALGEBRAS

Thus, α is surjective. As (ej∗)1 j n is a basis of E∗,thetensorsej∗ f,withf F ,span ⊗ ≤ ≤ n ⊗ ∈ E∗ F .Thus,everyelementofE∗ F is of the form e∗ f ,forsomef F . Assume ⊗ ⊗ i=1 i ⊗ i i ∈ n n α ( ei∗ fi)=α ( ei∗ fi)=f, ⊗ ⊗ ⊗ ⊗ i=1 i=1 for some f ,f F and some f Hom(E,F). Then for every x E, i i ∈ ∈ ∈ n n

ei∗(x)fi = ei∗(x)fi = f(x). i=1 i=1 Since the fi and fi are uniquely determined by the linear map, f,wemusthavefi = fi and α is injective. Therefore, α is a bijection. ⊗ ⊗ Note that in Proposition 22.9, the space F may have infinite dimension but E has finite dimension. In view of the canonical isomorphism Hom(E ,...,E ; F ) = Hom(E E ,F) 1 n ∼ 1 ⊗···⊗ n and the canonical isomorphism (E1 En)∗ ∼= E1∗ En∗,wheretheEi’s are finite- dimensional, Proposition 22.9 yields⊗ the··· canonical⊗ isomorphism⊗···⊗

Hom(E ,...,E ; F ) = E∗ E∗ F. 1 n ∼ 1 ⊗···⊗ n ⊗ 22.5 Tensor Algebras

The tensor product V V ⊗···⊗ m is also denoted as m m V or V ⊗

1 0 and is called the m-th tensor power ofV (with V ⊗ = V ,andV ⊗ = K). We can pack all the tensor powers of V into the “big” vector space,

m T (V )= V ⊗ , m 0 ≥ also denoted T •(V ), to avoid confusion with the tangent bundle. This is an interesting object because we can define a multiplication operation on it which makes it into an algebra called the tensor algebra of V . When V is of finite dimension n,thisspacecorrespondstothe algebra of polynomials with coefficients in K in n noncommuting variables. Let us recall the definition of an algebra over a field. Let K denote any (commutative) field, although for our purposes, we may assume that K = R (and occasionally, K = C). Since we will only be dealing with associative algebras with a multiplicative unit, we only define algebras of this kind. 22.5. TENSOR ALGEBRAS 599

Deﬁnition 22.3 Given a ﬁeld, K,aK-algebra is a K-vector space, A,togetherwitha bilinear operation, : A A A,calledmultiplication,whichmakesA into a ring with · × → unity, 1 (or 1A,whenwewanttobeveryprecise).Thismeansthat is associative and that there is a multiplicative identity element, 1, so that 1 a = a 1=·a,foralla A.Given two K-algebras A and B,aK-algebra homomorphism,· h: A · B,isalinearmapthatis∈ → also a ring homomorphism, with h(1A)=1B.

For example, the ring, M (K), of all n n matrices over a field, K,isaK-algebra. n × There is an obvious notion of ideal of a K-algebra: An ideal, A A,isalinearsubspace of A that is also a two-sided ideal with respect to multiplication⊆ in A.IfthefieldK is understood, we usually simply say an algebra instead of a K-algebra. We would like to define a multiplication operation on T (V )whichmakesitintoaK- algebra. As i T (V )= V ⊗ , i 0 ≥ n for every i 0, there is a natural injection ι : V ⊗ T (V ), and in particular, an injection ≥ n → ι0 : K T (V ). The multiplicative unit, 1,ofT (V )istheimage,ι0(1), in T (V )oftheunit, 1, of the→ field K.Sinceeveryv T (V )canbeexpressedasafinitesum ∈ v = ι (v )+ + ι (v ), n1 1 ··· nk k

ni where vi V ⊗ and the ni are natural numbers with ni = nj if i = j,todefinemultiplica- tion in T (∈V ), using bilinearity, it is enough to define multiplication operations, m n (m+n) n n : V ⊗ V ⊗ V ⊗ , which, using the isomorphisms, V ⊗ = ιn(V ⊗ ), yield multi- · × −→ m n (m+n) ∼ plication operations, : ιm(V ⊗ ) ιn(V ⊗ ) ιm+n(V ⊗ ). More precisely, we use the canonical isomorphism,· × −→ m n (m+n) V ⊗ V ⊗ = V ⊗ , ⊗ ∼ which defines a bilinear operation,

m n (m+n) V ⊗ V ⊗ V ⊗ , × −→ m n (m+n) which is taken as the multiplication operation. The isomorphism V ⊗ V ⊗ ∼= V ⊗ can be established by proving the isomorphisms ⊗

m n m V ⊗ V ⊗ = V ⊗ V V ⊗ ∼ ⊗ ⊗···⊗ n m (m+n) V ⊗ V V = V ⊗ , ⊗ ⊗···⊗ ∼ n which can be shown using methods similar to those used to proved associativity. Of course, m n (m+n) the multiplication, V ⊗ V ⊗ V ⊗ ,isdeﬁnedsothat × −→ (v v ) (w w )=v v w w . 1 ⊗···⊗ m · 1 ⊗···⊗ n 1 ⊗···⊗ m ⊗ 1 ⊗···⊗ n 600 CHAPTER 22. TENSOR ALGEBRAS

(This has to be made rigorous by using isomorphisms involving the associativity of tensor products, for details, see see Atiyah and Macdonald [9].)

Remark: It is important to note that multiplication in T (V )isnot commutative. Also, in all rigor, the unit, 1,ofT (V )isnot equal to 1, the unit of the ﬁeld K. However, in view of the injection ι0 : K T (V ), for the sake of notational simplicity, we will denote 1 by 1. → n n More generally, in view of the injections ιn : V ⊗ T (V ), we identify elements of V ⊗ with their images in T (V ). → The algebra, T (V ), satisﬁes a universal mapping property which shows that it is unique up to isomorphism. For simplicity of notation, let i: V T (V )bethenaturalinjectionof V into T (V ). →

Proposition 22.10 Given any K-algebra, A, for any linear map, f : V A, there is a unique K-algebra homomorphism, f : T (V ) A, so that → → f = f i, ◦ as in the diagram below: i V T (V )

f f A Proof .Leftananexercise(useTheorem22.5). Most algebras of interest arise as well-chosen quotients of the tensor algebra T (V ). This is true for the exterior algebra, (V )(alsocalledGrassmann algebra), where we take the quotient of T (V )modulotheidealgeneratedbyallelementsoftheformv v,wherev V , and for the symmetric algebra, Sym( V ), where we take the quotient of T⊗(V )modulothe∈ ideal generated by all elements of the form v w w v,wherev, w V . ⊗ − ⊗ ∈ Algebras such as T (V ) are graded, in the sense that there is a sequence of subspaces, n V ⊗ T (V ), such that ⊆ n T (V )= V ⊗ k 0 ≥ m n (m+n) and the multiplication, , behaves well w.r.t. the grading, i.e., : V ⊗ V ⊗ V ⊗ . Generally, a K-algebra,⊗E,issaidtobeagraded algebra iﬀthere⊗ is a sequence× of→ subspaces, En E,suchthat ⊆ E = En k 0 ≥ (E0 = K)andthemultiplication, ,respectsthegrading,thatis, : Em En Em+n. Elements in En are called homogeneous· elements of rank (or degree) ·n. × → In diﬀerential geometry and in physics it is necessary to consider slightly more general tensors. 22.5. TENSOR ALGEBRAS 601

Deﬁnition 22.4 Given a vector space, V ,foranypairofnonnegativeintegers,(r, s), the tensor space, T r,s(V ), of type (r, s), is the tensor product

r,s r s T (V )=V ⊗ (V ∗)⊗ = V V V ∗ V ∗, ⊗ ⊗···⊗ ⊗ ⊗···⊗ r s

0,0 , with T (V )=K. We also deﬁne the tensor algebra , T• •(V ), as the coproduct

, r,s T • •(V )= T (V ). r,s 0 ≥ Tensors in T r,s(V )arecalledhomogeneous of degree (r, s).

r,0 r , Note that tensors in T (V )arejustour“oldtensors”inV ⊗ .WemakeT • •(V )intoan algebra by deﬁning multiplication operations,

T r1,s1 (V ) T r2,s2 (V ) T r1+r2,s1+s2 (V ), × −→ in the usual way, namely: For u = u u u∗ u∗ and 1 ⊗···⊗ r1 ⊗ 1 ⊗···⊗ s1 v = v v v∗ v∗ ,let 1 ⊗···⊗ r2 ⊗ 1 ⊗···⊗ s2

u v = u u v v u∗ u∗ v∗ v∗ . ⊗ 1 ⊗···⊗ r1 ⊗ 1 ⊗···⊗ r2 ⊗ 1 ⊗···⊗ s1 ⊗ 1 ⊗···⊗ s2

r s r s Denote by Hom(V , (V ∗) ; W )thevectorspaceofallmultilinearmapsfromV (V ∗) to W .Then,wehavetheuniversalmappingpropertywhichassertsthatthereisacanonical× isomorphism r,s r s Hom(T (V ),W) ∼= Hom(V , (V ∗) ; W ). In particular, r,s r s (T (V ))∗ ∼= Hom(V , (V ∗) ; K). For ﬁnite dimensional vector spaces, the duality of Section 22.4 is also easily extended to the tensor spaces T r,s(V ). We deﬁne the pairing

r,s r,s T (V ∗) T (V ) K × −→ as follows: If r,s v∗ = v∗ v∗ u u T (V ∗) 1 ⊗···⊗ r ⊗ r+1 ⊗···⊗ r+s ∈ and r,s u = u u v∗ v∗ T (V ), 1 ⊗···⊗ r ⊗ r+1 ⊗···⊗ r+s ∈ then (v∗,u)=v∗(u ) v∗ (u ). 1 1 ··· r+s r+s This is a nondegenerate pairing and thus, we get a canonical isomorphism,

r,s r,s (T (V ))∗ ∼= T (V ∗). 602 CHAPTER 22. TENSOR ALGEBRAS

Consequently, we get a canonical isomorphism,

r,s r s T (V ∗) ∼= Hom(V , (V ∗) ; K).

r,s r r,s Remark: The tensor spaces, T (V )arealsodenotedTs (V ). A tensor, α T (V )is said to be contravariant in the ﬁrst r arguments and covariant in the last s∈arguments. This terminology refers to the way tensors behave under coordinate changes. Given a basis, r,s (e1,...,en), of V ,if(e1∗,...,en∗ ) denotes the dual basis, then every tensor α T (V )is given by an expression of the form ∈

i1,...,ir α = a e e e∗ e∗ . j1,...,js i1 ⊗···⊗ ir ⊗ j1 ⊗···⊗ js i1,...,ir j1,...,js The tradition in classical tensor notation is to use lower indices on vectors and upper indices on linear forms and in accordance to Einstein summation convention (or Einstein notation) the position of the indices on the coeﬃcients is reversed. Einstein summation convention is to assume that a summation is performed for all values of every index that appears simul- taneously once as an upper index and once as a lower index. According to this convention, the tensor α above is written

α = ai1,...,ir e e ej1 ejs . j1,...,js i1 ⊗···⊗ ir ⊗ ⊗···⊗ An older view of tensors is that they are multidimensional arrays of coeﬃcients,

i1,...,ir aj1,...,js , subject to the rules for changes of bases. Another operation on general tensors, contraction, is useful in diﬀerential geometry.

r,s r 1,s 1 Deﬁnition 22.5 For all r, s 1, the contraction, ci,j : T (V ) T − − (V ), with 1 i r and 1 j s,isthelinearmapdeﬁnedongeneratorsby≥ → ≤ ≤ ≤ ≤

c (u u v∗ v∗) i,j 1 ⊗···⊗ r ⊗ 1 ⊗···⊗ s = v∗(u ) u u u v∗ v∗ v∗, j i 1 ⊗···⊗ i ⊗···⊗ r ⊗ 1 ⊗···⊗ j ⊗···⊗ s where the hat over an argument means that it should be omitted.

1,1 Let us ﬁgure our what is c1,1 : T (V ) R,thatisc1,1 : V V ∗ R.If(e1,...,en) → ⊗ → is a basis of V and (e1∗,...,en∗ )isthedualbasis,everyh V V ∗ ∼= Hom(V,V )canbe expressed as ∈ ⊗ n h = a e e∗. ij i ⊗ j i,j=1 22.5. TENSOR ALGEBRAS 603

As c (e e∗)=δ , 1,1 i ⊗ j i,j we get n

c1,1(h)= aii =tr(h), i=1 where tr(h)isthetrace of h, where h is viewed as the linear map given by the matrix, (aij). Actually, since c1,1 is deﬁned independently of any basis, c1,1 provides an intrinsic deﬁnition of the trace of a linear map, h Hom(V,V ). ∈

Remark: Using the Einstein summation convention, if

α = ai1,...,ir e e ej1 ejs , j1,...,js i1 ⊗···⊗ ir ⊗ ⊗···⊗ then

i1,...,ik 1,i,ik+1...,ir j1 js − jl ck,l(α)=aj1,...,jl 1,i,jl+1,...,js ei1 eik eir e e e . − ⊗···⊗ ⊗···⊗ ⊗ ⊗···⊗ ⊗···⊗

If E and F are two K-algebras, we know that their tensor product,E F ,existsasa vector space. We can make E F into an algebra as well. Indeed, we have⊗ the multilinear map ⊗ E F E F E F × × × −→ ⊗ given by (a, b, c, d) (ac) (bd), where ac is the product of a and c in E and bd is the product of b and d in→F . By⊗ the universal mapping property, we get a linear map,

E F E F E F. ⊗ ⊗ ⊗ −→ ⊗ Using the isomorphism,

E F E F = (E F ) (E F ), ⊗ ⊗ ⊗ ∼ ⊗ ⊗ ⊗ we get a linear map, (E F ) (E F ) E F, ⊗ ⊗ ⊗ −→ ⊗ and thus, a bilinear map, (E F ) (E F ) E F, ⊗ × ⊗ −→ ⊗ which is our multiplication operation in E F .Thismultiplicationisdeterminedby ⊗ (a b) (c d)=(ac) (bd). ⊗ · ⊗ ⊗ One immediately checks that E F with this multiplication is a K-algebra. ⊗ We now turn to symmetric tensors. 604 CHAPTER 22. TENSOR ALGEBRAS 22.6 Symmetric Tensor Powers

Our goal is to come up with a notion of tensor product that will allow us to treat symmetric multilinear maps as linear maps. First, note that we have to restrict ourselves to a single vector space, E,ratherthenn vector spaces E1,...,En,sothatsymmetrymakessense. Recall that a multilinear map, f : En F ,issymmetric iﬀ →

f(uσ(1),...,uσ(n))=f(u1,...,un),

for all u E and all permutations, σ : 1,...,n 1,...,n .Thegroupofpermutations i ∈ { }→{ } on 1,...,n (the symmetric group)isdenotedSn.Thevectorspaceofallsymmetric multilinear{ maps,} f : En F ,isdenotedbySn(E; F ). Note that S1(E; F ) = Hom(E,F). → We could proceed directly as in Theorem 22.5, and construct symmetric tensor products from scratch. However, since we already have the notion of a tensor product, there is a more economical method. First, we deﬁne symmetric tensor powers.

Deﬁnition 22.6 An n-th symmetric tensor power of a vector space E,wheren 1, is a vector space S, together with a symmetric multilinear map ϕ: En S,suchthat,forevery≥ vector space F and for every symmetric multilinear map f : En F→,thereisauniquelinear map f : S F ,with → ⊙ → f(u1,...,un)=f (ϕ(u1,...,un)), ⊙ for all u1,...,un E,orforshort ∈ f = f ϕ. ⊙ ◦ Equivalently, there is a unique linear map f such that the following diagram commutes: ⊙ ϕ En S f f ⊙ F

First, we show that any two symmetric n-th tensor powers (S1,ϕ1)and(S2,ϕ2)forE, are isomorphic.

Proposition 22.11 Given any two symmetric n-th tensor powers (S1,ϕ1) and (S2,ϕ2) for E, there is an isomorphism h: S S such that 1 → 2 ϕ = h ϕ . 2 ◦ 1

Proof .Replacetensorproductbyn-th symmetric tensor power in the proof of Proposition 22.4. We now give a construction that produces a symmetric n-th tensor power of a vector space E. 22.6. SYMMETRIC TENSOR POWERS 605

Theorem 22.12 Given a vector space E, a symmetric n-th tensor power (Symn(E),ϕ) for E can be constructed (n 1). Furthermore, denoting ϕ(u1,...,un) as u1 un, ≥ n ⊙···⊙ the symmetric tensor power Sym (E) is generated by the vectors u1 un, where n ⊙···⊙ u1,...,un E, and for every symmetric multilinear map f : E F , the unique linear map f :Sym∈ n(E) F such that f = f ϕ, is deﬁned by → ⊙ → ⊙ ◦

f (u1 un)=f(u1,...,un), ⊙ ⊙···⊙ on the generators u u of Symn(E). 1 ⊙···⊙ n

n Proof .ThetensorpowerE⊗ is too big, and thus, we deﬁne an appropriate quotient. Let n C be the subspace of E⊗ generated by the vectors of the form

u u u u , 1 ⊗···⊗ n − σ(1) ⊗···⊗ σ(n)

for all ui E,andallpermutationsσ : 1,...,n 1,...,n .Weclaimthatthequotient ∈n { }→{ } space (E⊗ )/C does the job. n n n n Let p: E⊗ (E⊗ )/C be the quotient map. Let ϕ: E (E⊗ )/C be the map → → (u ,...,u ) p(u u ), 1 n → 1 ⊗···⊗ n or equivalently, ϕ = p ϕ ,whereϕ (u ,...,u )=u u . ◦ 0 0 1 n 1 ⊗···⊗ n Let us denote ϕ(u1,...,un)asu1 un.Itisclearthatϕ is symmetric. Since the n ⊙···⊙ vectors u1 un generate E⊗ ,andp is surjective, the vectors u1 un generate n ⊗···⊗ ⊙···⊙ (E⊗ )/C. n n Given any symmetric multilinear map f : E F , there is a linear map f : E⊗ F → ⊗ → such that f = f ϕ0,asinthediagrambelow: ⊗ ◦ n ϕ0 n E E⊗ f f ⊗ F

n However, since f is symmetric, we have f (z)=0foreveryz E⊗ .Thus,wegetan n ⊗ ∈ induced linear map h:(E⊗ )/C F ,suchthath([z]) = f (z), where [z] is the equivalence n n → ⊗ class in (E⊗ )/C of z E⊗ : ∈ n p ϕ0 n E ◦ (E⊗ )/C

h f F n However, if a linear map f :(E⊗ )/C F exists, since the vectors u1 un generate n ⊙ → ⊙···⊙ (E⊗ )/C,wemusthave f (u1 un)=f(u1,...,un), ⊙ ⊙···⊙ 606 CHAPTER 22. TENSOR ALGEBRAS

n n which shows that h and f agree. Thus, Sym (E)=(E⊗ )/C and ϕ constitute a symmetric ⊙ n-th tensor power of E. Again, the actual construction is not important. What is important is that the symmetric n-th power has the universal mapping property with respect to symmetric multilinear maps.

Remark: The notation for the commutative multiplication of symmetric tensor powers is not standard. Another notation⊙ commonly used is . We often abbreviate “symmetric tensor power” as “symmetric power”. The symmetric power,· Symn(E), is also denoted SymnE or S(E). To be consistent with the use of ,wecouldhaveusedthenotation n E.Clearly, 1 ⊙ 0 Sym (E) = E and it is convenient to set Sym (E)=K. ∼ The fact that the map ϕ: En Symn(E) is symmetric and multinear, can also be expressed as follows: →

u (u + v ) u =(u u u ) 1 ⊙···⊙ i i ⊙···⊙ n 1 ⊙···⊙ i ⊙···⊙ n +(u v u ), 1 ⊙···⊙ i ⊙···⊙ n u (λu ) u = λ(u u u ), 1 ⊙···⊙ i ⊙···⊙ n 1 ⊙···⊙ i ⊙···⊙ n u u = u u , σ(1) ⊙···⊙ σ(n) 1 ⊙···⊙ n for all permutations σ S . ∈ n The last identity shows that the “operation” is commutative. Thus, we can view the symmetric tensor u u as a multiset. ⊙ 1 ⊙···⊙ n Theorem 22.12 yields a canonical isomorphism

n n Hom(Sym (E),F) ∼= S(E ; F ), between the vector space of linear maps Hom(Symn(E),F), and the vector space of symmetric multilinear maps S(En; F ), via the linear map ϕ deﬁned by −◦ h h ϕ, → ◦ where h Hom(Symn(E),F). Indeed, h ϕ is clearly symmetric multilinear, and since by Theorem∈ 22.12, for every symmetric multilinear◦ map f S(En; F ), there is a unique linear map f Hom(Symn(E),F)suchthatf = f ϕ,themap∈ ϕ is bijective. As a matter ⊙ ⊙ of fact, its∈ inverse is the map ◦ −◦ f f . → ⊙ In particular, when F = K,wegetacanonicalisomorphism

n n (Sym (E))∗ ∼= S (E; K).

Symmetric tensors in Symn(E)arealsocalledsymmetric n-tensors,andtensorsofthe form u u ,whereu E,arecalledsimple (or decomposable) symmetric n-tensors. 1 ⊙···⊙ n i ∈ 22.7. BASES OF SYMMETRIC POWERS 607

Those symmetric n-tensors that are not simple are often called compound symmetric n- tensors. 2 Given two linear maps f : E E and g : E E,wecandeﬁneh: E E Sym (E) by → → × → h(u, v)=f(u) g(v). ⊙ It is immediately veriﬁed that h is symmetric bilinear, and thus, it induces a unique linear map 2 2 f g :Sym(E) Sym (E), ⊙ → such that (f g)(u v)=f(u) g(u). ⊙ ⊙ ⊙

If we also have linear maps f : E E and g : E E,wecaneasilyverifythat → →

(f f) (g g)=(f g) (f g). ◦ ⊙ ◦ ⊙ ◦ ⊙

The generalization to the symmetric tensor product f f of n 3linearmaps 1 ⊙···⊙ n ≥ f : E E is immediate, and left to the reader. i →

22.7 Bases of Symmetric Powers

n The vectors u1 un,whereu1,...,un E,generateSym (E), but they are not linearly independent. We⊙··· will⊙ prove a version of Proposition∈ 22.6 for symmetric tensor powers. For this, recall that a (finite) multiset over a set I is a function M : I N,suchthatM(i) =0 → for finitely many i I,andthatthesetofallmultisetsoverI is denoted as N(I).Welet ∈ dom(M)= i I M(i) =0 ,whichisafiniteset.Then,foranymultisetM N(I),note { ∈ | } ∈ that the sum i I M(i)makessense,since i I M(i)= i dom(M) M(i), and dom(M) ∈ ∈ ∈ is finite. For every multiset M (I),foranyn 2, we define the set J of functions N M η : 1,...,n dom(M), as follows:∈ ≥ { }→ 1 J = η η : 1,...,n dom(M), η− (i) = M(i),i dom(M), M(i)=n . M { | { }→ | | ∈ } i I ∈ 1 In other words, if i I M(i)=n and dom(M)= i1,...,ik , any function η JM specifies ∈ { } ∈ asequenceoflengthn,consistingofM(i1) occurrences of i1, M(i2) occurrences of i2,..., M(ik)occurrencesofik.Intuitively,anyη defines a “permutation” of the sequence (of length n) i ,...,i ,i,...,i ,...,i,...,i . 1 1 2 2 k k M(i1) M(i2) M(ik)

1Note that must have k n. ≤ 608 CHAPTER 22. TENSOR ALGEBRAS

Given any k 1, and any u E,wedenote ≥ ∈ u u ⊙···⊙ k

k as u⊙ . We can now prove the following Proposition.

Proposition 22.13 Given a vector space E, if (ui)i I is a basis for E, then the family of ∈ vectors

M(i1) M(i ) ⊙ ⊙ k ui1 uik ⊙···⊙ (I) M N , i I M(i)=n, i1,...,ik =dom(M) ∈ ∈ { } is a basis of the symmetric n-th tensor power Symn(E).

Proof .TheproofisverysimilartothatofProposition22.6.Foranynontrivialvectorspace F ,foranyfamilyofvectors

(w ) (I) , M M N , i I M(i)=n ∈ ∈ we show the existence of a symmetric multilinear map h:Symn(E) F ,suchthatforevery (I) → M N with i I M(i)=n,wehave ∈ ∈

M(i1) M(ik) h(u⊙ u⊙ )=wM , i1 ⊙···⊙ ik where i ,...,i =dom(M). We deﬁne the map f : En F as follows: { 1 k} →

f( v1 u1 ,..., vn un )= v1 vn w . j1 j1 jn jn η(1) ··· η(n) M j1 I jn I M N(I) η JM ∈ ∈ ∈ ∈ i I M(i)=n ∈ It is not diﬃcult to verify that f is symmetric and multilinear. By the universal mapping property of the symmetric tensor product, the linear map f :Symn(E) F such that ⊙ f = f ϕ,isthedesiredmaph.Then,byProposition22.3,itfollowsthatthefamily→ ⊙ ◦

M(i1) M(i ) ⊙ ⊙ k ui1 uik ⊙···⊙ (I) M N , i I M(i)=n, i1,...,ik =dom(M) ∈ ∈ { } is linearly independent. Using the commutativity of ,wecanalsoshowthatthesevectors generate Symn(E), and thus, they form a basis for⊙ Symn(E). The details are left as an exercise.

As a consequence, when I is ﬁnite, say of size p =dim(E), the dimension of Symn(E)is the number of ﬁnite multisets (j1,...,jp), such that j1 + + jp = n, jk 0. We leave as p+n 1 ··· ≥ an exercise to show that this number is n− .Thus,ifdim(E)=p,thenthedimensionof 22.8. SOME USEFUL ISOMORPHISMS FOR SYMMETRIC POWERS 609

n p+n 1 n n Sym (E)is n− . Compare with the dimension of E⊗ ,whichisp . In particular, when p =2,thedimensionofSymn(E)isn +1.Thiscanalsobeseendirectly. p+n 1 Remark: The number n− is also the number of homogeneous monomials Xj1 Xjp 1 ··· p of total degree n in p variables (we have j1 + + jp = n). This is not a coincidence! Symmetric tensor products are closely related to··· polynomials (for more on this, see the next remark).

Given a vector space E and a basis (ui)i I for E,Proposition22.13showsthatevery ∈ symmetric tensor z Symn(E)canbewritteninauniquewayas ∈ M(i1) M(ik) z = λM u⊙ u⊙ , i1 ⊙···⊙ ik M N(I) ∈ i I M(i)=n ∈ i1,...,i =dom(M) { k} for some unique family of scalars λ K,allzeroexceptforaﬁnitenumber. M ∈ This looks like a homogeneous polynomial of total degree n, where the monomials of total degree n are the symmetric tensors

M(i1) M(i ) u⊙ u⊙ k , i1 ⊙···⊙ ik in the “indeterminates” ui,wherei I (recall that M(i1)+ + M(ik)=n). Again, this is not a coincidence. Polynomials can∈ be deﬁned in terms of symmetric··· tensors.

22.8 Some Useful Isomorphisms for Symmetric Powers

We can show the following property of the symmetric tensor product, using the proof technique of Proposition 22.7:

n n k n k Sym (E F ) = Sym (E) Sym − (F ). ⊕ ∼ ⊗ k=0 22.9 Duality for Symmetric Powers

In this section, all vector spaces are assumed to have ﬁnite dimension. We deﬁne a nonde- n n generate pairing, Sym (E∗) Sym (E) K,asfollows:Considerthemultilinearmap, × −→ n n (E∗) E K, × −→ given by (v∗,...,v∗,u ,...,u ) v∗ (u ) v∗ (u ). 1 n 1 n → σ(1) 1 ··· σ(n) n σ Sn ∈ 610 CHAPTER 22. TENSOR ALGEBRAS

Note that the expression on the right-hand side is “almost” the determinant, det(vj∗(ui)), except that the sign sgn(σ)ismissing(wheresgn(σ)isthesignatureofthepermutation σ,thatis,theparityofthenumberoftranspositionsintowhichσ can be factored). Such an expression is called a permanent.Itiseasilycheckedthatthisexpressionissymmetric n w.r.t. the ui’s and also w.r.t. the vj∗.Foranyﬁxed(v1∗,...,vn∗) (E∗) ,wegetasymmetric multinear map, ∈

lv ,...,v :(u1,...,un) v∗ (u1) v∗ (un), 1∗ n∗ → σ(1) ··· σ(n) σ Sn ∈ n n from E to K.Themaplv1∗,...,vn∗ extends uniquely to a linear map, Lv1∗,...,vn∗ :Sym (E) K. Now, we also have the symmetric multilinear map, →

(v∗,...,v∗) Lv ,...,v , 1 n → 1∗ n∗ n n n from (E∗) to Hom(Sym (E),K), which extends to a linear map, L,fromSym(E∗)to Hom(Symn(E),K). However, in view of the isomorphism,

Hom(U V,W) = Hom(U, Hom(V,W)), ⊗ ∼ we can view L as a linear map,

n n L:Sym (E∗) Sym (E) K, ⊗ −→ which corresponds to a bilinear map,

n n Sym (E∗) Sym (E) K. × −→ Now, this pairing in nondegenerate. This can be done using bases and we leave it as an exercise to the reader (see Knapp [89], Appendix A). Therefore, we get a canonical isomorphism,

n n (Sym (E))∗ ∼= Sym (E∗). Since we also have an isomorphism

n n (Sym (E))∗ ∼= S (E,K), we get a canonical isomorphism

n n Sym (E∗) ∼= S (E,K) which allows us to interpret symmetric tensors over E∗ as symmetric multilinear maps.

n n Remark: The isomorphism, µ:Sym(E∗) ∼= S (E,K), discussed above can be described explicity as the linear extension of the map given by

µ(v∗ v∗)(u ,...,u )= v∗ (u ) v∗ (u ). 1 ⊙···⊙ n 1 n σ(1) 1 ··· σ(n) n σ Sn ∈ 22.10. SYMMETRIC ALGEBRAS 611

n n Now, the map from E to Sym (E)givenby(u1,...,un) u1 un yields a n n → ⊙···⊙ surjection, π : E⊗ Sym (E). Because we are dealing with vector spaces, this map has → n n some section, that is, there is some injection, ι:Sym (E) E⊗ ,withπ ι = id. If our field, K, has characteristic 0, then there is a special section having→ a natural◦ definition involving asymmetrizationprocessdefinedasfollows:Foreverypermutation,σ,wehavethemap, n n r : E E⊗ ,givenby σ → r (u ,...,u )=u u . σ 1 n σ(1) ⊗···⊗ σ(n) n n As rσ is clearly multilinear, rσ extends to a linear map, rσ : E⊗ E⊗ , and we get a map, n n → S E⊗ E⊗ ,namely, n × −→ σ z = r (z). · σ n It is immediately checked that this is a left action of the symmetric group, Sn,onE⊗ and n the tensors z E⊗ such that ∈ σ z = z, for all σ S · ∈ n n n are called symmetrized tensors. We define the map, ι: E E⊗ ,by → 1 1 ι(u ,...,u )= σ (u u )= u u . 1 n n! · 1 ⊗···⊗ n n! σ(1) ⊗···⊗ σ(n) σ Sn σ Sn ∈ ∈ n n As the right hand side is clearly symmetric, we get a linear map, ι:Sym(E) E⊗ . n n → Clearly, ι(Sym (E)) is the set of symmetrized tensors in E⊗ . If we consider the map, n n S = ι π : E⊗ E⊗ ,itiseasytocheckthatS S = S.Therefore,S is a projection and by linear◦ algebra,−→ we know that ◦

n n n E⊗ = S(E⊗ ) Ker S = ι(Sym (E)) Ker S. ⊕ ⊕ n It turns out that Ker S = E⊗ I =Kerπ,whereI is the two-sided ideal of T (E)generated ∩ 2 by all tensors of the form u v v u E⊗ (for example, see Knapp [89], Appendix A). Therefore, ι is injective, ⊗ − ⊗ ∈

n n n n E⊗ = ι(Sym (E)) E⊗ I = ι(Sym (E)) Ker π, ⊕ ∩ ⊕ n n and the symmetric tensor power, Sym (E), is naturally embedded into E⊗ .

22.10 Symmetric Algebras

As in the case of tensors, we can pack together all the symmetric powers, Symn(V ), into an algebra, Sym(V )= Symm(V ), m 0 ≥ 612 CHAPTER 22. TENSOR ALGEBRAS

called the symmetric tensor algebra of V . We could adapt what we did in Section 22.5 for general tensor powers to symmetric tensors but since we already have the algebra, T (V ), we can proceed faster. If I is the two-sided ideal generated by all tensors of the form 2 u v v u V ⊗ ,weset ⊗ − ⊗ ∈ Sym•(V )=T (V )/I.

Then, Sym•(V ) automatically inherits a multiplication operation which is commutative and since T (V )isgraded,thatis, m T (V )= V ⊗ , m 0 ≥ we have m m Sym•(V )= V ⊗ /(I V ⊗ ). ∩ m 0 ≥ However, it is easy to check that

m m m Sym (V ) = V ⊗ /(I V ⊗ ), ∼ ∩ so Sym•(V ) ∼= Sym(V ). When V is of finite dimension, n, T (V ) corresponds to the algebra of polynomials with coefficients in K in n variables (this can be seen from Proposition 22.13). When V is of infinite dimension and (ui)i I is a basis of V ,thealgebra,Sym(V ), corresponds to the ∈ algebra of polynomials in infinitely many variables in I. What’s nice about the symmetric tensor algebra, Sym(V ), is that it provides an intrinsic definition of a polynomial algebra in any set, I,ofvariables. It is also easy to see that Sym(V )satisfiesthefollowinguniversalmappingproperty: Proposition 22.14 Given any commutative K-algebra, A, for any linear map, f : V A, there is a unique K-algebra homomorphism, f :Sym(V ) A, so that → → f = f i, ◦ as in the diagram below: i V Sym(V )

f f A

n n Remark: If E is ﬁnite-dimensional, recall the isomorphism, µ:Sym(E∗) S (E,K), deﬁned as the linear extension of the map given by −→

µ(v∗ v∗)(u ,...,u )= v∗ (u ) v∗ (u ), 1 ⊙···⊙ n 1 n σ(1) 1 ··· σ(n) n σ Sn ∈ 22.11. EXTERIOR TENSOR POWERS 613

m n m+n Now, we have also a multiplication operation, Sym (E∗) Sym (E∗) Sym (E∗). The following question then arises: × −→ Can we deﬁne a multiplication, Sm(E,K) Sn(E,K) Sm+n(E,K), directly on symmetric multilinear forms, so that the following× diagram commutes:−→

m n m+n Sym (E∗) Sym (E∗) ⊙ Sym (E∗) × µ µ µ × Sm(E,K) Sn(E,K) · Sm+n(E,K). × The answer is yes!Thesolutionistodefinethismultiplicationsuchthat,forf Sm(E,K) and g Sn(E,K), ∈ ∈ (f g)(u ,...,u )= f(u ,...,u )g(u ,...,u ), · 1 m+n σ(1) σ(m) σ(m+1) σ(m+n) σ shuffle(m,n) ∈ where shuffle(m, n)consistsofall(m, n)-“shuffles”, that is, permutations, σ,of 1,...m+n , such that σ(1) < <σ(m)andσ(m +1)< <σ(m + n). We urge the reader{ to check} this fact. ··· ··· Another useful canonical isomorphim (of K-algebras) is Sym(E F ) = Sym(E) Sym(F ). ⊕ ∼ ⊗ 22.11 Exterior Tensor Powers

We now consider alternating (also called skew-symmetric)multilinearmapsandexterior tensor powers (also called alternating tensor powers), denoted n(E). In many respect, alternating multilinear maps and exterior tensor powers can be treated much like symmetric tensor powers except that the sign, sgn(σ), needs to be inserted in front of the formulae valid for symmetric powers. Roughly speaking, we are now in the world of determinants rather than in the world of permanents. However, there are also some fundamental differences, one of which being that the exterior tensor power, n(E), is the trivial vector space, (0), when E is finite-dimensional and when n>dim(E). As in the case of symmetric tensor powers, since we already have the tensor algebra, T (V ), we can proceed rather quickly. But first, let us review some basic definitions and facts. Definition 22.7 Let f : En F be a multilinear map. We say that f alternating iff f(u ,...,u )=0whenever u→= u , for some i with 1 i n 1, for all u E, 1 n i i+1 ≤ ≤ − i ∈ that is, f(u1,...,un)=0whenever two adjacent arguments are identical. We say that f is skew-symmetric (or anti-symmetric) iff

f(uσ(1),...,uσ(n)) = sgn(σ)f(u1,...,un), for every permutation, σ S , and all u E. ∈ n i ∈ 614 CHAPTER 22. TENSOR ALGEBRAS

For n =1,weagreethateverylinearmap,f : E F ,isalternating.Thevector space of all multilinear alternating maps, f : En F ,→ is denoted Altn(E; F ). Note that Alt1(E; F ) = Hom(E,F). The following basic proposition→ shows the relationship between alternation and skew-symmetry.

Proposition 22.15 Let f : En F be a multilinear map. If f is alternating, then the following properties hold: →

(1) For all i, with 1 i n 1, ≤ ≤ − f(...,u,u ,...)= f(...,u ,u,...). i i+1 − i+1 i (2) For every permutation, σ S , ∈ n

f(uσ(1),...,uσ(n))=sgn(σ)f(u1,...,un).

(3) For all i, j, with 1 i

f(...,ui,...uj,...)=0 whenever ui = uj.

Moreover, if our ﬁeld, K, has characteristic diﬀerent from 2, then every skew-symmetric multilinear map is alternating.

Proof .(i)Bymultilinearityappliedtwice,wehave

f(...,ui + ui+1,ui + ui+1,...)=f(...,ui,ui,...)+f(...,ui,ui+1,...)

+ f(...,ui+1,ui,...)+f(...,ui+1,ui+1,...).

Since f is alternating, we get

0=f(...,ui,ui+1,...)+f(...,ui+1,ui,...), that is, f(...,u,u ,...)= f(...,u ,u,...). i i+1 − i+1 i n (ii) Clearly, the symmetric group, Sn, acts on Alt (E; F )ontheleft,via

σ f(u ,...,u )=f(u ,...,u ). · 1 n σ(1) σ(n)

Consequently, as Sn is generated by the transpositions (permutations that swap exactly two elements), since for a transposition, (ii) is simply (i), we deduce (ii) by induction on the number of transpositions in σ.

(iii) There is a permutation, σ,thatsendsui and uj respectively to u1 and u2. As f is alternating, f(uσ(1),...,uσ(n))=0. 22.11. EXTERIOR TENSOR POWERS 615

However, by (ii), f(u1,...,un)=sgn(σ)f(uσ(1),...,uσ(n))=0.

Now, when f is skew-symmetric, if σ is the transposition swapping ui and ui+1 = ui,as sgn(σ)= 1, we get − f(...,u,u,...)= f(...,u,u,...), i i − i i so that 2f(...,ui,ui,...)=0,

and in every characteristic except 2, we conclude that f(...,ui,ui,...)=0,namely,f is alternating. Proposition 22.15 shows that in every characteristic except 2, alternating and skew- symmetric multilinear maps are identical. Using Proposition 22.15 we easily deduce the following crucial fact: Proposition 22.16 Let f : En F be an alternating multilinear map. For any families of vectors, (u ,...,u ) and (v ,...,v→ ), with u ,v E, if 1 n 1 n i i ∈ n v = a u , 1 j n, j ij i ≤ ≤ i=1 then

f(v ,...,v )= sgn(σ) a a f(u ,...,u )=det(A)f(u ,...,u ), 1 n σ(1),1 ··· σ(n),n 1 n 1 n σ Sn ∈ where A is the n n matrix, A =(a ). × ij Proof . Use property (ii) of Proposition 22.15. . We are now ready to deﬁne and construct exterior tensor powers. Deﬁnition 22.8 An n-th exterior tensor power of a vector space, E,wheren 1, is a vector space, A, together with an alternating multilinear map, ϕ: En A,suchthat,for≥ every vector space, F ,andforeveryalternatingmultilinearmap,f : E→n F ,thereisa unique linear map, f : A F ,with → ∧ → f(u1,...,un)=f (ϕ(u1,...,un)), ∧

for all u1,...,un E,orforshort ∈ f = f ϕ. ∧ ◦ Equivalently, there is a unique linear map f such that the following diagram commutes: ∧ ϕ En A

f f ∧ F 616 CHAPTER 22. TENSOR ALGEBRAS

First, we show that any two n-th exterior tensor powers (A1,ϕ1)and(A2,ϕ2)forE,are isomorphic.

Proposition 22.17 Given any two n-th exterior tensor powers (A1,ϕ1) and (A2,ϕ2) for E, there is an isomorphism h: A A such that 1 → 2 ϕ = h ϕ . 2 ◦ 1

Proof .Replacetensorproductbyn exterior tensor power in the proof of Proposition 22.4.

We now give a construction that produces an n-th exterior tensor power of a vector space E.

Theorem 22.18 Given a vector space E, an n-th exterior tensor power ( n(E),ϕ) for E can be constructed (n 1). Furthermore, denoting ϕ(u1,...,un) as u1 un, the exterior n ≥ ∧···∧ tensor power (E) is generated by the vectors u1 un, where u1,...,un E, and for every alternating multilinear map f : En F , the∧ unique···∧ linear map f : n(E∈) F such ∧ that f = f ϕ, is deﬁned by → → ∧ ◦

f (u1 un)=f(u1,...,un), ∧ ∧···∧ on the generators u u of n(E). 1 ∧···∧ n Proof sketch.Wecangiveaquickproofusingthetensoralgebra,T (E). let Ia be the 2 two-sided ideal of T (E)generatedbyalltensorsoftheformu u E⊗ .Then,let ⊗ ∈ n n n (E)=E⊗ /(I E⊗ ) a ∩ n n and let π be the projection, π : E⊗ (E). If we let u1 un = π(u1 un), it is easy to check that ( n(E), )satisﬁestheconditionsofTheorem22.18.→ ∧···∧ ⊗···⊗ ∧ Remark: We can also deﬁne

(E)=T (E)/Ia = (E), n 0 ≥ the exterior algebra of E. This is the skew-symmetric counterpart of Sym(E)andwewill study it a little later. For simplicity of notation, we may write n E for n(E). We also abbreviate “exterior 1 0 tensor power” as “exterior power”. Clearly, (E) ∼= E and it is convenient to set (E)= K. 22.11. EXTERIOR TENSOR POWERS 617

The fact that the map ϕ: En n(E)isalternatingandmultinear,canalsobeexpressed as follows: → u (u + v ) u =(u u u ) 1 ∧···∧ i i ∧···∧ n 1 ∧···∧ i ∧···∧ n +(u v u ), 1 ∧···∧ i ∧···∧ n u (λu ) u = λ(u u u ), 1 ∧···∧ i ∧···∧ n 1 ∧···∧ i ∧···∧ n u u =sgn(σ) u u , σ(1) ∧···∧ σ(n) 1 ∧···∧ n for all σ S . ∈ n Theorem 22.18 yields a canonical isomorphism n n Hom( (E),F) ∼= Alt (E; F ), between the vector space of linear maps Hom( n(E),F), and the vector space of alternating multilinear maps Altn(E; F ), via the linear map ϕ defined by −◦ h h ϕ, → ◦ where h Hom( n(E),F). In particular, when F = K,wegetacanonicalisomorphism ∈ n ∗ n (E) ∼= Alt (E; K). Tensors α n(E)arecalledalternating n-tensors or alternating tensors of degree n ∈ and we write deg(α)=n.Tensorsoftheformu1 un,whereui E,arecalledsimple (or decomposable) alternating n-tensors.Thosealternating∧···∧ n-tensors that∈ are not simple are n often called compound alternating n-tensors.Simpletensorsu1 un (E)arealso n ∧···∧ ∈ called n-vectors and tensors in (E∗)areoftencalled(alternating) n-forms. 2 Given two linear maps f : E E and g : E E,wecandefineh: E E (E)by → → × → h(u, v)=f(u) g(v). ∧ It is immediately verified that h is alternating bilinear, and thus, it induces a unique linear map 2 2 f g : (E) (E), ∧ → such that (f g)(u v)=f(u) g(u). ∧ ∧ ∧ If we also have linear maps f : E E and g : E E,wecaneasilyverifythat → → (f f) (g g)=(f g) (f g). ◦ ∧ ◦ ∧ ◦ ∧

The generalization to the alternating product f1 fn of n 3linearmapsfi : E E is immediate, and left to the reader. ∧···∧ ≥ → 618 CHAPTER 22. TENSOR ALGEBRAS 22.12 Bases of Exterior Powers

Let E be any vector space. For any basis, (ui)i Σ,forE,weassumethatsometotalordering, ∈ ,onΣ,hasbeenchosen.Callthepair((ui)i Σ, )anordered basis. Then, for any nonempty ∈ finite≤ subset, I Σ, let ≤ ⊆ u = u u , I i1 ∧···∧ im where I = i ,...,i ,withi < d, the exterior power n(E) is trivial, that is n(E)=(0). Otherwise, n for every ordered basis, ((ui)i Σ, ), the family, (uI ), is basis of (E), where I ranges over ∈ finite nonempty subsets of Σ of size≤ I =n. | | Proof . First, assume that E has finite dimension, d =dim(E)andthatn>d.Weknow that n(E)isgeneratedbythetensorsoftheformv v ,withv E.Ifu ,...,u 1 ∧···∧ n i ∈ 1 d is a basis of E,aseveryvi is a linear combination of the uj,whenweexpandv1 vn using multilinearity, we get a linear combination of the form ∧···∧

v v = λ u u , 1 ∧···∧ n (j1,...,jn) j1 ∧···∧ jn (j1,...,jn) where each (j ,...,j )issomesequenceofintegersj 1,...,d . As n>d,eachsequence 1 n k ∈{ } (j1,...,jn) must contain two identical elements. By alternation, uj1 ujn =0andso, v v =0.Itfollowsthat n(E)=(0). ∧···∧ 1 ∧···∧ n Now, assume that either dim(E)=d and that n d or that E is inﬁnite dimensional. ≤ The argument below shows that the uI are nonzero and linearly independent. As usual, let u∗ E∗ be the linear form given by i ∈

ui∗(uj)=δij.

For any nonempty subset, I = i1,...,in Σ, with i1 <

Note that when dim(E)=d and n d, the forms ui∗1 ,...,ui∗n are all distinct so, the above does hold. Since L (u )=1,weconcludethat≤ u = 0. Now, if we have a linear combination, I I I 

λI uI =0, I where the above sum is ﬁnite and involves nonempty ﬁnite subset, I Σ, with I = n,for ⊆ | | every such I,whenweapplyLI we get

λI =0, proving linear independence. As a corollary, if E is ﬁnite dimensional, say dim(E)=d and if 1 n d,thenwehave ≤ ≤ n n dim( (E)) = d and if n>d,thendim( n(E)) = 0.

Remark: When n =0,ifweset u =1,then(u )=(1)isabasisof 0(V )=K. ∅ ∅ It follows from Proposition 22.19 that the family, (uI )I ,whereI Σrangesoverﬁnite n ⊆ subsets of Σis a basis of (V )= n 0 (V ). ≥ As a corollary of Proposition 22.19 we obtain the following useful criterion for linear independence:

Proposition 22.20 For any vector space, E, the vectors, u1,...,un E, are linearly independent iﬀ u u =0. ∈ 1 ∧···∧ n Proof .Ifu1 un =0,thenu1,...,un must be linearly independent. Otherwise, some u would be∧ a··· linear∧ combination of the other u ’s (with j = i)andthen,asintheproof i j of Proposition 22.19, u1 un would be a linear combination of wedges in which two vectors are identical and∧ thus,···∧ zero.

Conversely, assume that u1,...,un are linearly independent. Then, we have the linear forms, u∗ E∗, such that i ∈ u∗(u )=δ 1 i, j n. i j i,j ≤ ≤ As in the proof of Proposition 22.19, we have a linear map, L : n(E) K,givenby u1,...,un → Lu ,...,u (v1 vn)=det(u∗(vi)), 1 n ∧···∧ j for all v v n(E). As, 1 ∧···∧ n ∈ Lu ,...,u (u1 un)=1, 1 n ∧···∧ we conclude that u u =0. 1 ∧···∧ n Proposition 22.20 shows that, geometrically, every nonzero wedge, u1 un,corresponds to some oriented version of an n-dimensional subspace of E. ∧···∧ 620 CHAPTER 22. TENSOR ALGEBRAS 22.13 Some Useful Isomorphisms for Exterior Powers

We can show the following property of the exterior tensor product, using the proof technique of Proposition 22.7: n n k n k − (E F ) = (E) (F ). ⊕ ∼ ⊗ k=0 22.14 Duality for Exterior Powers

In this section, all vector spaces are assumed to have ﬁnite dimension. We deﬁne a nonde- n n generate pairing, (E∗) (E) K,asfollows:Considerthemultilinearmap, × −→ n n (E∗) E K, × −→ given by

(v∗,...,v∗,u ,...,u ) sgn(σ) v∗ (u ) v∗ (u ) = det(v∗(u )). 1 n 1 n → σ(1) 1 ··· σ(n) n j i σ Sn ∈

It is easily checked that this expression is alternating w.r.t. the ui’s and also w.r.t. the vj∗. n For any ﬁxed (v∗,...,v∗) (E∗) ,wegetanalternatingmultinearmap, 1 n ∈

lv ,...,v :(u1,...,un) det(v∗(ui)), 1∗ n∗ → j from En to K.Bytheargumentusedinthesymmetriccase,wegetabilinearmap,

n n (E∗) (E) K. × −→ Now, this pairing in nondegenerate. This can be done using bases and we leave it as an exercise to the reader. Therefore, we get a canonical isomorphism,

n n ( (E))∗ ∼= (E∗). Since we also have a canonical isomorphism

n n ( (E))∗ ∼= Alt (E; K), we get a canonical isomorphism

n n (E∗) ∼= Alt (E; K) which allows us to interpret alternating tensors over E∗ as alternating multilinear maps. 22.14. DUALITY FOR EXTERIOR POWERS 621

n n The isomorphism, µ: (E∗) ∼= Alt (E; K), discussed above can be described explicity as the linear extension of the map given by 

µ(v∗ v∗)(u ,...,u )=det(v∗(u )). 1 ∧···∧ n 1 n j i

Remark: Variants of our isomorphism, µ,arefoundintheliterature.Forexample,there is a version, µ,where 1 µ = µ, n! 1 with the factor n! added in front of the determinant. Each version has its its own merits and inconvenients. Morita [114] uses µ because it is more convenient than µ when dealing with characteristic classes. On the other hand, when using µ,someextrafactorisneeded in deﬁning the wedge operation of alternating multilinear forms (see Section 22.15) and for exterior diﬀerentiation. The version µ is the one adopted by Warner [147], Knapp [89], Fulton and Harris [57] and Cartan [29, 30].

If f : E F is any linear map, by transposition we get a linear map, f : F ∗ E∗, given by → → f (v∗)=v∗ f, v∗ F ∗. ◦ ∈ Consequently, we have

f (v∗)(u)=v∗(f(u)), for all u E and all v∗ F ∗. ∈ ∈ For any p 1, the map, ≥ (u ,...,u ) f(u ) f(u ), 1 p → 1 ∧···∧ p from En to p F is multilinear alternating, so it induces a linear map, p f : p E p F , defined on generators by → p f (u u )=f(u ) f(u ). 1 ∧···∧ p 1 ∧···∧ p p p p p Combining and duality, we get a linear map, f : F ∗ E∗,definedongener- ators by → p f (v∗ v∗)=f (v∗) f (v∗). 1 ∧···∧ p 1 ∧···∧ p Proposition 22.21 If f : E F is any linear map between two finite-dimensional vector spaces, E and F , then →

p p µ f (ω) (u ,...,u )=µ(ω)(f(u ),...,f(u )),ω F ∗,u,...,u E. 1 p 1 p ∈ 1 p ∈ 622 CHAPTER 22. TENSOR ALGEBRAS

Proof .Itisenoughtoprovetheformulaongenerators.Bydeﬁnitionofµ, we have

p µ f (v∗ v∗) (u ,...,u )=µ(f (v∗) f (v∗))(u ,...,u ) 1 ∧···∧ p 1 p 1 ∧···∧ p 1 p =det(f (vj∗)(ui))

=det(vj∗(f(ui))) = µ(v∗ v∗)(f(u ),...,f(u )), 1 ∧···∧ p 1 p as claimed. p The map f is often denoted f ∗,althoughthisisanambiguousnotationsincep is p dropped. Proposition 22.21 gives us the behavior of f ∗ under the identification of E∗ and Altp(E; K) viathe isomorphism µ. n n As in the case of symmetric powers, the map from E to (E)givenby(u1,...,un) n n → u1 un yields a surjection, π : E⊗ (E). Now, this map has some section so there ∧···∧ n n → is some injection, ι: (E) E⊗ ,withπ ι =id.Ifourfield,K,hascharacteristic0, then there is a special section→ having a natural ◦ definition involving an antisymmetrization process.

n Recall that we have a left action of the symmetric group, Sn,onE⊗ .Thetensors, n z E⊗ ,suchthat ∈ σ z =sgn(σ) z, for all σ S · ∈ n n n are called antisymmetrized tensors. We deﬁne the map, ι: E E⊗ ,by → 1 ι(u ,...,u )= sgn(σ) u u . 1 n n! σ(1) ⊗···⊗ σ(n) σ Sn ∈ n n As the right hand side is clearly an alternating map, we get a linear map, ι: (E) E⊗ . n n → Clearly, ι( (E)) is the set of antisymmetrized tensors in E⊗ .Ifweconsiderthemap, n n A = ι π : E⊗ E⊗ ,itiseasytocheckthatA A = A.Therefore,A is a projection and by◦ linear algebra,−→ we know that ◦

n n n E⊗ = A(E⊗ ) Ker A = ι( (A)) Ker A. ⊕ ⊕ n It turns out that Ker A = E⊗ Ia =Kerπ,whereIa is the two-sided ideal of T (E) ∩ 2 generated by all tensors of the form u u E⊗ (for example, see Knapp [89], Appendix A). Therefore, ι is injective, ⊗ ∈

n n n n E⊗ = ι( (E)) E⊗ I = ι( (E)) Ker π, ⊕ ∩ ⊕

n n and the exterior tensor power, (E), is naturally embedded into E⊗ . 22.15. EXTERIOR ALGEBRAS 623 22.15 Exterior Algebras

As in the case of symmetric tensors, we can pack together all the exterior powers, n(V ), into an algebra, m (V )= (V ), m 0 ≥ called the exterior algebra (or Grassmann algebra) of V .Wemimictheprocedureused for symmetric powers. If Ia is the two-sided ideal generated by all tensors of the form 2 u u V ⊗ ,weset ⊗ ∈ • (V )=T (V )/Ia.

Then, •(V ) automatically inherits a multiplication operation, called wedge product,and since T (V )isgraded,thatis, m T (V )= V ⊗ , m 0 ≥ we have • m m (V )= V ⊗ /(I V ⊗ ). a ∩ m 0 ≥ However, it is easy to check that

m m m (V ) = V ⊗ /(I V ⊗ ), ∼ a ∩ so • (V ) ∼= (V ). When V has ﬁnite dimension, d,weactuallyhaveaﬁnitecoproduct 

d m (V )= (V ), m=0 m d and since each (V )hasdimension, m ,wededucethat dim( (V )) = 2d =2dim(V ). The multiplication, : m(V ) n(V ) m+n(V ), is skew-symmetric in the following precise sense: ∧ × → Proposition 22.22 For all α m(V ) and all β n(V ), we have ∈ ∈ β α =( 1)mnα β. ∧ − ∧ 624 CHAPTER 22. TENSOR ALGEBRAS

Proof .Sincev u = u v for all u, v V , Proposition 22.22 follows by induction. ∧ − ∧ ∈ Since α α =0foreverysimple tensor, α = u1 un,itseemsnaturaltoinferthat α α =0for∧ every tensor α (V ). If we consider∧··· the∧ case where dim(V ) 3, we can indeed∧ prove the above assertion.∈ However, if dim(V ) 4, the above fact is generally≤ false! ≥ For example, when dim(V )=4,ifu1,u2,u3,u4 are a basis for V ,forα = u1 u2 + u3 u4, we check that ∧ ∧ α α =2u u u u , ∧ 1 ∧ 2 ∧ 3 ∧ 4 which is nonzero. The above discussion suggests that it might be useful to know when an alternating tensor is simple, that is, decomposable. It can be shown that for tensors, α 2(V ), α α =0iﬀ α is simple. A general criterion for decomposability can be given in terms∈ of some∧ operations known as left hook and right hook (also called interior products), see Section 22.17. It is easy to see that (V )satisﬁesthefollowinguniversalmappingproperty: Proposition 22.23 Given any K-algebra, A, for any linear map, f : V A, if (f(v))2 =0 for all v V , then there is a unique K-algebra homomorphism, f : (V→) A, so that ∈ → f = f i, ◦ as in the diagram below: i V (V )

f f A n n When E is finite-dimensional, recall the isomorphism, µ: (E∗) Alt (E; K), defined as the linear extension of the map given by −→ µ(v∗ v∗)(u ,...,u ) = det(u∗(u )). 1 ∧···∧ n 1 n j i m n m+n Now, we have also a multiplication operation, (E∗) (E∗) (E∗). The following question then arises: × −→ Can we define a multiplication, Altm(E; K) Altn(E; K) Altm+n(E; K), directly on alternating multilinear forms, so that the following× diagram commutes:−→

m n m+n (E∗) (E∗) ∧ (E∗) × µ µ µ × m n m+n Alt (E; K) Alt (E; K) ∧ Alt (E; K). × As in the symmetric case, the answer is yes! The solution is to deﬁne this multiplication such that, for f Altm(E; K)andg Altn(E; K), ∈ ∈ (f g)(u ,...,u )= sgn(σ) f(u ,...,u )g(u ,...,u ), ∧ 1 m+n σ(1) σ(m) σ(m+1) σ(m+n) σ shuﬄe(m,n) ∈ 22.15. EXTERIOR ALGEBRAS 625

where shuﬄe(m, n)consistsofall(m, n)-“shuﬄes”, that is, permutations, σ,of 1,...m+n , such that σ(1) < <σ(m)andσ(m+1) < <σ(m+n). For example, when{ m = n =1,} we have ··· ··· (f g)(u, v)=f(u)g(v) g(u)f(v). ∧ − When m =1andn 2, check that ≥ m+1 i 1 (f g)(u ,...,u )= ( 1) − f(u )g(u ,...,u ,...,u ), ∧ 1 m+1 − i 1 i m+1 i=1 where the hat over the argument ui means that it should be omitted. As a result of all this, the coproduct

Alt(E)= Altn(E; K) n 0 ≥

is an algebra under the above multiplication and this algebra is isomorphic to (E∗). For the record, we state n n Proposition 22.24 When E is ﬁnite dimensional, the maps, µ: (E∗) Alt (E; K), induced by the linear extensions of the maps given by −→

µ(v∗ v∗)(u ,...,u ) = det(u∗(u )) 1 ∧···∧ n 1 n j i

yield a canonical isomorphism of algebras, µ: (E∗) Alt(E), where the multiplication in Alt(E) is defined by the maps, : Altm(E; K) Alt−→n(E; K) Altm+n(E; K), with ∧ × −→ (f g)(u ,...,u )= sgn(σ) f(u ,...,u )g(u ,...,u ), ∧ 1 m+n σ(1) σ(m) σ(m+1) σ(m+n) σ shuffle(m,n) ∈ where shuffle(m, n) consists of all (m, n)-“shuffles”, that is, permutations, σ, of 1,...m+n , such that σ(1) < <σ(m) and σ(m +1)< <σ(m + n). { } ··· ···

Remark: The algebra, (E) is a graded algebra. Given two graded algebras, E and F ,we can make a new tensor product, E F ,whereE F is equal to E F as a vector space, but with a skew-commutative multiplication⊗ given⊗ by ⊗ (a b) (c d)=( 1)deg(b)deg(c)(ac) (bd), ⊗ ∧ ⊗ − ⊗ where a Em,b F p, c En,d F q.Then,itcanbeshownthat ∈ ∈ ∈ ∈ (E F ) = (E) (F ). ⊕ ∼ ⊗ 626 CHAPTER 22. TENSOR ALGEBRAS 22.16 The Hodge -Operator ∗ In order to define a generalization of the Laplacian that will apply to differential forms on a Riemannian manifold, we need to define isomorphisms,

k n k − V V, −→ for any Euclidean vector space, V ,ofdimension n and any k,with0 k n.If , denotes the inner product on V ,wedeﬁneaninnerproducton k V ,alsodenoted≤ ≤ −, −, by setting − − u u ,v v =det(u ,v), 1 ∧···∧ k 1 ∧···∧ k i j for all u ,v V and extending , by bilinearity. i i ∈ − − k It is easy to show that if (e1,...,en) is an orthonormal basis of V ,thenthebasisof V consisting of the eI (where I = i1,...,ik ,with1 i1 < orientation of a vector space, V ,ofdimensionn is given by the choice of some basis, (e1,...,en). We say that a basis, (u1,...,un), of V is positively oriented iﬀ det(u1,...,un) > 0(wheredet(u1,...,un)denotesthedeterminantofthematrixwhosejth column consists of the coordinates of uj over the basis (e1,...,en)), otherwise it is negatively oriented. An oriented vector space is a vector space, V ,togetherwithanorientationofV .

If V is oriented by the basis (e1,...,en), then V ∗ is oriented by the dual basis, (e1∗,...,en∗ ). If V is an oriented vector space of dimension n,thenwecandeﬁnealinearmap,

k n k − : V V, ∗ → called the Hodge -operator,asfollows:Foranychoiceofapositivelyorientedorthonormal ∗ basis, (e1,...,en), of V ,set (e e )=e e . ∗ 1 ∧···∧ k k+1 ∧···∧ n In particular, for k =0andk = n,wehave

(1) = e e ∗ 1 ∧···∧ n (e e )=1. ∗ 1 ∧···∧ n It is easy to see that the deﬁnition of does not depend on the choice of positively oriented orthonormal basis. ∗ k n k The Hodge -operators, : V − V ,inducesalinearbijection, ∗ ∗ → k n k : (V ) (V ). We also have Hodge -operators, : V ∗ − V ∗. ∗ → ∗ ∗ → The following proposition is easy to show: 22.17. TESTING DECOMPOSABILITY; LEFT AND RIGHT HOOKS 627

Proposition 22.25 If V is any oriented vector space of dimension n, for every k, with 0 k n, we have ≤ ≤ k(n k) (i) =( id) − . ∗∗ − (ii) x, y = (x y)= (y x), for all x, y k V . ∗ ∧∗ ∗ ∧∗ ∈ If (e1,...,en)isanorthonormalbasisofV and (v1,...,vn)isanyotherbasisofV ,itis easy to see that v v = det( v ,v ) e e , 1 ∧···∧ n i j 1 ∧···∧ n from which it follows that 1 (1) = v1 vn ∗ det( v ,v ) ∧···∧ i j (see Jost [83], Chapter 2, Lemma 2.1.3).

22.17 Testing Decomposability; Left and Right Hooks

In this section, all vector spaces are assumed to have ﬁnite dimension. Say dim(E)=n. Using our nonsingular pairing,

p p , : E∗ E K (1 p n), − − × −→ ≤ ≤ deﬁned on generators by

u∗ u∗,v u =det(u∗(v )), 1 ∧···∧ p 1 ∧···∧ p i j we deﬁne various contraction operations,

p p+q q : E E∗ E∗ (left hook) × −→ and p+q p q : E∗ E E∗ (right hook), × −→ as well as the versions obtained by replacing Eby E∗ and E∗∗ by E. We begin with the left interior product or left hook, . Let u p E.Foranyq such that p + q n,multiplicationontherightbyu is a linear map ∈ ≤ q p+q (u): E E, ∧R −→ 628 CHAPTER 22. TENSOR ALGEBRAS

given by v v u → ∧ where v q E.Thetransposeof (u)yieldsalinearmap, ∈ ∧R p+q q t ( (u)) :( E)∗ ( E)∗, ∧R −→ p+q p+q q q which, using the isomorphisms ( E)∗ ∼= E∗ and ( E)∗ ∼= E∗ can be viewed as amap p+q q t ( (u)) : E∗ E∗, ∧R −→ given by z∗ z∗ (u), → ◦∧R p+q where z∗ E∗. ∈ We denote z∗ (u)by ◦∧R u z∗. In terms of our pairing, the q-vector u z∗ is uniquely deﬁned by p q p+q u z∗,v = z∗,v u , for all u E, v E and z∗ E∗. ∧ ∈ ∈ ∈ It is immediately veriﬁed that

(u v) z∗ = u (v z∗), ∧ so deﬁnes a left action p p+q q : E E∗ E∗. × −→ By interchanging E and E∗ and using the isomorphism, 

k k ( F )∗ ∼= F ∗, we can also deﬁne a left action p p+q q : E∗ E E. × −→ In terms of our pairing, u∗ z is uniquely deﬁned by p q p+q v∗,u∗ z = v∗ u∗,z , for all u∗ E∗, v∗ E∗ and z E. ∧ ∈ ∈ ∈ In order to proceed any further, we need some combinatorial properties of the basis of p E constructed from a basis, (e1,...,en), of E.Recallthatforany(nonempty)subset, I 1,...,n ,welet ⊆{ } e = e e , I i1 ∧···∧ ip 22.17. TESTING DECOMPOSABILITY; LEFT AND RIGHT HOOKS 629

where I = i1,...,ip with i1 < l . |{ | ∈ × }|

Proposition 22.26 For any basis, (e1,...,en), of E the following properties hold:

(1) If H L = , H = h, and L = l, then ∩ ∅ | | | | ρ ρ =( 1)hl. H,L L,H −

(2) For H, L 1,...,m , we have ⊆{ }

eH eL = ρH,LeH L. ∧ ∪

(3) For the left hook, p p+q q : E E∗ E∗, × −→ we have 

e e∗ =0 if H L H L ⊆ eH eL∗ = ρL H,HeL∗ H if H L. − − ⊆

p p+q q Similar formulae hold for : E∗ E E. Using Proposition 22.26, we have the × −→ Proposition 22.27 For the left hook,

p p+q q : E E∗ E∗, × −→ for every u E, we have ∈ s u (x∗ y∗)=( 1) (u x∗) y∗ + x∗ (u y∗), ∧ − ∧ ∧ s where y E∗. ∈ 630 CHAPTER 22. TENSOR ALGEBRAS

Proof .Wecanprovetheaboveidentityassumingthatx∗ and y∗ are of the form eI∗ and eJ∗ using Proposition 22.26 but this is rather tedious. There is also a proof involving determinants, see Warner [147], Chapter 2.

Thus, is almost an anti-derivation, except that the sign, ( 1)s is applied to the wrong factor. −

It is also possible to deﬁne a right interior product or right hook, , using multiplication on the left rather than multiplication on the right. Then, deﬁnes a right action,

p+q p q : E∗ E E∗, × −→ such that

p q p+q z∗,u v = z∗ u, v , for all u E, v E,andz∗ E∗. ∧ ∈ ∈ ∈ Similarly, we have the right action 

p+q p q : E E∗ E, × −→ such that

p q p+q u∗ v∗,z = v∗,z u∗ , for all u∗ E∗, v∗ E∗,andz E. ∧ ∈ ∈ ∈ p p+q q Since the left hook, : E E∗ E∗,isdeﬁnedby × −→ p q p+q u z∗,v = z∗,v u , for all u E, v E and z∗ E∗, ∧ ∈ ∈ ∈ the right hook, p+q p q : E∗ E E∗, × −→ by p q p+q z∗ u, v = z∗,u v , for all u E, v E,andz∗ E∗, ∧ ∈ ∈ ∈ and v u =( 1)pqu v,weconcludethat ∧ − ∧ pq u z∗ =( 1) z∗ u, − p p+q where u E and z E∗. ∈ ∈ Using the above property and Proposition 22.27 we get the following version of Proposi- tion 22.27 for the right hook: 22.17. TESTING DECOMPOSABILITY; LEFT AND RIGHT HOOKS 631

Proposition 22.28 For the right hook,

p+q p q : E∗ E E∗, × −→ for every u E, we have ∈ r (x∗ y∗) u =(x∗ u) y∗ +( 1) x∗ (y∗ u), ∧ ∧ − ∧ r where x∗ E∗. ∈ Thus, is an anti-derivation. For u E,therighthook,z∗ u,isalsodenoted,i(u)z∗,andcalledinsertion operator or interior product∈ .Thisoperatorplaysanimportantroleindiﬀerentialgeometry.Ifweview n+1 n+1 n z∗ (E∗) as an alternating multilinear map in Alt (E; K), then i(u)z∗ Alt (E; K) is given∈ by ∈ (i(u)z∗)(v1,...,vn)=z∗(u, v1,...,vn).

Note that certain authors, such as Shafarevitch [138], denote our right hook z∗ u (which is also the right hook in Bourbaki [21] and Fulton and Harris [57]) by u z∗.

p n p Using the two versions of ,wecandeﬁnelinearmapsγ : E − E∗ and p n p → δ : E∗ − E.Foranybasis(e1,...,en)ofE, if we let M = 1,...,n , e = e1 en, → { } ∧···∧ and e∗ = e∗ e∗ ,then 1 ∧···∧ n γ(u)=u e∗ and δ(v)=v∗ e,

p p for all u E and all v∗ E∗.Thefollowingpropositioniseasilyshown. ∈ ∈ p n p p n p Proposition 22.29 The linear maps γ : E − E∗ and δ : E∗ − E are isomorphims. The isomorphisms γ and δ map decomposable→ vectors to decomposable→ vectors. p p Furthermore, if z E is decomposable, then γ(z),z =0, and similarly for z E∗. ∈ p n p p ∈ n p If (e1 ,...,en ) is any other basis of E and γ : E − E∗ and δ : E∗ − E → 1 → are the corresponding isomorphisms, then γ = λγ and δ = λ− δ for some nonzero λ Ω. ∈ Proof . Using Proposition 22.26, for any subset J 1,...,n = M such that J = p,we have ⊆{ } | | γ(eJ )=eJ e∗ = ρM J,J eM∗ J and δ(eJ∗ )=eJ∗ e = ρM J,J eM J . − − − − Thus, p(n p) δ γ(eJ )=ρM J,J ρJ,M J eJ =( 1) − eJ . ◦ − − − Asimilarresultholdsforγ δ.Thisimpliesthat ◦ p(n p) p(n p) δ γ =( 1) − id and γ δ =( 1) − id. ◦ − ◦ − 632 CHAPTER 22. TENSOR ALGEBRAS

Thus, γ and δ are isomorphisms. If z p E is decomposable, then z = u u where ∈ 1 ∧···∧ p u1,...,up are linearly independent since z =0,andwecanpickabasisofE of the form (u1,...,un). Then, the above formulae show that

γ(z)= u∗ u∗ . ± p+1 ∧···∧ n Clearly γ(z),z =0. m If (e1 ,...,en )isanyotherbasisofE, because E has dimension 1, we have

e e = λe1 en 1 ∧···∧ n ∧···∧ for some nonnull λ Ω, and the rest is trivial. ∈ We are now ready to tacke the problem of ﬁnding criteria for decomposability. We need afewpreliminaryresults. Proposition 22.30 Given z p E, with z =0, the smallest vector space W E such that z p W is generated by∈ the vectors of the form ⊆ ∈ p 1 u∗ z, with u∗ − E∗. ∈ Proof . First, let W be any subspace such that z p(E)andlet(e ,...,e ,e ,...,e )be ∈ 1 r r+1 n abasisofE such that (e1,...,er)isabasisofW .Then,u∗ = e∗,whereI 1,...,n I I ⊆{ } and I = p 1, and z = eJ ,whereJ 1,...,r and J = p r.Itfollowsimmediately | | − J ⊆{ } | | ≤ from the formula of Proposition 22.26 (3) that u∗ z W . ∈ Next, we prove that if W is the smallest subspace of E such that z p(W ), then W p 1 ∈ is generated by the vectors of the form u∗ z,whereu∗ − E∗.Supposenot,thenthe p 1 ∈ vectors u∗ z with u∗ − E∗ span a proper subspace, U,ofW .Weprovethatforevery ∈ subspace, W ,ofW ,withdim(W ) = dim(W ) 1=r 1, it is not possible that u∗ z W p 1 − − ∈ for all u∗ − E∗.Butthen,asU is a proper subspace of W ,itiscontainedinsome ∈ subspace, W ,withdim(W )=r 1andwehaveacontradiction. − Let w W W and pick a basis of W formed by a basis (e1,...,er 1)ofW and w.We ∈ − p p 1 − can write z = z + w z,wherez W and z − W ,andsinceW is the smallest ∧ ∈ ∈ subspace containing z,wehavez =0.Consequently,ifwewritez = eI in terms of I the basis (e1,...,er 1)ofW ,thereissomeeI ,withI 1,...,r 1 and I = p 1, so − ⊆{ − } | | − that the coeﬃcient λI is nonzero. Now, using any basis of E containing (e1,...,er 1,w), by − Proposition 22.26 (3), we see that

e∗ (w e )=λw, λ = 1. I ∧ I ± It follows that

e∗ z = e∗ (z + w z)=e∗ z + e∗ (w z)=e∗ z + λw, I I ∧ I I ∧ I with eI∗ z W ,whichshowsthateI∗ z/W .Therefore,W is indeed generated by the ∈ p 1∈ vectors of the form u∗ z,whereu∗ − E∗. ∈ 22.17. TESTING DECOMPOSABILITY; LEFT AND RIGHT HOOKS 633

Proposition 22.31 Any nonzero z p E is decomposable iﬀ ∈ p 1 (u∗ z) z =0, for all u∗ − E∗. ∧ ∈ Proof .Clearly,z p E is decomposable iﬀthe smallest vector space, W ,suchthatz p ∈ ∈ W has dimension p.Ifdim(W )=p,wehavez = e1 ep where e1,...,ep form a p 1 ∧···∧ basis of W .ByProposition22.30,foreveryu∗ − E∗,wehaveu∗ z W ,soeachu∗ z ∈ ∈ is a linear combination of the ei’s and (u∗ z) z =(u∗ z) e1 ep =0. ∧ ∧ ∧···∧ p 1 Now, assume that (u∗ z) z =0forallu∗ − E∗ and that dim(W )=n>p.If ∧ ∈ e1,...,en is a basis of W ,thenwehavez = λI eI ,whereI 1,...,n and I = p. I ⊆{ } | | Recall that z =0,andso,someλI is nonzero. By Proposition 22.30, each ei can be written p 1 p 1 as u∗ z for some u∗ − E∗ and since (u∗ z) z =0forallu∗ − E∗,weget ∈ ∧ ∈ e z =0 for j =1,...,n. j ∧

By wedging z = I λI eI with each ej,asn>p,wededuceλI =0forallI,soz =0,a contradiction. Therefore, n = p and z is decomposable. p 1 In Proposition 22.31, we can let u∗ range over a basis of − E∗,andthen,theconditions are (e∗ z) z =0 H ∧ p+1 for all H 1,...,n ,with H = p 1. Since (e∗ z) z E,thisisequivalentto ⊆{ } | | − H ∧ ∈

e∗ ((e∗ z) z)=0 J H ∧

for all H, J 1,...,n ,with H = p 1 and J = p +1.Then,forallI,I 1,...,n ⊆{ } | | − | | ⊆{ } with I = I = p,wecanshowthat | | | |

e∗ ((e∗ e ) e )=0, J H I ∧ I unless there is some i 1,...,n such that ∈{ }

I H = i ,J I = i . − { } − { } In this case,

eJ∗ (eH∗ eH i ) eJ i = ρ i ,H ρ i ,J i . ∪{ } ∧ −{ } { } { } −{ } If we let i,J,H = ρ i ,H ρ i ,J i , { } { } −{ } we have i,J,H =+1iftheparityofthenumberofj J such that j

p Proposition 22.32 (Grassmann-Pl¨ucker’s Equations) For z = I λI eI E, the condi- tions for z =0to be decomposable are ∈ 

i,J,H λH i λJ i =0, ∪{ } −{ } i J H ∈− for all H, J 1,...,n such that H = p 1 and J = p +1. ⊆{ } | | − | | Using these criteria, it is a good exercise to prove that if dim(E)=n,theneverytensor n 1 in − (E)isdecomposable.Thiscanalsobeshowndirectly. It should be noted that the equations given by Proposition 22.32 are not independent. For example, when dim(E)=n =4andp =2,theseequationsreducetothesingleequation

λ λ λ λ + λ λ =0. 12 34 − 13 24 14 23 When the ﬁeld, K,istheﬁeldofcomplexnumbers,thisisthehomogeneousequationofa 5 quadric in CP known as the Klein quadric.Thepointsonthisquadricareinone-to-one 3 correspondence with the lines in CP .

22.18 Vector-Valued Alternating Forms

In this section, the vector space, E,isassumedtohaveﬁnitedimension.Weknowthat n n there is a canonical isomorphism, (E∗) ∼= Alt (E; K), between alternating n-forms and alternating multilinear maps. As in the case of general tensors, the isomorphisms, n n Alt (E; F ) ∼= Hom( (E),F) n n Hom( (E),F) = ( (E))∗ F ∼ ⊗ n n ( (E))∗ ∼= (E∗) yield a canonical isomorphism

n n Alt (E; F ) ∼= (E∗) F. ⊗ Note that F may have inﬁnite dimension. This isomorphism allows us to view the tensors in n (E∗) F as vector valued alternating forms,apointofviewthatisusefulindiﬀerential × n geometry. If (f1,...,fr)isabasisofF ,everytensor,ω (E∗) F can be written as some linear combination ∈ × r ω = α f , i ⊗ i i=1 22.18. VECTOR-VALUED ALTERNATING FORMS 635

n with α (E∗). We also let i ∈ n (E; F )= (E∗) F = (E) F. ⊗ ⊗ n=0 Given three vector spaces, F, G, H,ifwehavesomebilinearmap,Φ:F G H,then we can deﬁne a multiplication operation, ⊗ → : (E; F ) (E; G) (E; H), ∧Φ × → as follows: For every pair, (m, n), we deﬁne the multiplication,

m n m+n

Φ : (E∗) F (E∗) G (E∗) H, ∧ ⊗ × ⊗ −→ ⊗ by (α f) (β g)=(α β) Φ(f,g). ⊗ ∧Φ ⊗ ∧ ⊗ As in Section 22.15 (following H. Cartan [30]) we can also define a multiplication, : Altm(E; F ) Altm(E; G) Altm+n(E; H), ∧Φ × −→ directly on alternating multilinear maps as follows: For f Altm(E; F )andg Altn(E; G), ∈ ∈ (f g)(u ,...,u )= sgn(σ)Φ(f(u ,...,u ),g(u ,...,u )), ∧Φ 1 m+n σ(1) σ(m) σ(m+1) σ(m+n) σ shuffle(m,n) ∈ where shuffle(m, n)consistsofall(m, n)-“shuffles”, that is, permutations, σ,of 1,...m+n , such that σ(1) < <σ(m)andσ(m +1)< <σ(m + n). { } ··· ··· In general, not much can be said about Φ unless Φhas some additional properties. In particular, is generally not associative. We∧ also have the map, ∧Φ n n µ: (E∗) F Alt (E; F ), ⊗ −→ defined on generators by

µ((v∗ v∗) a)(u ,...,u )=(det(v∗(u ))a. 1 ∧···∧ n ⊗ 1 n j i Proposition 22.33 The map n n µ: (E∗) F Alt (E; F ), ⊗ −→ deﬁned as above is a canonical isomorphism for every n 0. Furthermore, given any three ≥ n vector spaces, F, G, H, and any bilinear map, Φ: F G H, for all ω ( (E∗)) F and n × → ∈ ⊗ all η ( (E∗)) G, ∈ ⊗ µ(α Φ β)=µ(α) Φ µ(β). ∧ ∧ 636 CHAPTER 22. TENSOR ALGEBRAS

n n Proof .Sincewealreadyknowthat( (E∗)) F and Alt (E; F )areisomorphic,itisenough n ⊗ to show that µ maps some basis of ( (E∗)) F to linearly independent elements. Pick ⊗ some bases, (e1,...,ep)inE and (fj)j J in F . Then, we know that the vectors, eI∗ fj,where ∈ n ⊗ I 1,...,p and I = n form a basis of ( (E∗)) F .Ifwehavealineardependence, ⊆{ } | | ⊗

λ µ(e∗ f )=0, I,j I ⊗ j I,j 

applying the above combination to each (ei1 ,...,ein )(I = i1,...,in , i1 <

λI,jfj =0, j and by linear independence of the fj’s, we get λI,j =0,forallI and all j.Therefore,the µ(eI∗ fj)arelinearlyindependentandwearedone.Thesecondpartofthepropositionis easily⊗ checked (a simple computation). AspecialcaseofinterestisthecasewhereF = G = H is a Lie algebra and Φ(a, b)=[a, b], is the Lie bracket of F . In this case, using a base, (f ,...,f ), of F if we write ω = α f 1 r i i ⊗ i and η = βj fj, we have j ⊗ [ω,η]= α β [f ,f ]. i ∧ j ⊗ i j i,j Consequently, [η,ω]=( 1)mn+1[ω,η]. − The following proposition will be useful in dealing with vector-valued diﬀerential forms:

n Proposition 22.34 If (e1,...,ep) is any basis of E, then every element, ω ( (E∗)) F , can be written in a unique way as ∈ ⊗ 

ω = e∗ f ,fF, I ⊗ I I ∈ I 

where the eI∗ are deﬁned as in Section 22.12.

n Proof .Since,byProposition22.19,theeI∗ form a basis of (E∗), elements of the form n eI∗ f span ( (E∗)) F . Now, if we apply µ(ω)to(ei1 ,...,ein ), where I = i1,...,in 1,...,p⊗ ,weget ⊗ { }⊆ { } µ(ω)(e ,...,e )=µ(e∗ f )(e ,...,e )=f . i1 in I ⊗ I i1 in I

Therefore, the fI are uniquely determined by f. Proposition can also be formulated in terms of alternating multilinear maps, a fact that will be useful to deal with diﬀerential forms. 22.19. TENSOR PRODUCTS OF MODULES OVER A COMMMUTATIVE RING 637

n n n Deﬁne the product, : Alt (E; R) F Alt (E; F ), as follows: For all ω Alt (E; R) and all f F , · × → ∈ ∈ (ω f)(u ,...,u )=ω(u ,...,u )f, · 1 n 1 n n for all u1,...,un E.Then,itisimmediatelyveriﬁedthatforeveryω ( (E∗)) F of the form ∈ ∈ ⊗ ω = u∗ u∗ f, 1 ∧···∧ n ⊗ we have µ(u∗ u∗ f)=µ(u∗ u∗ ) f. 1 ∧···∧ n ⊗ 1 ∧···∧ n · Then, Proposition 22.34 yields

n Proposition 22.35 If (e1,...,ep) is any basis of E, then every element, ω Alt (E; F ), can be written in a unique way as ∈

ω = e∗ f ,fF, I · I I ∈ I 

where the eI∗ are deﬁned as in Section 22.12.

22.19 Tensor Products of Modules over a Commmutative Ring

If R is a commutative ring with identity (say 1), recall that a module over R (or R-module) is an abelian group, M,withascalarmultiplication, : R M M,andalltheaxiomsof avectorspacearesatisfied. · × → At first glance, a module does not seem any different from a vector space but the lack of multiplicative inverses in R has drastic consequences, one being that unlike vector spaces, modules are generally not free, that is, have no bases. Furthermore, a module may have torsion elements,thatis,elements,m M,suchthatλ m =0,eventhoughm =0and λ =0. ∈ · Nevertheless, it is possible to define tensor products of modules over a ring, just as in Section 22.1 and the results of this section continue to hold. The results of Section 22.3 also continue to hold since they are based on the universal mapping property. However, the results of Section 22.2 on bases generally fail, except for free modules. Similarly, the results of Section 22.4 on duality generally fail. Tensor algebras can be defined for modules, as in Section 22.5. Symmetric tensor and alternating tensors can be defined for modules but again, results involving bases generally fail. Tensor products of modules have some unexpected properties. For example, if p and q are relatively prime integers, then

Z/pZ Z/qZ =(0). ⊗Z 638 CHAPTER 22. TENSOR ALGEBRAS

It is possible to salvage certain properties of tensor products holding for vector spaces by restricting the class of modules under consideration. For example, projective modules,have aprettygoodbehaviorw.r.t.tensorproducts.

AfreeR-module, F ,isamodulethathasabasis(i.e.,thereisafamily,(ei)i I ,of ∈ linearly independent vectors in F that span F ). Projective modules have many equivalent characterizations. Here is one that is best suited for our needs:

Deﬁnition 22.9 An R-module, P ,isprojective if it is a summand of a free module, that is, if there is a free R-module, F ,andsomeR-module, Q,sothat

F = P Q. ⊕

Given any R-module, M,weletM ∗ = HomR(M,R)beitsdual.Wehavethefollowing proposition:

Proposition 22.36 For any ﬁnitely-generated projective R-modules, P , and any R-module, Q, we have the isomorphisms:

P ∗∗ ∼= P Hom (P, Q) = P ∗ Q. R ∼ ⊗R Sketch of proof .Weonlyconsiderthesecondisomorphism.SinceP is projective, we have some R-modules, P1,F,with P P = F, ⊕ 1 where F is some free module. Now, we know that for any R-modules, U, V, W , we have

Hom (U V,W) = Hom (U, W ) Hom (V,W) = Hom (U, W ) Hom (V,W), R ⊕ ∼ R R ∼ R ⊕ R so P ∗ P ∗ = F ∗, Hom (P, Q) Hom (P ,Q) = Hom (F, Q). ⊕ 1 ∼ R ⊕ R 1 ∼ R By tensoring with Q and using the fact that tensor distributes w.r.t. coproducts, we get

(P ∗ Q) (P ∗ Q) = (P ∗ P ∗) Q = F ∗ Q. ⊗R ⊕ 1 ⊗ ∼ ⊕ 1 ⊗R ∼ ⊗R Now, the proof of Proposition 22.9 goes through because F is free and ﬁnitely generated, so

α :(P ∗ R Q) (P1∗ Q) = F ∗ R Q HomR(F, Q) = HomR(P, Q) HomR(P1,Q) ⊗ ⊗ ⊕ ⊗ ∼ ⊗ −→ ∼ ⊕ is an isomorphism and as αα maps P ∗ R Q to HomR(P, Q), it yields an isomorphism between these two spaces. ⊗

The isomorphism α : P ∗ R Q = HomR(P, Q)ofProposition22.36isstillgivenby ⊗ ⊗ ∼

α (u∗ f)(x)=u∗(x)f, u∗ P ∗,f Q, x P. ⊗ ⊗ ∈ ∈ ∈ 22.20. THE PFAFFIAN POLYNOMIAL 639

It is convenient to introduce the evaluation map,Evx : P ∗ R Q Q,definedforevery x P by ⊗ → ∈ Ev (u∗ f)=u∗(x)f, u∗ P ∗,f Q. x ⊗ ∈ ∈ In Section 11.2 we will need to consider a slightly weaker version of the universal mapping property of tensor products. The situation is this: We have a commutative R-algebra, S, where R is a field (or even a commutative ring), we have two R-modules, U and V ,and moreover, U is a right S-module and V is a left S-module. In Section 11.2, this corresponds i to R = R, S = C∞(B), U = (ξ)andV =Γ(ξ), where ξ is a vector bundle. Then, we can A form the tensor product, U R V ,andweletU S V be the quotient module, (U R V )/W , where W is the submodule⊗ of U V generated⊗ by the elements of the form ⊗ ⊗R us v u sv. ⊗R − ⊗R As S is commutative, we can make U V into an S-module by defining the action of S via ⊗S s(u v)=us v. ⊗S ⊗S It is immediately verified that this S-module is isomorphic to the tensor product of U and V as S-modules and the following universal mapping property holds:

Proposition 22.37 For every, R-bilinear map, f : U V Z, if f satisﬁes the property × → f(us, v)=f(u, sv), for all u U, v V, s S, ∈ ∈ ∈ then f induces a unique R-linear map, f : U V Z, such that ⊗S → f(u, v)=f(u v), for all u U, v V. ⊗S ∈ ∈

Note that the linear map, f : U V Z,isonlyR-linear, it is not S-linear in general. ⊗S → 22.20 The Pfaﬃan Polynomial

Let so(2n)denotethevectorspace(actually,Liealgebra)of2n 2n real skew-symmetric × matrices. It is well-known that every matrix, A so(2n), can be written as ∈

A = PDP,

where P is an orthogonal matrix and where D is a block diagonal matrix

D1 D2 D =   ...    Dn     640 CHAPTER 22. TENSOR ALGEBRAS

consisting of 2 2blocksoftheform × 0 a D = i . i a −0 i For a proof, see see Horn and Johnson [79], Corollary 2.5.14, Gantmacher [61], Chapter IX, or Gallier [58], Chapter 11. 2 Since det(D )=a and det(A) = det(PDP)=det(D)=det(D ) det(D ), we get i i 1 ··· n det(A)=(a a )2. 1 ··· n The Pfaﬃan is a polynomial function, Pf(A), in skew-symmetric 2n 2n matrices, A,(a polynomial in (2n 1)n variables) such that × − Pf(A)2 =det(A)

and for every arbitrary matrix, B,

Pf(BAB)=Pf(A)det(B).

The Pfaffian shows up in the definition of the Euler class of a vector bundle. There is a simple way to define the Pfaffian using some exterior algebra. Let (e1,...,e2n)beanybasis of R2n.Foranymatrix,A so(2n), let ∈ ω(A)= a e e , ij i ∧ j isymmetric matrix, A so(2n), the Pfaffian polynomial or Pfaffian is the degree n polynomial, Pf(A), defined by∈

n ω(A)=n!Pf(A) e e e . 1 ∧ 2 ∧···∧ 2n Clearly, Pf(A) is independent of the basis chosen. If A is the block diagonal matrix D, asimplecalculationshowsthat

ω(D)= (a1e1 e2 + a2e3 e4 + + ane2n 1 e2n) − ∧ ∧ ··· − ∧ and that n ω(D)=( 1)nn! a a e e e , − 1 ··· n 1 ∧ 2 ∧···∧ 2n and so Pf(D)=( 1)na a . − 1 ··· n Since Pf(D)2 =(a a )2 =det(A), we seem to be on the right track. 1 ··· n 22.20. THE PFAFFIAN POLYNOMIAL 641

Proposition 22.38 For every skew symmetric matrix, A so(2n) and every arbitrary matrix, B, we have: ∈ (i) Pf(A)2 =det(A)

(ii) Pf(BAB)=Pf(A)det(B).

Proof . If we assume that (ii) is proved then, since we can write A = PDP for some orthogonal matrix, P , and some block diagonal matrix, D, as above, as det(P )= 1and Pf(D)2 =det(A), we get ±

2 2 2 2 Pf(A) =Pf(PDP) =Pf(D) det(P ) =det(A),

which is (i). Therefore, it remains to prove (ii). 2n Let fi = Bei,fori =1,...,2n,where(e1,...,e2n)isanybasisofR . Since fi = k bkiek, we have τ = a f f = b a b e e = (BAB) e e , ij i ∧ j ki ij lj k ∧ l kl k ∧ l i,j i,j k,l k,l and so, as BAB is skew symmetric and e e = e e ,weget k ∧ l − l ∧ k τ =2ω(BAB).

Consequently,

n n n n τ =2 ω(BAB)=2 n!Pf(BAB) e e e . 1 ∧ 2 ∧···∧ 2n Now, n τ = Cf f f , 1 ∧ 2 ∧···∧ 2n for some C R.IfB is singular, then the fi are linearly dependent which implies that f f ∈ f =0,inwhichcase, 1 ∧ 2 ∧···∧ 2n Pf(BAB)=0,

as e e e =0.Therefore,ifB is singular, det(B)=0and 1 ∧ 2 ∧···∧ 2n Pf(BAB)=0=Pf(A)det(B).

If B is invertible, as τ = a f f =2 a f f , we have i,j ij i ∧ j i

so n τ =2nn!Pf(A)det(B) e e e 1 ∧ 2 ∧···∧ 2n and as n n τ =2 n!Pf(BAB) e e e , 1 ∧ 2 ∧···∧ 2n we get Pf(BAB)=Pf(A)det(B), as claimed.

Remark: It can be shown that the polynomial, Pf(A), is the unique polynomial with integer coefficients such that Pf(A)2 =det(A)andPf(diag(S, . . . , S)) = +1, where 01 S = , 10 − see Milnor and Stasheff[110] (Appendix C, Lemma 9). There is also an explicit formula for Pf(A), namely: 1 n Pf(A)= sgn(σ) aσ(2i 1) σ(2i). 2nn! − σ S2n i=1 ∈ Beware, some authors use a different sign convention and require the Pfaffian to have the value +1 on the matrix diag(S,...,S), where 0 1 S = . 10− For example, if R2n is equipped with an inner product, , ,thensomeauthorsdefine ω(A)as − − ω(A)= Ae ,e e e , i j i ∧ j i

We will also need another property of Pfaﬃans. Recall that the ring, Mn(C), of n n × matrices over C is embedded in the ring, M2n(R), of 2n 2n matrices with real coeﬃcients, × using the injective homomorphism that maps every entry z = a + ib C to the 2 2matrix ∈ × a b . ba− If A Mn(C), let A M2n(R) denote the real matrix obtained by the above process. ∈ R ∈ Observe that every skew Hermitian matrix, A u(n), (i.e.,withA∗ = A = A)yieldsa ∈ − matrix A so(2n). R ∈ 22.20. THE PFAFFIAN POLYNOMIAL 643

Proposition 22.39 For every skew Hermitian matrix, A u(n), we have ∈ n Pf(AR)=i det(A).

Proof . It is well-known that a skew Hermitian matrix can be diagonalized with respect to a unitary matrix, U, and that the eigenvalues are pure imaginary or zero, so we can write

A = U diag(ia1, . . . , ian)U ∗,

for some reals, ai R.Consequently,weget ∈

AR = UR diag(D1,...,Dn)UR, where 0 a D = i i a −0 i and Pf(A )=Pf(diag(D ,...,D )) = ( 1)n a a , R 1 n − 1 ··· n as we saw before. On the other hand,

det(A) = det(diag(ia , . . . , ia )) = in a a , 1 n 1 ··· n and as ( 1)n = inin,weget − n Pf(AR)=i det(A), as claimed.

Madsen and Tornehave [100] state Proposition 22.39 using the factor ( i)n,whichis wrong. − 644 CHAPTER 22. TENSOR ALGEBRAS Bibliography

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