Lie algebras from other algebras

Lie algebras from other algebras

1

Lie algebras from associative algebras

Given an

associative

(unital)

algebra

A

over a

commutative ring

R

, we define

A

-

as the

R

-module

A

together with a new

multiplication

[

,

]

:

A

×

A

→

A

derived from the associative multiplication as follows:

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

.

This

operation

is commonly called the

commutator bracket

on

A

.

Proposition 1

.

A

-

is a

Lie algebra

.

Proof.

We already know

A

-

is a module so we need simply to confirm that the commutator
bracket is a

bilinear mapping

and then demonstrate that it is

alternating

and

satisfies

the

Jacobi identity

.

Given

a

,

b

,

c

∈

A

, and

l

∈

R

then

[

l

⁢

a

+

b

,

c

]

=

(

l

⁢

a

+

b

)

⁢

c

-

c

⁢

(

l

⁢

a

+

b

)

=

l

⁢

(

a

⁢

c

-

c

⁢

a

)

+

(

b

⁢

c

-

c

⁢

b

)

=

l

⁢

[

a

,

c

]

+

[

b

,

c

]

.

The

similar

argument

in the second

variable

shows that the operation is

bilinear

.

Next,

[

x

,

x

]

=

x

⁢

x

-

x

⁢

x

=

0

so

[

,

]

is alternating. Finally for the Jacobi
identity we compute directly.

[

[

a

,

b

]

,

c

]

+

[

[

b

,

c

]

,

a

]

+

[

[

c

,

a

]

,

b

]

=

(

a

⁢

b

-

b

⁢

a

)

⁢

c

-

c

⁢

(

a

⁢

b

-

b

⁢

a

)

+

(

b

⁢

c

-

c

⁢

b

)

⁢

a

-

a

⁢

(

b

⁢

c

-

c

⁢

b

)

+

(

c

⁢

a

-

a

⁢

c

)

⁢

b

-

b

⁢

(

c

⁢

a

-

a

⁢

c

)

=

a

⁢

b

⁢

c

-

b

⁢

a

⁢

c

-

c

⁢

a

⁢

b

+

c

⁢

b

⁢

a

+

b

⁢

c

⁢

a

-

c

⁢

b

⁢

a

-

a

⁢

b

⁢

c

+

a

⁢

c

⁢

b

+

c

⁢

a

⁢

b

-

a

⁢

c

⁢

b

-

b

⁢

c

⁢

a

+

b

⁢

a

⁢

c

=

0

.

∎

We notice this produces a

functor

from the

category

of

associative algebras

to the category of Lie algebras. However, to every

commutative algebra

A

,

A

-

is a trivial Lie algebra, and so this functor is not

faithful

.
More generally, the center of an arbitrary associative algebra

A

is lost
to the Lie algebra

structure

A

-

.

We do observe some relationships between the

algebraic structure

of

A

and
that of

A

-

.

Theorem 2

.

If

I

⊴

A

then

I

⊲

A

-

.

Proof.

We observe that a

submodule

of

A

is a submodule of

A

-

as the two are
identitcal as modules. It remains to show

[

I

,

A

-

]

≤

I

. So given

a

∈

I

and

b

∈

A

, then

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

and as

a

⁢

b

,

b

⁢

a

∈

I

we conclude

[

a

,

b

]

∈

I

.
∎

2

Associative envelopes

Given a Lie algebra

𝔤

it is often desirable to reverse the process
described above, that is, to provide an associative algebra

A

for which

𝔤

=

A

-

. In general this is impossible as we will now explain.

Let

V

be a

vector space

and

A

the

endomorphism

algebra on

V

. Then we
give the name

𝔤

⁢

𝔩

⁢

(

V

)

to the Lie algebra

A

-

(noting that

A

is
associative under the

composition

of functions operation.) Then we can also
define a

subalgebra

𝔰

⁢

𝔩

⁢

(

V

)

as the set of

linear transformations

with trace 0.

Now we claim that

𝔰

⁢

𝔩

2

⁢

(

ℂ

)

is not equal to

B

-

for any associative (unital) algebra

B

. For it is easy to see

𝔰

⁢

𝔩

2

⁢

(

ℂ

)

has a basis of
three

elements

:

e

=

[

0

1

0

0

]

,

h

=

[

1

0

0

-

1

]

,

f

=

[

0

0

1

0

]

.

Therefore

B

would also be 3-dimensional. We also know that

𝔰

⁢

𝔩

2

⁢

(

ℂ

)

is a

simple Lie algebra

, that is, it has no

proper ideals

. Therefore by

Theorem

2

,

B

can have no
ideals either, so

B

must be simple. However the finite dimensional

simple rings

over

ℂ

are

isomorphic

to

matrix rings

(by the

Wedderburn-Artin theorem

)

M

n

⁢

(

ℂ

)

and thus cannot have

dimension

3.

This forces the weaker question as to whether a Lie algebra can be embedded
in

A

-

for some associative algebra

A

. We call such

embeddings

associative envelopes

of the Lie aglebra. The

existence

of associative
envelopes of arbitrary Lie algebras is answered by a corollary
to the

Poincare-Birkhoff-Witt theorem

.

Theorem 3

.

Every Lie algebra

g

embeds in the

universal enveloping algebra

U

⁢

(

g

)

-

, where

U

⁢

(

g

)

is an associative algebra.

Finite dimensional analogues also exist, some of which are simpler to
observe. For

instance

, a Lie aglebra

𝔤

can be represented in

𝔤

⁢

𝔩

⁢

(

𝔤

)

by the

adjoint representation

. The representation is
not faithful unless the center of

𝔤

is

trivial

. However,
for semi-simple Lie algebras, the adjoint representation thus suffices
as an associative envelope.

Remark 4

.

This result is in contrast to

Jordan algebras

where there are

isomorphism

types (for example

3

×

3

matrices over the

octonions

) which cannot be embedded
in

A

+

for any associative algebra

A

. [

A

+

is the derived algebra
of

A

under the

product

a

.

b

=

a

⁢

b

+

b

⁢

a

.]

2.1

Lie algebra from non-associative algebras

If

A

is not an associative algebra to begin with then we may still determine
the commutator bracket is bilinear and alternating. However, the Jacobi
identity is in question. If we define the

associator

bracket

as

[

a

,

b

,

c

]

=

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

then we can write the computation for the Jacobi
identity as:

[

[

a

,

b

]

,

c

]

+

[

[

b

,

c

]

,

a

]

+

[

[

c

,

a

]

,

b

]

=

(

a

⁢

b

-

b

⁢

a

)

⁢

c

-

c

⁢

(

a

⁢

b

-

b

⁢

a

)

+

(

b

⁢

c

-

c

⁢

b

)

⁢

a

-

a

⁢

(

b

⁢

c

-

c

⁢

b

)

+

(

c

⁢

a

-

a

⁢

c

)

⁢

b

-

b

⁢

(

c

⁢

a

-

a

⁢

c

)

=

(

a

⁢

b

)

⁢

c

-

(

b

⁢

a

)

⁢

c

-

c

⁢

(

a

⁢

b

)

+

c

⁢

(

b

⁢

a

)

+

(

b

⁢

c

)

⁢

a

-

(

c

⁢

b

)

⁢

a

-

a

⁢

(

b

⁢

c

)

+

a

⁢

(

c

⁢

b

)

+

(

c

⁢

a

)

⁢

b

-

(

a

⁢

c

)

⁢

b

-

b

⁢

(

c

⁢

a

)

+

b

⁢

(

a

⁢

c

)

=

(

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

)

+

(

(

b

⁢

c

)

⁢

a

-

b

⁢

(

c

⁢

a

)

)

+

(

(

c

⁢

a

)

⁢

b

-

c

⁢

(

a

⁢

b

)

)

-

(

(

b

⁢

a

)

⁢

c

-

b

⁢

(

a

⁢

c

)

)

-

(

(

c

⁢

b

)

⁢

a

-

c

⁢

(

b

⁢

a

)

)

-

(

(

a

⁢

c

)

⁢

b

-

a

⁢

(

c

⁢

b

)

)

=

[

a

,

b

,

c

]

+

[

b

,

c

,

a

]

+

[

c

,

a

,

b

]

-

[

b

,

a

,

c

]

-

[

c

,

b

,

a

]

-

[

a

,

c

,

b

]

.

We can write this

right hand side

using

permutations

on the set

{

a

,

b

,

c

}

as:

∑

σ

∈

Alt

⁡

(

{

a

,

b

,

c

}

)

[

[

a

σ

,

b

σ

]

,

c

σ

]

=

[

a

,

b

]

,

c

]

+

[

[

b

,

c

]

,

a

]

+

[

[

c

,

a

]

,

b

]

=

∑

σ

∈

Sym

⁡

(

{

a

,

b

,

c

}

)

sign

(

σ

)

[

a

σ

,

b

σ

,

c

σ

]

.

That is, in a

non-associative algebra

the corresponding Jacobi identity
is the possibly non-trivial sum over all permutations of associators.
We consider a few non-associative examples.

•

If

A

is a

commutative

non-associative algebra (perhaps a Jordan algebra) then

[

a

,

b

,

c

]

=

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

=

(

b

⁢

a

)

⁢

c

-

(

b

⁢

c

)

⁢

a

=

c

⁢

(

b

⁢

a

)

-

(

c

⁢

b

)

⁢

a

=

[

c

,

b

,

a

]

so the Jacobi identity holds. However, if

A

is commutative
then

[

a

,

b

]

=

0

to begin with so the associated Lie algebra product is
trivial.

•

If

A

is an

alternative algebra

, so

[

a

,

b

,

c

]

=

-

[

b

,

a

,

c

]

, then again
the Jacobi identity holds. So

A

-

is a Lie algebra. The typicall non-associative examples of an alternative algebra are the

octonion algebras

.
These produce a non-trivial Lie algebra.

•

We can also consider beginning with a Lie algebra

A

and producing

A

-

. To avoid confusing the bracket of

A

and that of

A

-

we let the
multiplication of

A

be denoted by

juxtaposition

,

a

⁢

b

,

a

,

b

∈

A

. Recall
that in a Lie algebra of

characteristic

0 or odd then

a

⁢

b

=

-

b

⁢

a

so that

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

=

2

⁢

a

⁢

b

in

A

-

. So we have simply scaled the original product
of

A

by

2

. To see the Jacobi identity still holds we note

[

a

,

b

,

c

]

=

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

=

-

(

b

⁢

a

)

⁢

c

+

(

b

⁢

c

)

⁢

a

=

c

⁢

(

b

⁢

a

)

-

(

c

⁢

b

)

⁢

a

=

[

c

,

b

,

a

]

.

So once again the associators cancel.

Title

Lie algebras from other algebras

Canonical name

LieAlgebrasFromOtherAlgebras

Date of creation

2013-03-22 16:37:33

Last modified on

2013-03-22 16:37:33

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Example

Classification

msc 17B99

Related topic

Algebras2

Defines

associative envelope

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