Derivations - Maple Help
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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

LBx⋅y = LBx⋅y+ x⋅LBy  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

> 

with⁡DifferentialGeometry:with⁡LieAlgebras:

 

Example 1.

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

 

> 

L1≔_DG⁡LieAlgebra,Alg1,4,1,4,1,1,2,4,1,1,2,4,2,1,3,4,3,1:

> 

DGsetup⁡L1:

Alg1 > 

MultiplicationTable⁡LieBracket

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 > 

Inner≔Derivations⁡Inner

Alg1 > 

Der≔Derivations⁡Full

Alg1 > 

Outer≔Derivations⁡Outer

 

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 > 

Basis≔op⁡Inner,op⁡Outer:

Alg1 > 

L2≔LieAlgebraData⁡Basis,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

(2.2)
Alg1 > 

DGsetup⁡L2,E,a:

 

We see that the derivation algebra is solvable.

DerAlg > 

Query⁡Solvable

true

(2.3)

 

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

DerAlg > 

Query⁡E1,E2,E3,E4,Ideal

true

(2.4)

 

We compute the quotient algebra of outer derivations.

DerAlg > 

L3≔QuotientAlgebra⁡E1,E2,E3,E4,E5,E6,E7,E8,OuterAlg

L3:=e1,e3=e3,e1,e4=−e4,e3,e4=e2

(2.5)
DerAlg > 

DGsetup⁡L3

Lie algebra: OuterAlg

(2.6)

 

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).

 

> 

L4≔AlgebraLibraryData⁡Octonions,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

(2.7)
> 

DGsetup⁡L4:

 

We find that the derivation algebra is 14-dimensional

> 

Der≔Derivations⁡Oct

> 

nops⁡Der

14

(2.8)

 

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

Oct > 

L5≔LieAlgebraData⁡Der,Alg5

L5:=e1,e2=−e7,e1,e3=−e8,e1,e4=−e5−e9,e1,e5=e4−e10,e1,e6=−e11,e1,e7=e2,e1,e8=e3,e1,e9=e4−e10,e1,e10=e5+e9,e1,e11=e6,e1,e12=−e13,e1,e13=e12,e2,e3=−e5,e2,e4=−2⁢e6,e2,e5=e3,e2,e6=2⁢e4,e2,e7=−e1,e2,e8=e4,e2,e9=−e3−e11,e2,e10=−e6,e2,e11=e5+e9,e2,e12=−e14,e2,e14=e12,e3,e4=−e12,e3,e5=−2⁢e2−2⁢e13,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=−2⁢e2,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=e6−e8,e5,e8=e7−e12,e5,e10=−e1+e14,e5,e11=−e2−e13,e5,e12=e6,e5,e13=−e3,e5,e14=−e4,e6,e7=−e5−e9,e6,e9=e7−e12,e6,e10=e2,e6,e11=−e1,e6,e12=−e5,e6,e13=e4,e6,e14=−e3,e7,e8=e9,e7,e9=−e8,e7,e10=−2⁢e11,e7,e11=2⁢e10,e7,e13=−e14,e7,e14=e13,e8,e9=2⁢e7−2⁢e12,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=−2⁢e7,e10,e12=e11,e10,e13=−e8,e10,e14=−e9,e11,e12=−e10,e11,e13=e9,e11,e14=−e8,e12,e13=−2⁢e14,e12,e14=2⁢e13,e13,e14=−2⁢e12

(2.9)
Oct > 

DGsetup⁡L5

Lie algebra: Alg5

(2.10)
Oct > 

Query⁡Semisimple

true

(2.11)

 

 

 

 

See Also

DifferentialGeometry

LieAlgebras

Adjoint

Query

Query[Derivation]

Query[Ideal]

Query[Solvable]

QuotientAlgebra