trace forms on algebras

trace forms on algebras

Given an

finite dimensional

algebra

A

over a field

k

we define the
left(right)

regular representation

of

A

as the map

L

:

A

→

End

k

⁡

A

given by

L

a

⁢

b

:=

a

⁢

b

(

R

a

⁢

b

:=

b

⁢

a

).

Example 1

.

In a

Lie algebra

the left

representation

is called the

adjoint

representation and denoted

ad

⁡

x

and defined

(

ad

⁡

x

)

⁢

(

y

)

=

[

x

,

y

]

. Because

[

x

,

y

]

=

-

[

y

,

x

]

in

characteristic

not 2, there is generally no distinction of
left/right adjoint representations.

The

trace form

of

A

is defined as

⟨

,

⟩

:

T

×

T

→

k

:

⟨

a

,

b

⟩

:=

tr

⁡

(

L

a

⁢

L

b

)

.

Proposition 2

.

The trace form is a

symmetric bilinear form

.

Proof.

Given

a

,

b

,

x

∈

A

and

l

∈

k

then

L

a

+

l

⁢

b

⁢

x

=

(

a

+

l

⁢

b

)

⁢

x

=

a

⁢

x

+

l

⁢

b

⁢

x

=

L

a

⁢

x

+

l

⁢

L

b

⁢

x

.
So

L

a

+

l

⁢

b

=

L

a

+

l

⁢

L

b

. So we have

⟨

a

+

l

⁢

b

,

x

⟩

=

tr

⁡

(

L

a

+

l

⁢

b

⁢

L

x

)

=

tr

⁡

(

L

a

⁢

L

x

+

l

⁢

L

b

⁢

L

x

)

=

tr

⁡

(

L

a

⁢

L

x

)

+

l

⁢

tr

⁡

(

L

b

⁢

L

x

)

=

⟨

a

,

x

⟩

+

l

⁢

⟨

b

,

x

⟩

.

Furthermore,

tr

⁡

(

f

⁢

g

)

=

tr

⁡

(

g

⁢

f

)

is general property of traces, thus

⟨

a

,

b

⟩

=

tr

⁡

(

L

a

⁢

L

b

)

=

tr

⁡

(

L

b

⁢

L

a

)

=

⟨

b

,

a

⟩

.

So the trace form is a symmetric bilinear form.
∎

The

symmetric

property can be interpreted as a weak form of

commutativity

of
the product:

a

,

b

∈

A

commute within their trace from. A more essential
property arises for certain algebras and can be interpreted as “the product
is

associative

within the trace” and written as

⟨

a

⁢

b

,

c

⟩

=

⟨

a

,

b

⁢

c

⟩

.

(1)

We shall call such an algebra

weakly associative

though the term is not standard.

This property is clear for all

associative algebras

as:

L

a

⁢

b

⁢

L

c

⁢

(

x

)

=

(

(

a

⁢

b

)

⁢

c

)

⁢

x

=

(

a

⁢

(

b

⁢

c

)

)

⁢

x

=

L

a

⁢

L

b

⁢

c

⁢

x

.

When we use a Lie algebra, the trace form is commonly called the

Killing form

which has property (

1

). A result of Koecher shows that

Jordan algebras

also have this property.

Proposition 3

.

Given a weakly associative algebra, then the

radical

of the trace form
is an ideal of the algebra.

Proof.

We know the radical of form

R

is a

subspace

so we must simply show that

R

is an ideal. Given

x

∈

R

and

y

∈

A

then for all

z

∈

A

,

⟨

x

⁢

y

,

z

⟩

=

⟨

x

,

y

⁢

z

⟩

=

0

. Thus

x

⁢

y

∈

R

. Likewise

y

⁢

z

∈

R

so

R

is a

two-sided ideal

of

A

.
∎

From this result many authors define an algebra to be

semi-simple

if its trace
form is

non-degenerate

. In this way,

A

/

R

,

R

the radical of

A

, is semi-simple. [Some variations on this definition are often required over small
fields/characteristics, especially when characteristic is 2.]

More can be said when ideals are considered.

Proposition 4

.

Given a weakly associative algebra

A

, then
if

I

is an ideal of

A

then so is

I

⟂

.

Proof.

Given

a

∈

I

⟂

, then for all

b

∈

A

and

c

∈

I

, then

b

⁢

c

∈

I

as

I

is an ideal and so

⟨

a

⁢

b

,

c

⟩

=

⟨

a

,

b

⁢

c

⟩

=

0

as

a

∈

I

⟂

. This makes

a

⁢

b

∈

I

⟂

so

I

is a

right ideal

.
Likewise

⟨

c

,

b

⁢

a

⟩

=

⟨

c

⁢

b

,

a

⟩

=

0

so

b

⁢

a

∈

I

⟂

and
thus

I

⟂

is an ideal of

A

.
∎

To proceed one factors out the radical so that

A

is

semisimple

. Then
given an ideal

I

of

A

, if

I

∩

I

⟂

=

0

then as the trace form
is a non-degenerate

bilinear

from,

A

=

I

⊕

I

⟂

, and so by iterating
we produce a

decomposition

of

A

into minimal ideals:

A

=

A

1

⊕

⋯

⊕

A

s

.

Hence we arrive at the alternative definition of a

semisimple algebra

: that the
algebra be a

direct product

of

simple algebras

. To obtain the property

I

∩

I

⟂

=

0

it is sufficient to assume

A

has not ideal

I

such that

I

2

=

0

. This is the content of the proof in

Theorem 5

.

[

1

, Thm III.3]

Let

A

be a

finite-dimensional

weakly associative (trace) semisimple
algebra over a field

k

in which no ideal

I

≠

0

of

A

has

I

2

=

0

,
then

A

is a direct product of minimal ideals, that is, of simple
algebras.

Alternatively any bilinear form with (

1

) can be used. However,
the trace form is always definable and the desired properties are easily translated into implications about the

multiplication

of the algebra.

References

1

Jacobson, Nathan

Lie Algebras

, Interscience Publishers, New York, 1962.

2

Koecher, Max,

The Minnesota notes on Jordan algebras and their applications

.
Edited and annotated by Aloys Krieg and Sebastian Walcher.
[B] Lecture Notes in Mathematics 1710. Berlin: Springer. (1999).

Title

trace forms on algebras

Canonical name

TraceFormsOnAlgebras

Date of creation

2013-03-22 16:28:01

Last modified on

2013-03-22 16:28:01

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Topic

Classification

msc 17A01

Defines

regular representation

Defines

trace form

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