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LieAlgebras[Derivations] - find the derivations of a Lie algebra, find the derivations of a general non-commutative algebra

Calling Sequences

     Derivations(Algname, "keyword")

Parameters

     Algname   - (optional) name or string, the name of a Lie algebra

     keyword   - one of the 3 keywords "Inner", "Full", or "Outer"

 

Description

Examples

Description

• 

 Let 𝔤 be a n-dimensional Lie algebra. An n × n matrix B is a derivation for 𝔤  if the associated linear transformation mapping LB : 𝔤 𝔤 satisfies

LBx,y = LBx, y + x, LABy)  for all x, y 𝔤.

The set of all derivations defines a matrix Lie algebra denoted by Der𝔤. For each x 𝔤, the adjoint matrix adx is a derivation -- these are the inner derivations InnDer(𝔤). The inner derivations define an ideal in Der(𝔤)and the quotient Lie algebra Der(𝔤)/InnDer(𝔤) is the Lie algebra of outer derivations.

• 

Let 𝔸 be a n-dimensional Lie algebra (such as the octonion, a Jordan algebra, or a Clifford algebra. See AlgebraLibraryData). An n × n matrix B is a derivation for 𝔸 if the associated linear transformation mapping LB : 𝔸 𝔸 satisfies

LBxy = LBxy+ xLBy  for all x, y 𝔸.

• 

Derivations(Algname, "Inner") returns a list of linearly independent matrices which defines a basis for the Lie algebra of inner derivations for the Lie algebra Algname.

• 

Derivations(Algname) or Derivations(Algname, "Full") returns a list of linearly independent matrices which defines a basis for the Lie algebra of all derivations for the Lie algebra Algname.

• 

Derivations(Algname, "Outer") returns a list of linearly independent matrices which gives a representative list of the outer derivations for the Lie algebra Algname.

• 

If Algname is a general non-commutative algebra, then Derivations(Algname) computes the derivations of this algebra.

• 

The command Derivations is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Derivations(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Derivations(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.

 

L1_DGLieAlgebra,Alg1,4,1,4,1,1,2,4,1,1,2,4,2,1,3,4,3,1:

DGsetupL1:

Alg1 > 

MultiplicationTableLieBracket

e1,e4=e1,e2,e4=e1+e2,e3,e4=e3

(2.1)

 

For the Lie algebra Alg1 we find that Derivations(Alg1, "Inner") is 4 dimensional and Derivations(Alg1) is 8 dimensional.

Alg1 > 

InnerDerivationsInner

Inner:=1100010000100000,0001000000000000,0000000100000000,0000000000010000

(2.2)
Alg1 > 

DerDerivationsFull

Der:=1000010000000000,0100000000000000,0010000000000000,0001000000000000,0000000100000000,0000000001000000,0000000000100000,0000000000010000

(2.3)
Alg1 > 

OuterDerivationsOuter

Outer:=1000010000000000,0100000000000000,0010000000000000,0000000001000000

(2.4)

 

We can study the properties of Derivations(Alg1) by initializing these matrices as a Lie algebra. We use as a basis for Derivations(Alg1) the inner and outer derivations.

Alg1 > 

BasisopInner,opOuter:

Alg1 > 

L2LieAlgebraDataBasis,DerAlg

L2:=e1,e2=e2,e1,e3=e2+e3,e1,e4=e4,e2,e5=e2,e3,e5=e3,e3,e6=e2,e3,e8=e4,e4,e7=e2,e5,e7=e7,e5,e8=e8,e7,e8=e6

(2.5)
Alg1 > 

DGsetupL2,E,a:

 

We see that the derivation algebra is solvable.

DerAlg > 

QuerySolvable

true

(2.6)

 

We check that the span of the vectors E1, E2, E3, E4(corresponding to the inner derivations) define an ideal.

DerAlg > 

QueryE1,E2,E3,E4,Ideal

true

(2.7)

 

We compute the quotient algebra of outer derivations.

DerAlg > 

L3QuotientAlgebraE1,E2,E3,E4,E5,E6,E7,E8,OuterAlg

L3:=e1,e3=e3,e1,e4=e4,e3,e4=e2

(2.8)
DerAlg > 

DGsetupL3

Lie algebra: OuterAlg

(2.9)

 

Example 2.

We show that the derivations of the octonions form a 14-dimensional semi-simple Lie algebra (which can be seen to be compact real form of the exceptional Lie algebra g2).

 

L4AlgebraLibraryDataOctonions,Oct

L4:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e1.e5=e5,e1.e6=e6,e1.e7=e7,e1.e8=e8,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e2.e5=e6,e2.e6=e5,e2.e7=e8,e2.e8=e7,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e3.e5=e7,e3.e6=e8,e3.e7=e5,e3.e8=e6,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1,e4.e5=e8,e4.e6=e7,e4.e7=e6,e4.e8=e5,e5.e1=e5,e5.e2=e6,e5.e3=e7,e5.e4=e8,e52=e1,e5.e6=e2,e5.e7=e3,e5.e8=e4,e6.e1=e6,e6.e2=e5,e6.e3=e8,e6.e4=e7,e6.e5=e2,e62=e1,e6.e7=e4,e6.e8=e3,e7.e1=e7,e7.e2=e8,e7.e3=e5,e7.e4=e6,e7.e5=e3,e7.e6=e4,e72=e1,e7.e8=e2,e8.e1=e8,e8.e2=e7,e8.e3=e6,e8.e4=e5,e8.e5=e4,e8.e6=e3,e8.e7=e2,e82=e1

(2.10)

DGsetupL4:

 

We find that the derivation algebra is 14-dimensional

DerDerivationsOct

Der:=0000000000100000010000000000000000000000000000100000010000000000,0000000000010000000000000100000000000000000000010000000000000100,0000000000001000000000000000001001000000000000000001000000000000,0000000000000100000000000000000100000000010000000000000000010000,0000000000000010000000000000100000010000000000000100000000000000,0000000000000001000000000000010000000000000100000000000001000000,0000000000000000000100000010000000000000000000000000000100000010,0000000000000000000010000000010000100000000100000000000000000000,0000000000000000000001000000100000010000001000000000000000000000,0000000000000000000000100000000100000000000000000010000000010000,0000000000000000000000010000001000000000000000000001000000100000,0000000000000000000000000000000000000100000010000000000100000010,0000000000000000000000000000000000000010000000010000100000000100,0000000000000000000000000000000000000001000000100000010000001000

(2.11)

nopsDer

14

(2.12)

 

Calculate the structure equations for the derivations, initialize ,and check that the derivation algebra is semi-simple.

Oct > 

L5LieAlgebraDataDer,Alg5

L5:=e1,e2=e7,e1,e3=e8,e1,e4=e5e9,e1,e5=e4e10,e1,e6=e11,e1,e7=e2,e1,e8=e3,e1,e9=e4e10,e1,e10=e5+e9,e1,e11=e6,e1,e12=e13,e1,e13=e12,e2,e3=e5,e2,e4=2e6,e2,e5=e3,e2,e6=2e4,e2,e7=e1,e2,e8=e4,e2,e9=e3e11,e2,e10=e6,e2,e11=e5+e9,e2,e12=e14,e2,e14=e12,e3,e4=e12,e3,e5=2e22e13,e3,e6=e14,e3,e7=e10,e3,e8=e1,e3,e9=e2+e13,e3,e10=e7,e3,e12=e4,e3,e13=e5,e3,e14=e6,e4,e5=e14,e4,e6=2e2,e4,e7=e11,e4,e8=e2,e4,e9=e1+e14,e4,e11=e7,e4,e12=e3,e4,e13=e6,e4,e14=e5,e5,e6=e12,e5,e7=e6e8,e5,e8=e7e12,e5,e10=e1+e14,e5,e11=e2e13,e5,e12=e6,e5,e13=e3,e5,e14=e4,e6,e7=e5e9,e6,e9=e7e12,e6,e10=e2,e6,e11=e1,e6,e12=e5,e6,e13=e4,e6,e14=e3,e7,e8=e9,e7,e9=e8,e7,e10=2e11,e7,e11=2e10,e7,e13=e14,e7,e14=e13,e8,e9=2e72e12,e8,e10=e13,e8,e11=e14,e8,e12=e9,e8,e13=e10,e8,e14=e11,e9,e10=e14,e9,e11=e13,e9,e12=e8,e9,e13=e11,e9,e14=e10,e10,e11=2e7,e10,e12=e11,e10,e13=e8,e10,e14=e9,e11,e12=e10,e11,e13=e9,e11,e14=e8,e12,e13=2e14,e12,e14=2e13,e13,e14=2e12

(2.13)
Oct > 

DGsetupL5

Lie algebra: Alg5

(2.14)
Oct > 

QuerySemisimple

true

(2.15)

 

 

 

 

See Also

DifferentialGeometry

LieAlgebras

Adjoint

Query

Query[Derivation]

Query[Ideal]

Query[Solvable]

QuotientAlgebra

 


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