Tensor product of vector spaces

Te obstructions to the global solvability of (∗) for closed 1-forms α are of topologlical nature. This results from

Homotopy invariance of the line integral of closed 1-forms:

[0,1]×[a, b]−→^C1 M : (s, t)→c_s(t) homotopy with fixed end pointsc₀(a) := const, c₀(b) :=

const

dα= 0⇒s→ Z

cs

α const proof: Exercise (use local potential)

Whether the line integral of closed forms is path independent is related to the possibility of deforming (homotoping) paths into each other (fixing endpoints). In case of simple topology: closed ⇒ exact (no global obstruction).

11.12.2012

Whether the line integral of closed 1–forms is path independent is related to the possibility of deforming paths inM into each other. This is related to thefundamental group π₁(M). Precisely, if all closed paths in M are nullhomotopic (i.e. homotopic relative to their endpoints to the constant path), then the line integral of closed 1–forms is path independent, and closed 1–forms are therefore exact. Short: Ifπ₁(M) = 0, then H¹(M) = 0, where H¹ is the first de–Rham cohomology group.

Thus, if M has sufficiently simple topology, then there are no global obstructions to exactness.

Example (Poincaré–Lemma for 1–forms). ForU ⊂Rⁿ open star–shaped the following holds: Ifα ∈Ω¹(U) is closed, then it is exact.

Example. Consider polar coordinates on R²r{0}. The 1–form dθ⊂Ω¹(R²r{0}) is well–defined and closed, hence locally exact. But it is not globally exact since for the circle γ around the origin, we have

Z

γ

dθ= 2π6= 0

For any bilinear map β:U ×V → W there exists a unique linear map λ:U ⊗V →W with β=λ◦ ⊗.

In other words, the natural homomorphism of vector spaces Hom(U ⊗V, W)→Bil(U, V;W), λ→λ◦ ⊗ is an isomorphism.

Theorem 1.50. For any vector spacesU, V the tensor product⊗:U×V →U⊗V exists and is unique up to natural isomorphism. We can therefore speak of the tensor product of U and V.

Proof. The uniqueness follows from the universal property: It yields linear maps λand λe with

λ(u⊗v) =u⊗e v λ(u^e ⊗e v) =u⊗v ⇒ (λe◦λ)(u⊗v) =u⊗v U ⊗V

λ

U×V

⊗ 88

⊗e %%

U⊗Ve eλ

JJ

Both id_U⊗V and λe◦λsolve the mapping problem U×V ^{⊗ //}

⊗

U⊗V

U⊗V

id 66 eλ◦λ

@@

and uniqueness of its solution implies eλ◦λ= id_U⊗V. Analogously, λ◦eλ= id_U

⊗Ve . This shows that there are natural isomorphisms between and two tensor products of U and V.

For existence, we first consider a vector space E with basis U ×V, i.e. E consists of all finite linera combinations ^P_iai(ui, vi) with ui ∈ U, vi ∈ V and ai ∈ K. In order to make the canonical map U×V →E bilinear, we devide out corresponding relations:

LetR⊂E be the vector subspace spanned by the elements

(u1, v) + (u2, v)−(u1+u2, v) ∀u₁, u2 ∈U, v∈V (u, v1) + (u, v2)−(u, v1+v2) ∀u∈U, v1, v2 ∈V

(αu, v)−α(u, v) ∀u∈U, v∈V, α∈K (u, αv)−α(u, v) ∀u∈U, v∈V, α∈K Then the composition

U ×V ^//

⊗

22E ^////E/R=U⊗V

because, for instance u1 ⊗v +u2 ⊗v −(u1 +u2) ⊗v = 0. It satisfies the required universal property: Given a bilinear map β: U ×V → W, there exists a unique linear mapλb:E →W with bλ◦ι=β. The bilinearity ofβ is equivalent toλ(R) = 0, becauseb e.g.

bλ((αu, v)−α(u, v)) =^bλ((αu, v))−αλ((u, v)) =^b β(αu, v)−αβ(u, v) = 0

E ^{bλ //}

##

W

U ×V

ι

ff

β 77

⊗ E/R=U⊗V

λ

EE

Therefore bλdescends to a linear mapλ:U⊗V →W which satisfies as desired:

λ(u⊗v) =^bλ((u, v)) =β(u, v) ⇒ β=λ◦ ⊗

This determines the mapλuniquely, because by our construction theu⊗vgenerate the vector spaceU ⊗V.

The following result gives a more concrete idea of the tensor product.

Lemma 1.51. If {e_i | i ∈ I} is a basis of U and {f_i | i ∈ J} is a basis of V, then {e_i ⊗fj | (i, j) ∈ I ×J} is a basis of U ⊗V. In particular, in the case of finite dimensions, we get

dimU ⊗V = dimU·dimV

Proof. From the construction of the tensor product in the preceding proof we know that the elementsu⊗v generateU⊗V. They are linear combinations of thee_i⊗f_j because of the bilinearity of the tensor product:

X

i

a_ie_i⊗ ^X

i

b_if_i=^X

i,j

a_ib_je_i⊗f_j

To verify their linear independence, we construct (using the universal property) linear forms ‘seperating’ them, i.e. taking independent values on them. The bilinear form (wheree^∗ and f^∗ denote the dual basis elements, i.e. e^∗_k(ei) =δik,f_l^∗(fj) =δjl)

U ×V →K, (u, v)7→e^∗_k(u)·f_l^∗(v) induces the linear form

U ⊗V →K, u⊗v7→e^∗_k(u)·f_l^∗(v) e_i⊗f_j 7→δ_ikδ_jl. For a linear relation

X

i,j

eijei⊗fj = 0

where eij = 0 for almost all (i, j) follows by applying the above linear form, that 0 =^X

i,j

c_ij =δ_ikδ_jl=c_kl ∀k, l Thus, the e_i⊗f_j form a basis.

13.12.2012

Remark: To construct the tensor product of vector spaces, one can also proceed more directly and, after choosing bases {e_i} of U and {f_j} of V, construct U ⊗V as the vector space consisting of the symbols ei⊗fj and define the natural bilinear map U ×V −→^⊗ U ⊗V by

(^X

i

a_ie_i)⊗(^X

j

b_jf_j) :=^X

i,j

a_ib_j e_i⊗f_j.

The universal property can easily be verified (Exercise).

The universal proberty implies the base independence for our second construction of the tensor product.

the abstract construction is more general. One can in the same way construct the tensor product ⊗_R of mudules over a fixed ring R (e.g. R = Z: abelian groups) also if the modules are not free, e.g. have torsion.

Change of bases:

e˜_r=^P

i

g_ri e_i in U f˜_s=^P

j

h_sj f_j in V )

⇒e˜r⊗f˜s=^X

i,j

grjfsj ei⊗fj

(˜er=g_rⁱei, f˜s=h^j_sfj, e˜r⊗f˜s=g_rⁱh^j_s ei⊗fj)

Example: For vector spaces U and V , the bilinear map U^∗×V −→Hom(U, V)

(u^∗, v)−→(u−→u^∗(u)·v) induces a natural homomorphism (Ex: injective!)

U^∗⊗V −→Hom(U, V) (∗)

(the image constists of homomorphismsU →V with finite-dim image) U^∗×V

⊗ //Hom(U, V)

U^∗⊗V

55

If dimU,dimV <∞, then it is an isomorphism. Namely, if{e_i} is a basis of U and{e^∗_i} the dual basis ofU^∗, then for arbitrary elementsvi ∈V the element^P

i

e^∗_i ⊗vi ∈U^∗⊗V

corresponds to the homomophismU →V mappingei →vi.

Hence (∗) is surjective and therefore an isomorphism by dimesion reasons.

If{f_j} is a basis for V, then X

j,i

a_jif_j e^∗_i ⊗f_j a_ji∈K

corresponds to the homomorphismU →V which is given relative to the chosen basis by the matrix (aji)_j,i because it maps ei −→^P

j

aji fj (←−i-th column).

In particular, the natural homomorphism U^∗ ⊗U −→ End(U) is an isomorphism if dimU <∞. (Note that then ^P

i

e^∗_i ⊗e_i corresponds to id_U.)

Analogously to the twofold one defines and constructs the multiple tensor productU₁⊗

· · · ⊗Un. It satisfies the universal property.

U₁×...×U_n

⊗ multilin.

multilin. //W

U1⊗...⊗Un

∃! lin (unique)

55

Lemma 1.52. (Associativity:) For vector spaces U1, . . . , Un+m (n, m ≥1) there exists an unique isomophism of vector spaces with

(U1⊗ · · · ⊗Un)⊗(Un+1⊗ · · · ⊗Un+m)−→U1⊗ · · · ⊗Un+m

(u1⊗ · · · ⊗un)⊗(un+1⊗ · · · ⊗un+m)−→u1⊗ · · · ⊗un+m

Proof. The multilinear map

U₁× · · · ×U_n+m−→(U₁⊗ · · · ⊗U_n)⊗(U_n+1⊗ · · · ⊗U_n+m) u₁× · · · ×un+m −→(u₁⊗ · · · ⊗un)⊗(un+1⊗ · · · ⊗un+m) induces the homomorphism

U₁⊗ · · · ⊗U_n+m −→(U₁⊗ · · · ⊗U_n)⊗(U_n+1⊗ · · · ⊗U_n+m) u₁⊗ · · · ⊗u_n+m−→(u₁⊗ · · · ⊗u_n)⊗(u_n+1⊗ · · · ⊗u_n+m) (∗)

To see that it is an isomorphism, one can choose bases and observe that (∗) sends the induced basis bijectively to the induced basis. (Alternatively, one can use the universal property also to construct an inverse map).

Permutation of factors: For vector spaces U₁, . . . U_n and a permutationσ ∈S_n, there is the natural isomorphism of vector spaces:

U1⊗ · · · ⊗Un−→U_σ(1)⊗ · · · ⊗U_σ(n) u₁⊗ · · · ⊗u_n−→u_σ(1)⊗ · · · ⊗u_σ(n)

Functoriality: Homomorphisms of vector spaces U1 α1

−→ V1, . . . , Un αn

−−→ Vn induce the natural homomorphism

U1⊗ · · · ⊗Un α1⊗···⊗αn

−−−−−−→V1⊗ · · · ⊗Vn

u1⊗ · · · ⊗un−→α1(u1)⊗ · · · ⊗αn(un) Tensor algebra: We denote byT_m(U) :=^NmU :=U⊗ · · · ⊗U

| {z }

m

the m-fold tensor product of U with itself, m∈N0.

Convention: T₀(U) :=K. Of course, T₁(U) =U. We consider the graded vector space T∗(U) :=

∞

M

m=0

Tm(U).

There are natural bilinear maps (by associativity) Tm(U)×Tn(U)−→^⊗

bil Tm+n(U) (u₁⊗ · · · ⊗u_m, v₁⊗ · · · ⊗v_n)−→u₁⊗ · · · ⊗v_n i.e. ⊗defines (after bilinear extension) a product

T∗(U)×T∗(U)−→^⊗

bil T∗(U)

T∗(U) becomes a graded associative K-algebra with unit element, the covariant tensor algebra.

Covariant, because the functor U −→ T∗(U) from vector spaces to algebras is covari-ant, i.e. a homomorphism of vector spaces U → V induces an algebra homomorphism T∗(U)−→T∗(V) in the same direction. Also the tensor algebra can be characterized by a universal property:

T∗(U) is the “largest associative K-algebra with 1 generated by U”, i.e. for every ho-momorphism of vector spaces U −→^α A into an associative K-algebra with 1 exists an unique extension to an algebra homomorphismT∗(U)→A. It satisfiesu1⊗ · · · ⊗un−→

α(u1)·...·α(un).

U

∼=

α

V-sp. homom. //A T₁(U)⊂T∗(U)

∃! alg. homom.

55

The covariant tensor algebra of U is defined as

T^∗(U) :=T∗(U^∗) (monomials of degree nu^∗₁⊗ · · · ⊗u⁺_n ∈Tⁿ(U))

A homomorphism of vector spaces U → V induces a homomorphism of vector spaces V^∗ → U^∗ in the opposite direction (pull-back of linear forms) and hence an algebra homomorphism T^∗(U)→T(U^∗).

We will need mixed tensors which contain co- and contravariant components. We there-fore consider

T_r^s(U) :=U ⊗ · · · ⊗U

| {z }

r

⊗U^∗⊗ · · · ⊗U^∗

| {z }

s

r, s∈N0

with the convention T₀⁰(U) :=K, and put T(U) :=

∞

M

r,s=0

T_r^s(U).

Again, the natural bilinear maps

T_r^s₁¹(U)×T_r^s₂²(U)−→T_r^s₁¹_+r^+s₂²(U)

(u₁⊗ · · · ⊗u_r1⊗u^∗₁⊗ · · · ⊗u^∗_s1, v₁⊗. . .)−→u₁⊗ · · · ⊗v₁⊗ · · · ⊗u^∗₁⊗ · · · ⊗v^∗₁⊗. . . define a product and makeT(U) a bigraded associated K-algebra with 1, the tensor algebra of U. We habe natural inclusions T∗(U) ⊂T(U), T^∗(U)⊂T(U).Elements in T(U) are called tensors, elements inT_r^s(U) are called tensors of type (r,s), e.g. vectors (1,0), linear forms (0,1) , endomorphisms (1,1), scalar products (0,2).

Trace and contraction: From now on, let our “initial” vector spaces U . . . be finite di-mensional. Then the natural inclusionU ,→U^∗∗ is an isomorphism and induces natural isomorphisms

T_r^s(U^∗)∼=T_s^r(U), T(U^∗)∼=T(U).

The natural bilinear pairing

U ×U^∗ −→K, (u, u^∗)−→u^∗(u) induces the linear form

T₁¹(U) =U⊗U^∗∼= End(U)−^tr→K

Indeed, if {e_i} is a basis of U and {e^∗_i} the dual basis of U^∗, then the endomorphism A=^P

i,j

a_ije_i⊗e^∗_j

| {z }

→δ_ij

given by the matrix (a_ij) is mapped to ^P

i,j

a_ijδ_ij =^P

i

a_ii= tr(A).

18.12.2012

More generally one can pair the i–th covariant component of a homogeneous tensor of type≥(i, j) with thej–th contravariant component and thus obtain the contraction homomorphisms

C_i^j:T_r^s(U)→T_r−1^s−1(U) given by

u₁⊗ · · · ⊗u_r⊗u^∗₁⊗ · · · ⊗u^∗_s 7→u^∗_j(u_i)·u₁⊗ · · · ⊗u_bi⊗ · · · ⊗u_r⊗u^∗₁⊗ · · · ⊗cu^∗_j⊗ · · · ⊗u^∗_s. A natural non–degenerate pairing

T_r^s(U)×T_s^r(U)→K can be obtained e.g. as the composition

T_r^s(U)×T_s^r(U)→T_r^s(U)⊗T_s^r(U)∼=T_r+s^r+s(U)→K,

which is

(u1⊗ · · · ⊗ur⊗u^∗₁⊗ · · · ⊗u^∗_s, v1⊗ · · · ⊗vs⊗v^∗₁⊗ · · · ⊗v^∗_r)7→

r

Y

i=1 s

Y

j=1

v_i^∗(ui)u^∗_j(vj).

The bases of T_r^s(U) and T_s^r(U) induced by a basis of U are wrt. this pairing dual to each other. The pairing induces a natural isomorphism:

(T_r^s(U))^∗∼=T_s^r(U)∼=T_r^s(U^∗)

In particular, we can interprete homogeneous contravariant tensors as multilinear forms and vice versa.

Mult_s(U)∼= (T_s⁰(U))^∗ ∼=T₀^s(U)∼=T_s⁰(U^∗) To the (0, s)–tensor u^∗₁⊗ · · · ⊗u^∗_s corresponds the multilinear form

(u1, . . . , us)7→

s

X

i=1

u^∗_i(ui) If{e_i} is a basis of U, then

{e^∗_i

1 ⊗ · · · ⊗e^∗_is |1≤i1, . . . , is≤dimU}

is a basis of Mult_s∼=T₀^s(U). A multilinear form µ∈Mult_s(U) can be represented wrt.

this basis as

µ= ^X

i1,...,is

µ(ei1, . . . , eis)e^∗_i1 ⊗ · · · ⊗e^∗_is

Im Dokument Differential Geometry (Seite 38-45)