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
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(...).
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:
MultiplicationTableLieBracket
e1,e4=e1,e2,e4=e1+e2,e3,e4=e3
For the Lie algebra Alg1 we find that Derivations(Alg1, "Inner") is 4 dimensional and Derivations(Alg1) is 8 dimensional.
Inner≔DerivationsInner
Der≔DerivationsFull
Outer≔DerivationsOuter
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.
Basis≔opInner,opOuter:
L2≔LieAlgebraDataBasis,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
DGsetupL2,E,a:
We see that the derivation algebra is solvable.
QuerySolvable
true
We check that the span of the vectors E1, E2, E3, E4(corresponding to the inner derivations) define an ideal.
QueryE1,E2,E3,E4,Ideal
We compute the quotient algebra of outer derivations.
L3≔QuotientAlgebraE1,E2,E3,E4,E5,E6,E7,E8,OuterAlg
L3:=e1,e3=e3,e1,e4=−e4,e3,e4=e2
DGsetupL3
Lie algebra: OuterAlg
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≔AlgebraLibraryDataOctonions,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
DGsetupL4:
We find that the derivation algebra is 14-dimensional
Der≔DerivationsOct
nopsDer
14
Calculate the structure equations for the derivations, initialize ,and check that the derivation algebra is semi-simple.
L5≔LieAlgebraDataDer,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=−2e6,e2,e5=e3,e2,e6=2e4,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=−2e2−2e13,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=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=−2e11,e7,e11=2e10,e7,e13=−e14,e7,e14=e13,e8,e9=2e7−2e12,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
DGsetupL5
Lie algebra: Alg5
QuerySemisimple
See Also
DifferentialGeometry
LieAlgebras
Adjoint
Query
Query[Derivation]
Query[Ideal]
Query[Solvable]
QuotientAlgebra
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