7.4 Tensor, Symmetric and Exterior Algebra of a Vector Space

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Let F be a field and let V be an F-vector space. We define a sequence Tr(V )

of F-vector spaces by setting T0(V ) = F, T1(V ) = V , and in general,

Tr(V ) =

Or

i=1

V = V F · · · F V (r factors ).

Note that “” gives a natural “multiplication:”

 : Tr(V ) × Ts(V ) −! Tr+s(V ),

where (_, _) 7! _  _. As a result, if we set

T(V ) =

1M

r=0

Tr(V ),

we have a natural F-algebra structure on T(V ), with multiplication given

by . The algebra T(V ) so determined is called the tensor algebra of V .

If we denote by i : V ! T(V ) the composition V = T1(V ) ,! T(V ), then

we have the following universal mapping property. If A is any F-algebra,

and if f : V ! A is any linear transformation, then there exists a unique

F-algebra homomorphism _ : T(V ) ! A that extends f. In other words, we

have the commutative triangle below:

V A,

T(V )

-

􀀀

􀀀

􀀀

􀀀

􀀀_ @

@

@

@

@R

f

i _

where i : V ! T(V ) is the inclusion map.

In order to facilitate discussions of the symmetric and exterior algebras

of the vector space V , we pause to make a few more comments concerning

166 CHAPTER 7. TENSOR PRODUCTS

the tensor algebra T(V ) of V . First of all if A is any F-algebra admitting a

direct sum decomposition A =

L1

i=0 Ar such that for all indices r, s we have

ArAs _ Ar+s, then we call A a graded algebra. Elements of Ai are called

homogeneous elements of degree r. Therefore, it is clear that the tensor

algebra T(V ) is a graded algebra.

Next, if A is a graded algebra and if I _ A is a 2-sided ideal in A,

we say that I is a homogeneous ideal (sometimes called a graded ideal) , if

I =

L1

r=0 Ar \ I.

The following is pretty routine:

Proposition 7.4.1 Let A be a graded algebra and let I be a 2-sided ideal

generated by homogeneous elements. Then I is a homogeneous ideal. In

this case A/I =

L1

r=0 Ar/(Ar \ I) is a graded algebra.

With the above in place, we now define the symmetric algebra of the

vector space V as the quotient algebra S(V ) = T(V )/I, where I is the

homogeneous ideal generated by tensors of the form vw−wv, v,w 2 V .

By Proposition 7.4.1, S(V ) =

L

Sr(V ) is a graded algebra, where Sr(V ) =

Tr(V )/(Tr(V ) \ I).

Multiplication in S(V ) is usually denoted by juxtaposition; in particular,

if v,w 2 V _ S(V ), then vw is the product of v and w. Equivalently vw

is just the coset: vw = v  w + I, and vw = wv v,w 2 V . As a result, if

{v1, v2, . . . , vn} is a basis of V , then Sr(V ) is spanned by elements of the

form ve1

1 ve2

2 · · · ven

n , where e1+e2+· · ·+en = r. In fact, these elements form

a basis of Sr(V ); see Proposition 7.4.2, below.

Note that there is a very natural isomorphism i : V

_=!

S1(V ) ,! S(V ).

The symmetric algebra S(V ) then enjoys the following universal property.

If A is any commutative F-algebra, and if f : V ! A is any linear transformation,

then there exists a unique F-algebra homomorphism   : S(V ) ! A

that extends f. In other words, we have the commutative triangle below:

V A,

S(V )

-

􀀀

􀀀

􀀀

􀀀

􀀀_ @

@

@

@

@R

f

i 

7.4. TENSOR, SYMMETRIC AND EXTERIOR ALGEBRA 167

Actually the symmetric algebra is a pretty familiar object:

Proposition 7.4.2 Let V have F-dimension n, and set A = F[x1, x2, . . . , xn],

where x1, x2, . . . xn are indeterminates over F. Then S(V ) _= A.

Finally, we turn to the so-called exterior algebra of the vector space

V . This time we start with the homogeneous ideal J _ T(V ) of T(V )

generated by homogeneous elements v  v, v 2 V . By Proposition 7.4.1, if

we set E(V ) = T(V )/J,

Vr(V ) = Tr(V )/(Tr L (V ) \ J), then then E(V ) = 1

r=0

Vr(V ) is a graded algebra (sometimes denoted

V

V ).

Again, we have a natural inclusion i : V ,! E(V ), and E(V ) has the

predictable universal mapping property: If If A is any F-algebra, and if

f : V ! A is any linear transformation satisfying f(v)2 = 0 for all v 2 V ,

then there exists a unique F-algebra homomorphism _ : E(V ) ! A that

extends f. In other words, we have the commutative triangle below:

V A,

E(V )

-

􀀀

􀀀

􀀀

􀀀

􀀀_ @

@

@

@

@R

f

i _

If we regard V as a subspace of E(V ) via the map i above, and if v,w 2 V ,

we denote the product of v and w by v ^ w; again, this is just the coset

v ^wvw+J. Therefore, it is clear that v ^v = 0, and if v,w 2 V we have

(v + w) ^ (v + w) = 0, which implies that v ^ w = −w ^ v. In particular, if

dim V = n and if v1, v2, . . . vr 2 V , where r > n, then v1 ^ v2 ^ · · · ^ vr = 0.

Therefore, r > n implies that

Vr(V ) = 0 for all m > n.

Proposition 7.4.3 Assume that V is finite dimensional and that A =

{v1, . . . , vn} is a basis of V . Let R = {i1, . . . , ir}, where 1 _ i1 < · · · < ir _

n, set N = {1, 2, . . . , n}, and set vR = vi1 ^ · · · ^ vir . Then {vR| R _ A}

spans E(V ) as a vector space. In particular dim E(V ) _ 2n.

In fact, in the above proposition, we get equality: dim E(V ) = 2n. To

prove this, it suffices to prove that dim

Vr V = (nr

) . The method of doing

168 CHAPTER 7. TENSOR PRODUCTS

this is interesting in its own right; we sketch the argument here. First of all,

let f1, f2, . . . , fr 2 V _ (the F-dual of V ), and define

F = F(f1,f2,...,fr) : V × V × · · · × V −! F

by setting F(w1,w2, . . . ,wr) = f1(w1)f2(w2) · · · fr(wr). It is routine to check

that F is multilinear; thus there exists a unique linear map

_ = _(f1,f2,...,fr) : V  V  · · ·  V −! F

satisfying _(w1  w2 . . .  wr) = f1(w1)f2(w2) · · · fr(wr).

Now let {v1, v2, . . . , vn} be the above basis of V , and let f1, f2, . . . , fn be

the dual functionals, i.e., satisfying fi(vj) = _ij . Let R = {i1, . . . , ir}, where

1 _ i1 < · · · < ir _ n, and define the linear map

_R =

X

_2Sr

sgn(_)_(fi_(1),fi_(2),···fi_(r)) : V  V  · · ·  V −! F.

It is easy to check that _R factors through

Vr V , giving a linear map

fK :

Vr V −! F,

satisfying

fR(w1 ^ · · · ^ wr) =

X

_2Sr

sgn(_)fi_(1)(w1)fi_(2)(w2) · · · fi_(r)(wr).

From the above, it follows immediately that fR(vR0) = _RR0 , which implies

that the set {vR| |R| = r} is a linearly independent subset of

Vr V. This

proves what we wanted, viz.,

Theorem 7.4.4 The exterior algebra E(V ) of the n-dimensional vector

space V has dimension 2n.

Exercises 7.4

1. Assume that the F-vector space V has dimension n. For each r _ 0,

compute the F-dimension of Sr(V ).

7.4. TENSOR, SYMMETRIC AND EXTERIOR ALGEBRA 169

2. Let V andW be F-vector spaces. An n-linear map f : V ×V ×· · ·×V !

W is called alternating if for any v 2 V , we have

f(. . . , v, . . . , v, . . .) = 0.

Prove that in this case there exists an F-linear map ˆ f :

Vn V ! W

such that

f(v1, v2, . . . , vn) = ˆ f(v1 ^ v2 ^ · · · ^ vn).

In particular, how can the determinant be interpreted as a linear functional

on

Vn V ?

3. Let T : V ! V be a linear transformation, and let r be a non-negative

integer. Show that there exists a unique linear transformation

Vr V T : r V !

Vr V satisfying

Vr T(v1 ^v2 ^· · ·^vr) = T(v1)^T(v2)^· · ·^

T(vr), where v1, v2, . . . , vr 2 V .

4. Let T : V ! V be a linear transformation, and assume that dim V =

n. Show that

Vn T = det T · 1Vn V :

Vn V !

Vn V.

5. Let G be a group represented on the F-vector space V (see page 146).

Show that the mapping G ! GLF(

Vr V ) given by g 7!

Vr g defines a

group representation on

Vr V, r _ 0.

6. VLet V be a vector space and let v 2 V . Define the linear map · ^ v : r V !

Vr+1 V by ! 7! ! ^ v. If dim V = n, compute the dimension

of the kernel of · ^ v.

7. Let V be an n-dimensional F-vector space. If d _ n, define the (n, d)-

Grassmann space, Gd(V ) as the set of all d-dimensional subspaces of

V . In particular, if d = 1, the set G1(V ) is more frequently called the

projective space on V , and is denoted by P(V ). We define a mapping

_ : Gd(V ) −! P(

Vd V ),

as follows. If U 2 Gd(V ), let {u1, . . . , ud} be a basis of U, and let _(U)

be the 1-space in P(

Vd V ) spanned by u1 ^ · · · ^ ud. Prove that _ :

Gd(V ) ! P(

Vd V ) is a well-defined injection of Gd(V ) into P(

Vd V ).

(This mapping is called the Pl¨ucker embedding .)

8. Let V be an n-dimensional over the finite field Fq. Show that the

Pl¨ucker embedding _ : Gn−1(V ) −! P(

Vn−1 V ) is surjective. This

170 CHAPTER 7. TENSOR PRODUCTS

implies that every element of z 2

Vn−1 V can be written as a “decomposable

element” of the form z = v1^v2^· · ·^vn−1 for suitable vectors

v1, v2, . . . , vn−1 2 V . (Actually this result is true independently of the

field F; see, e.g., M. Marcus, Finite Dimensional Linear Algebra, part

II, Pure and Applied Mathematics, Marcel Dekker, Inc., New York,

1975, page 7. An alternative approach, suggested to me by Ernie Shult,

is sketched in the exercise below.)

9. Let G = GL(V ) acting naturally on the n-dimensional vector space V .

(a) Show that the recipe g(f) = det g ·f _g−1, g 2 G, f 2 V _ defines

a representation of G on V _, the dual space of V .

(b) Show that in the above action, G acts transitively on the non-zero

vectors of V _.

(c) Fix any isomorphism

Vn V _= F; show that the map

Vn−1 V !

V _ given by ! 7! ! ^ · is a G-equivarient isomorphism. (See

page 7.)

(d) Since G clearly acts on the set of decomposable vectors in

Vn−1 V ,

conclude that every vector is decomposable.

10. Let V,W be F-vector spaces. Prove that there is an isomorphism

M

i+j=r

Vi V _

Vj W −!

Vr(V _W).

11. Let V be an F-vector space, where char F 6= 2. Define the linear

transformation S : V  V ! V  V by setting S(v  w) = w  v.

(a) Prove that S has minimal polynomial mS(x) = (x − 1)(x + 1).

(b) If V1 = ker(S − I), V−1 = ker(S + I), conclude that V  V =

V1 _ V−1.

(c) Prove that V1

_= S2(V ), V−1

_=

V2(V ).

(d) If T : V ! V is any linear transformation, prove that V1 and V−1

are T  T-invariant subspaces of V  V .

12. Let V an n-dimensional F-vector space.

(a) Prove that E(V ) is graded-commutative in the sense of Exercise 5

of Section 7.3.

7.4. TENSOR, SYMMETRIC AND EXTERIOR ALGEBRA 171

(b) If L is a one-dimensional F-vector space, prove that as gradedcommutative

algebras,

E(V ) _= E(L)  E(L)  · · ·  E(L) (n factors).

172 CHAPTER 7. TENSOR PRODUCTS

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