exterior algebra

exterior algebra

1

Introductory remarks.

We begin with some informal remarks to motivate the formal definitions
found in the next

section

. Throughout,

V

is a

vector space

over a
field

K

. Many of the

concepts

and constructions discussed below
apply verbatim to modules over

commutative rings

, but we will stick to
vector spaces to keep things simple.

The

exterior product

, commonly denoted by the wedge symbol

∧

and also known as the

wedge product

, is an

antisymmetric

variant of
the

tensor product

. The former, like the latter is an

associative

,

bilinear operation

. Thus, for all

u

,

v

∈

V

and

a

,

b

,

c

,

d

∈

K

,
we have

(

a

⁢

u

+

b

⁢

v

)

∧

(

c

⁢

u

+

d

⁢

v

)

=

a

⁢

c

⁢

(

u

∧

u

)

+

a

⁢

d

⁢

(

u

∧

v

)

+

b

⁢

c

⁢

(

v

∧

u

)

+

b

⁢

d

⁢

(

v

∧

v

)

(1)

(

a

⁢

u

+

b

⁢

v

)

⊗

(

c

⁢

u

+

d

⁢

v

)

=

a

⁢

c

⁢

(

u

⊗

u

)

+

a

⁢

d

⁢

(

u

⊗

v

)

+

b

⁢

c

⁢

(

v

⊗

u

)

+

b

⁢

d

⁢

(

v

⊗

v

)

.

(2)

The essential

difference

between the two

operations

is that all
squares formed using the exterior product vanish, by definition.
Thus,

v

∧

v

=

0

,

(3)

whereas

v

⊗

v

≠

0

. Hence, the expressions in
(

1

) are equal to

(

a

⁢

d

-

b

⁢

c

)

⁢

u

∧

v

, but there is no
way to simplify further the right-hand side of (

2

).

A

polarization

argument

shows
that for

u

,

v

∈

V

we have

0

=

(

u

+

v

)

∧

(

u

+

v

)

=

u

∧

u

+

u

∧

v

+

v

∧

u

+

v

∧

v

=

u

∧

v

+

v

∧

u

.

Therefore, if the

characteristic

(

http://planetmath.org/characteristic

) of the underlying field

K

is not equal to

2

, that is if

1

+

1

≠

0

, then the key

postulate

(

3

) is

logically equivalent

to the antisymmetry condition

u

∧

v

=

-

v

∧

u

,

u

,

v

∈

V

.

(4)

However, if the characteristic is 2, that is if

K

is a field where

1

=

-

1

, then (

3

) does not, necessarily, follow from
(

4

). Therefore, to keep things as general as possible,
we must use (

3

) to formulate the essential

identity

satisfied by the exterior product.

So far so good, but we have not yet given a meaning to the symbol

u

∧

v

. The geometric

interpretation

of

u

∧

v

is that of
an oriented area element in the plane

spanned by

u

and

v

. Without
additional

structure

, there is no way to assign a area measurement to
a parallelogram in a vector space. However, parallelograms that lie
in the same plane are commensurate. If we adopt the parallelogram
spanned by

u

and

v

as the standard area, we can say that the
oriented area of another parallelogram, say one that is spanned by

a

⁢

u

+

b

⁢

v

and

c

⁢

u

+

d

⁢

v

, has an area that is

a

⁢

d

-

b

⁢

c

times the area of the
first parallelogram. The exterior product allows us to express this
algebraically. To wit,

(

a

⁢

u

+

b

⁢

v

)

∧

(

c

⁢

u

+

d

⁢

v

)

=

(

a

⁢

d

-

b

⁢

c

)

⁢

(

u

∧

v

)

,

u

,

v

∈

V

,

a

,

b

,

c

,

d

∈

K

.

The analogous interpretation for vectors is that of
an oriented length element on a line. For this reason, the object

u

∧

v

is referred to as a bivector.

From a more

algebraic

point of view, a bivector

u

∧

v

can be
considered as a formal antisymmetric

product

of vectors

u

and

v

,
in much the same way that

u

⊗

v

can be regarded as a formal

non-commutative

product of two vectors. Such descriptions can hardly
serve as

rigorous

definitions, but an explicit construction is not
really the way to go here.

Take the case of the tensor product. Formal sums of formal products

u

⊗

v

, where

u

,

v

∈

V

, form a certain vector space, which we
denote as

V

⊗

V

. However, rather than saying that

V

⊗

V

is

such and such a thing, it is better to state a certain

universal property

that describes

V

⊗

V

up to vector space

isomorphism

. The property in question is that every

bilinear

map

f

:

V

×

V

→

W

determines a unique

linear

map from

g

:

V

⊗

V

→

W

such that

f

⁢

(

u

,

v

)

=

g

⁢

(

u

⊗

v

)

,

u

,

v

∈

V

.

Similarly, formal sums of
bivectors constitute a vector space

Λ

2

⁢

(

V

)

, called the second

exterior power

of

V

. This vector space is defined, up to
isomorphism, by the condition that every antisymmetric, bilinear map

f

:

V

×

V

→

W

determines a unique

linear map

g

:

Λ

2

⁢

(

V

)

→

W

with

f

⁢

(

u

,

v

)

=

g

⁢

(

u

∧

v

)

.

Thus, in the same way that the tensor
product replaces bilinear maps with a certain kind of linear map,
the exterior product replaces bilinear, antisymmetric maps with linear
maps from

Λ

2

⁢

V

.

More generally,

k

-multivectors are

k

-fold products

v

1

∧

⋯

∧

v

k

, and the

k

th

exterior power,

Λ

k

⁢

(

V

)

, is the vector space of formal sums of

k

-multivectors. The product of a

k

-multivector and an

ℓ

-multivector is a

(

k

+

ℓ

)

-multivector. So, the

direct sum

⊕

k

Λ

k

⁢

(

V

)

forms an

associative algebra

, which is
closed with respect to the wedge product. This

algebra

, commonly
denoted by

Λ

⁢

(

V

)

, is called the

exterior algebra

of

V

.

Again, the

analogy

with the tensor product is useful. The

tensor
algebra

T

⁢

(

V

)

can be characterized as the associative,
non-commutative algebra

freely generated

by

V

. If the
characteristic of

k

is not 2, then the wedge product

satisfies

the
supercommutativity

relations

α

∧

β

=

(

-

1

)

k

+

ℓ

⁢

β

∧

α

,

α

∈

Λ

k

⁢

(

V

)

,

β

∈

Λ

ℓ

⁢

(

V

)

.

Thus,

Λ

⁢

(

V

)

can be characterized as the

supercommutative algebra

which is freely
generated by

V

.

2

Formal definitions.

Supercommutative algebras.

For the purposes of this discussion, we define a supercommutative
algebra to be an associative, unital

K

-algebra

A

with an

ℤ

2

-

grading

,

A

=

A

+

⊕

A

-

, such that for all odd

a

∈

A

-

we have

a

2

=

0

,

and such that for all even

b

∈

A

+

and all

a

∈

A

, we have

a

⁢

b

=

b

⁢

a

.

Using a polarization argument we see that the first
condition implies that for all odd

a

,

b

∈

A

-

we have

a

⁢

b

=

-

b

⁢

a

.

If the characteristic of

K

is different from

2

, then the

converse

is true, and we recover the usual definition of supercommutativity,
namely that

a

⁢

b

=

±

b

⁢

a

,

a

∈

A

±

,

b

∈

A

±

,

with the minus sign employed if both

a

and

b

or odd, and with

+

employed otherwise.

Exterior algebra.

Let

E

be a supercommutative algebra and

ι

:

V

→

E

-

a linear
map. We will say that

(

E

,

ι

)

is a model for the exterior algebra
of

V

, if every linear map

f

:

V

→

A

-

, where

A

a supercommutative
algebra, to a unique algebra homomorphism

g

:

E

→

A

, where
“lifts” means that

f

=

g

∘

ι

.
Diagrammatically:

The above condition on

E

is a universal property; this implies that
all models are isomorphic as algebras. Thus, when we speak of

Λ

⁢

(

V

)

, the exterior algebra of

V

, we are referring to the
isomorphism class of all such models. It is also common to identify

V

with its image

ι

⁢

(

V

)

, and to write

v

rather than

ι

⁢

(

v

)

.

Exterior powers.

For the purposes of the present entry, we define an antisymmetric map
to be a

k

-multilinear map

f

:

V

×

k

→

W

such that

f

⁢

(

…

,

v

,

v

,

…

)

=

0

for all

v

∈

V

. A polarization argument then implies the
usual antisymmetry condition, namely that for every

permutation

π

of

{

1

,

2

,

…

,

k

}

we have

f

⁢

(

v

π

1

,

…

,

v

π

k

)

=

sgn

⁡

(

π

)

⁢

f

⁢

(

v

1

,

…

,

v

k

)

,

v

1

,

…

,

v

k

∈

V

.

As usual, if the characteristic of

K

is different from

2

, the two
assertions are

equivalent

. However if

1

=

-

1

, then the first
assertion is stronger, and that is why we adopt it as the definition
of antisymmetry.

We now define a model of the

k

th

exterior power of

V

to be a
vector space

E

k

and an antisymmetric map

∧

:

V

×

k

→

E

k

such that every antisymmetric map

f

:

V

×

k

→

W

lifts to a unique linear map

g

:

E

k

→

W

, where “lifts”
means that

f

⁢

(

v

1

,

…

,

v

k

)

=

g

⁢

(

v

1

∧

⋯

∧

v

k

)

,

v

1

,

…

,

v

k

∈

V

.

As above, all models are isomorphic as vector spaces, and we use

Λ

k

⁢

(

V

)

to denote the isomorphism class of all such.

The standard model.

A model of the exterior algebra

Λ

⁢

(

V

)

, and the exterior powers

Λ

k

⁢

(

V

)

can be easily
constructed as the antisymmetrized

quotients

of the tensor algebra

T

⁢

(

V

)

=

⊕

k

=

0

∞

V

⊗

k

,

V

⊗

k

=

V

⊗

⋯

⊗

V

⁢

(k times)

.

To that end, let

S

⁢

(

V

)

denote the two sided ideal of

T

⁢

(

V

)

generated by
elements of the form

v

⊗

v

,

v

∈

V

. Then

S

⁢

(

V

)

=

⊕

k

=

0

∞

S

k

⁢

(

V

)

,

where

S

k

⁢

(

V

)

=

S

⁢

(

V

)

∩

V

⊗

k

,

and let

E

⁢

(

V

)

=

T

⁢

(

V

)

/

S

⁢

(

V

)

,

E

k

⁢

(

V

)

=

V

⊗

k

/

S

k

⁢

(

V

)

denote the
indicated quotients, with

a

:

T

⁢

(

V

)

→

E

⁢

(

V

)

and

a

k

:

V

⊗

k

→

E

k

⁢

(

V

)

denoting the corresponding antisymmetrization

surjections

. It
is easy to see that

S

1

⁢

(

V

)

is the

trivial vector space

, and hence
that

E

1

⁢

(

V

)

≅

V

. We leave it as an exercise for the reader to
show that

E

k

⁢

(

V

)

,

k

≥

2

is a model of the

k

th

exterior
power, while

E

⁢

(

V

)

together with the map

V

→

E

1

⁢

(

V

)

is a model of
the full exterior algebra.

The canonical grading.

An inspection of the above
construction reveals that

E

⁢

(

V

)

=

⊕

k

=

0

∞

E

k

⁢

(

V

)

.

Indeed, every model of
exterior algebra carries a canonical grading. Let

E

be a particular
model of the exterior algebra of

V

. For

k

=

1

,

2

,

…

, we will call

α

∈

E

, a

k

-

primitive element

if

α

=

v

1

∧

⋯

∧

v

k

,

for some

v

i

∈

V

. We now let

E

k

⊂

E

denote the vector space spanned by all

k

-primitive
elements, and let

E

0

=

K

.

Proposition 1

.

The

subspace

E

k

is a model for the

k

𝑡ℎ

exterior power of

V

. Furthermore,

E

=

⊕

k

=

0

∞

E

k

.

Categorical formulation.

The above definition of exterior product has a very appealing

categorical

formulation. Let

𝕊

denote the

category

of
supercommutative

K

-algebras, let

𝕍

denote category of vector
spaces over

K

, and let

(

⋅

)

-

:

𝕊

→

𝕍

denote the

forgetful functor

A

↦

A

-

. We may now say that the
exterior algebra

function

Λ

:

𝕍

→

𝕊

is the

left
adjoint

of

(

⋅

)

-

. In other words,

Hom

𝕍

⁡

(

V

,

A

-

)

≅

Hom

𝕊

⁡

(

Λ

⁢

(

V

)

,

A

)

,

V

∈

𝕍

,

A

∈

𝕊

,

with the
isomorphism natural in

V

and

A

.

It is useful to compare the above definition to the categorical
definition of the tensor algebra. Let

𝔸

denote the category of
associative, unital

K

-algebras, and let

F

:

𝔸

→

𝕍

be the
forgetful functor that gives the underlying vector space structure of
a

K

-algebra. We can then define the tensor algebra

T

⁢

(

V

)

of a
vector space

V

by saying that

T

:

𝕍

→

𝔸

is the left-adjoint
of

F

:

𝔸

→

𝕍

. Thus, whereas

T

⁢

(

V

)

as the

associative
algebra

freely generated by

V

, the exterior algebra

Λ

⁢

(

V

)

is the

supercommutative algebra

freely generated by

V

. The
antisymmetrization
quotient map

a

V

:

T

⁢

(

V

)

→

Λ

⁢

(

V

)

is a

natural transformation

between
these two

functors

.

3

Finite dimensional models.

Basis models.

If

V

is an

n

-dimensional vector space, there are some
down-to-earth constructions of

Λ

⁢

(

V

)

that go a long way to
illuminate the nature of the exterior product. Suppose then, that

V

is

n

-dimensional, and let

e

1

,

…

,

e

n

be a basis of

V

. For
every ascending

sequence

0

≤

i

1

<

i

2

<

⋯

<

i

k

≤

n

let us
introduce the symbol

e

I

=

e

i

1

⁢

…

⁢

i

k

to

represent

the

primitive

k

-multivector

e

i

1

∧

…

∧

e

i

k

. If

I

is the
empty sequence, we let

e

I

denote the unit element of the field

K

.

Proposition 2

.

The

(

n

k

)

-dimensional vector space spanned by

e

i

1

⁢

…

⁢

i

k

is a model of

Λ

k

⁢

(

V

)

.

Note that

Λ

1

⁢

(

V

)

is just the

n

-dimensional space spanned by
the basis symbols

e

1

,

…

,

e

n

. As such,

Λ

1

⁢

(

V

)

is
naturally isomorphic to

V

. For

disjoint

sequences

I

and

J

, let
us define

e

I

∧

e

J

=

sgn

⁡

(

I

⁢

J

)

⁢

e

[

I

⁢

J

]

,

where

[

I

⁢

J

]

denotes the
ascending sequence composed of the union of

I

and

J

, and where

sgn

⁡

(

I

⁢

J

)

=

±

1

denotes the parity of the permutation that takes the
sorted list

[

I

⁢

J

]

to the unsorted

concatenation

I

⁢

J

. If

I

and

J

have one or more

elements

in common, we define

e

I

∧

e

J

=

0

.

Here are some examples:

e

3

∧

e

12

=

e

123

,

e

2

∧

e

14

=

-

e

124

,

e

14

∧

e

23

=

e

1234

,

e

24

∧

e

13

=

-

e

1234

,

e

24

∧

e

14

=

0

.

Proposition 3

.

The

2

n

dimensional vector spanned by the symbols

e

I

, together
with the above product and the

linear isomorphism

from

V

to

Λ

1

⁢

(

V

)

is a model of the exterior algebra

Λ

⁢

(

V

)

.

Evidently, any list of numbers between

1

and

n

with length greater
than

n

will contain duplicates. Thus, an immediate

consequence

of
this construction is that

Λ

k

⁢

(

V

)

=

0

for

k

>

n

, and hence that

Λ

⁢

(

V

)

=

⊕

k

=

0

n

Λ

n

⁢

(

V

)

.

Alternating forms.

If

V

is

finite-dimensional

, we have the

natural isomorphism

between

V

and the double-dual

V

*

*

. We can exploit this natural
isomorphism to

construct

the following model of exterior algebra. Let

A

k

⁢

(

V

)

denote the vector space of

k

-multilinear mappings from

(

V

*

)

×

⋯

×

(

V

*

)

(

k

times) to

K

. Such an

mapping

is known as an

alternating

k

-form. Using the above duality we can
prove that

A

k

⁢

(

V

)

is a model for the

k

th

exterior power of

V

.

Given alternating forms

u

∈

A

k

⁢

(

V

)

and

v

∈

A

ℓ

⁢

(

V

)

,
let us define

u

∧

v

∈

A

k

+

ℓ

⁢

(

V

)

according to

(

u

∧

v

)

⁢

(

a

1

,

…

,

a

k

+

ℓ

)

=

∑

π

sgn

⁡

(

π

)

⁢

u

⁢

(

a

π

1

,

…

,

a

π

k

)

⁢

v

⁢

(

a

π

k

+

1

,

…

,

a

π

k

+

ℓ

)

,

where

a

1

,

…

,

a

k

+

ℓ

∈

V

*

,

and where the sum is taken over
all permutations

π

of

{

1

,

2

,

…

,

k

+

ℓ

}

such that

π

1

<

π

2

<

⋯

<

π

k

and

π

k

+

1

<

⋯

<

π

k

+

ℓ

, and
where

sgn

⁡

π

=

±

1

according to whether

π

is an
even or

odd permutation

. With this definition, we can show that

A

⁢

(

V

)

=

⊕

k

=

0

n

A

k

⁢

(

V

)

together with the above product, and the linear isomorphism

V

→

A

1

⁢

(

V

)

≅

V

*

*

is a model for the exterior algebra

Λ

⁢

(

V

)

.

4

Historical Notes.

The exterior algebra is also known as the Grassmann
algebra after its inventor

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Grassmann.html

Hermann Grassmann
who created it to give algebraic treatment of linear
geometry. Grassmann was also one of the first people to talk about
the geometry of an

n

-dimensional space with

n

an arbitrary

natural
number

. The axiomatics of the exterior product are needed to define
differential forms and therefore play an essential role in the theory
of integration on manifolds. Exterior algebra is also an essential
prerequisite to understanding de Rham’s theory of differential
cohomology.

Title

exterior algebra

Canonical name

ExteriorAlgebra

Date of creation

2013-03-22 12:34:14

Last modified on

2013-03-22 12:34:14

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

35

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A75

Synonym

Grassmann algebra

Related topic

AntiSymmetric

Defines

exterior product

Defines

wedge product

Defines

multivector

Defines

exterior power

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