Lie superalgebra

Lie superalgebra

Definition 1

.

A

Lie superalgebra

is a vector superspace equipped with a

bilinear map

[

⋅

,

⋅

]

:

V

⊗

V

→

V

,

v

⊗

w

↦

[

v

,

w

]

,

(1)

satisfying the following properties:

1.

If

v

and

w

are

homogeneous

vectors, then

[

v

,

w

]

is a homogeneous vector of degree

|

v

|

+

|

w

|

(

mod

2

)

,

2.

For any homogeneous vectors

v

,

w

,

[

v

,

w

]

=

(

-

1

)

|

v

|

⁢

|

w

|

+

1

⁢

[

w

,

v

]

,

3.

For any homogeneous vectors

u

,

v

,

w

,

(

-

1

)

|

u

|

⁢

|

w

|

⁢

[

u

,

[

v

,

w

]

]

+

(

-

1

)

|

v

|

⁢

|

u

|

⁢

[

v

,

[

w

,

u

]

]

+

(

-

1

)

|

w

|

⁢

|

v

|

⁢

[

w

,

[

u

,

v

]

]

= 0.

The map

[

⋅

,

⋅

]

is called a

Lie superbracket

.

Example 1

.

A

Lie algebra

V

can be considered as a Lie superalgebra by setting

V

=

V

0

and, therefore,

V

1

=

{

0

}

.

Example 2

.

Any associative

superalgebra

A

has a Lie superalgebra structure where, for any

homogeneous elements

a

,

b

∈

A

, the Lie superbracket is defined by the equation

[

a

,

b

]

=

a

⁢

b

-

(

-

1

)

|

a

|

⁢

|

b

|

⁢

b

⁢

a

.

(2)

The Lie superbracket (

2

) is called the

supercommutator bracket

on

A

.

Example 3

.

The space of graded derivations of a

supercommutative

superalgebra, equipped with the supercommutator bracket, is a Lie superalgebra.

Definition 2

.

A

vector superspace

is a

vector space

V

equipped with a decomposition

V

=

V

0

⊕

V

1

.

Let

V

=

V

0

⊕

V

1

be a vector superspace. Then any element of

V

0

is said to be

even

, and any element of

V

1

is said to be

odd

. By the definition of the

direct sum

, any element

v

of

V

can be uniquely written as

v

=

v

0

+

v

1

, where

v

0

∈

V

0

and

v

1

∈

V

1

.

Definition 3

.

A vector

v

∈

V

is

homogeneous

of degree

i

if

v

∈

V

i

for

i

=

0

or

1

.

If

v

∈

V

is homogeneous, then the degree of

v

is denoted by

|

v

|

. In other words, if

v

∈

V

i

, then

|

v

|

=

i

by definition.

Remark

.

The vector

0

is homogeneous of both degree

0

and

1

, and thus

|

0

|

is not

well-defined

.

Title

Lie superalgebra

Canonical name

LieSuperalgebra

Date of creation

2013-03-22 15:35:44

Last modified on

2013-03-22 15:35:44

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81R50

Classification

msc 17B60

Classification

msc 17B01

Classification

msc 81Q60

Synonym

Lie super algebra

Synonym

graded Lie algebra

Related topic

CartanCalculus

Related topic

Superalgebra

Related topic

GradedAlgebra

Related topic

LieAlgebroids

Related topic

SuperfieldsSuperspace

Related topic

SupersymmetryOrSupersymmetries

Related topic

LieAlgebroids

Related topic

JordanBanachAndJordanLieAlgebras

Related topic

LieAlgebra

Related topic

LieAlgebraCohomology

Related topic

SuperAlgebra

Related topic

CartanCalculus

Related topic

QuantumGravityTheories

Related topic

Fu

Defines

vector superspace

Defines

Lie superbracket

Defines

supercommutator bracket

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