find the ad Matrix for a vector in a Lie algebra - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : DifferentialGeometry/LieAlgebras/Adjoint

LieAlgebras[Adjoint] - find the ad Matrix for a vector in a Lie algebra

LieAlgebras[AdjointExp] - find the Ad Matrix for a vector in a Lie algebra

Calling Sequences

     Adjoint(alg)

     Adjoint(alg, keyword)

     Adjoint(x, h, k)

     AdjointExp(x)

Parameters

     alg      - (optional) the name of a Lie algebra 𝔤

     keyword  - (optional) the keyword argument representationspace = framename, where framename is the name of an initialized frame

     x        - a vector in a Lie algebra g

     h        - (optional) a list of vectors defining a basis for a subspace h in a Lie algebra 𝔤

     k        - (optional) a list of vectors defining a complementary basis in 𝔤 to h

 

 

Description

Examples

Description

• 

 Let 𝔤 be a Lie algebra and x  𝔤. Then the adjoint transformation defined by x is the linear transformation adx : 𝔤  𝔤 defined by adxy = x, y for all y  𝔤. The transformation adxalways defines a derivation on 𝔤, that is, adxy, z = adxy, z + y, adxz. The mapping xadx defines a representation of 𝔤. The exponential of adx, usually denoted by Ad(x), is a Lie algebra isomorphism.

• 

Adjoint(x) returns the matrix representing the linear transformation adx.

• 

AdjointExp(x) returns the matrix representing the linear transformation Ad(x) = exp(adx).

• 

Adjoint() returns the list of adjoint matrices for the basis vectors of the current algebra 𝔤.

• 

Adjoint(alg) returns the list of adjoint matrices for the basis vectors of the algebra alg.

• 

Adjoint(alg , representationspace = V) returns the adjoint representation of 𝔤, with representation space V.

• 

Adjoint(x, h) calculates the restriction of ad(x) to the subspace h (h must be an ad(x) invariant subspace).

• 

Adjoint(x, h, k) calculates Adjoint(x) on the vector space quotient g/k with respect to the basis determined by h (k must be an ad(x) invariant subspace).

• 

The commands Adjoint and AdjointExp are part of the DifferentialGeometry:-LieAlgebras package. They can be used in the form Adjoint(...) and AdjointExp(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Adjoint(...) and DifferentialGeometry:-LieAlgebras:-AdjointExp(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra.

L1 ≔ _DGLieAlgebra,Alg1,4,1,3,1,1,2,4,1,1,1,4,2,1,2,3,2,1

L1:=e1,e3=e1,e1,e4=e2,e2,e3=e2,e2,e4=e1

(2.1)

DGsetupL1:

Alg1 > 

Adjointte4

0t00t00000000000

(2.2)

 

AdjointExp(t*e4) is given by the Matrix exponential of Adjoint(t*e4).

Alg1 > 

AdjointExpte4

costsint00sintcost0000100001

(2.3)
Alg1 > 

Adjointe1+2e3

2010020100000000

(2.4)

 

Calculate the restriction of Adjoint(e3) to the subspace defined by [e1, e2].

Alg1 > 

Adjointe3,Adjointe3,e1,e2

1000010000000000,1001

(2.5)

 

Calculate the linear transformation induced by Adjoint(e4 + 2*e3) on the quotient of [e1, e2, e3, e4] by the subspace defined by [e3, e4] with respect to the basis [e1, e2].

Alg1 > 

Adjointe4+2e3,Adjointe4+2e3,e1,e2,e3,e4

2100120000000000,2112

(2.6)

Calculate the adjoint representation of Alg1. First define the representation space.

DGsetupx1,x2,x3,x4,V

frame name: V

(2.7)

ρ ≔ AdjointAlg1,representationspace=V

ρ:=e1,0010000100000000,e2,0001001000000000,e3,1000010000000000,e4,0100100000000000

(2.8)
Alg1 > 

Queryρ,Representation

true

(2.9)

See Also

DifferentialGeometry

LieAlgebras

LinearAlgebra[MatrixExponential]

Representation

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam