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LieAlgebras[Representation] - define a representation of a Lie algebra

LieAlgebras[ApplyRepresentation] - apply a representation of a Lie algebra to a vector in the Lie algebra

Calling Sequences

     Representation(A, V, M)

     ApplyRepresentation(ρ, x, y)

Parameters

     A         - a Maple name or string, the name of a defined Lie Algebra 𝔤

     V         - a Maple name or string, the name of the defined frame used as the vector space V  for the representation

     M         - a list of matrices  each defining a linear transformation from V to V

     ρ         - a representation of a Lie algebra g on a vector space V, as constructed with the Representation command

     x         - a vector in g

     y         - (optional) a vector in V

 

Description

Examples

Description

• 

Let g be a Lie algebra and V a vector space.  A representation of g on V is a linear transformation rho: g -> gl(V) (where gl(V) is the Lie algebra of all linear transformations on V) such that rho([x, y]) = [rho(x), rho(y)] (*) for all x, y in g.  Here [x, y] denotes the Lie bracket of two vectors in g while [rho(x), rho(y)] denotes the commutator of two linear transformations in gl(V).

• 

The command Representation(A, V, M) creates a data structure which defines the representation rho : g -> gl(V) such that rho(e_i) = M_i.

• 

The command Query can be used to verify that the map rho does in fact satisfy the representation property (*).

• 

If  x=c1e1+c2e2+ then ApplyRepresentation(ρ, x) returns the matrix ρx=c1e1+c2e2+. If  y  is vector in the representation space V, then Representation(ρ,x, y) returns the vector ρxyV.

Examples

withDifferentialGeometry:withLieAlgebras:withLibrary:withGroupActions:

 

Example 1.

First we retrieve a Lie algebra from the DifferentialGeometry Library and initialize it.

LRetrieveWinternitz,1,4,7,Alg1

L:=e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3

(2.1)

DGsetupL:

 

The center of Alg1 is trivial and therefore the adjoint representation is faithful.  The command Adjoint will return the list of adjoint matrices for the Lie algebra Alg1.

Alg1 > 

Center

(2.2)
Alg1 > 

MAdjoint

M:=0002000000000000,0010000100000000,0100000100010000,2000011000100000

(2.3)

 

To define a representation of Alg1 using the matrices M, we shall need to define a representation space V.

Alg1 > 

DGsetupx1,x2,x3,x4,V:

V > 

ρRepresentationAlg1,V,M

ρ:=e1,0002000000000000,e2,0010000100000000,e3,0100000100010000,e4,2000011000100000

(2.4)

 

We can verify that this is really a representation of Alg1 with the Query command.

Alg1 > 

Queryρ,Representation

true

(2.5)

 

Apply the representation to some vectors in Alg1:

Alg1 > 

A1ApplyRepresentationρ,e1

A1:=0002000000000000

(2.6)
Alg1 > 

A2ApplyRepresentationρ,e1+2e2+3e4

A2:=6022033200300000

(2.7)

 

Apply the linear transformations to some vectors in V:

Alg1 > 

ApplyRepresentationρ,e1,D_x4

2D_x1

(2.8)
V > 

ApplyRepresentationρ,e1+2e2+3e4,D_x3

2D_x13D_x23D_x3

(2.9)

 

Example 2.

The infinitesimal automorphisms or derivations of a Lie algebra g define a matrix Lie algebra which is automatically a representation.  In this case we can take g to be the representation space.

V > 

LRetrieveWinternitz,1,3,2,Alg2

L:=e1,e3=e1,e2,e3=e1+e2

(2.10)
V > 

DGsetupL:

 

The Derivations command calculates the derivations on the Lie algebra Alg2.

Alg2 > 

MDerivationsFull

M:=100010000,010000000,001000000,000001000

(2.11)

 

We calculate the structure equations for this matrix algebra and initialize the result.

Alg2 > 

ALieAlgebraDataM,Alg2a

A:=e1,e3=e3,e1,e4=e4,e2,e4=e3

(2.12)
Alg2 > 

DGsetupA

Lie algebra: Alg2a

(2.13)
Alg2a > 

ρ2RepresentationAlg2a,Alg2,M

ρ2:=e1,100010000,e2,010000000,e3,001000000,e4,000001000

(2.14)
Alg2a > 

Queryρ2,Representation

true

(2.15)

 

Example 3.

If Gamma is a Lie algebra of vector fields on a manifold M, then the isotropy subalgebra at a given point admits a natural representation, defined by the Lie bracket, on the tangent space of M. To illustrate the definition of this representation, we first obtain a Lie algebra of vector fields from the DifferentialGeometry Library with the Retrieve command. Then we use the IsotropySubalgebra from the GroupActions package to calculate the isotropy subalgebra and its representation.

Alg2a > 

DGsetupx,y,M:

M > 

GammaRetrieveGonzalez-Lopez,1,7,manifold=M

Γ:=D_x,D_y,xD_x+yD_y,yD_xxD_y,x2y2D_x+2xyD_y,2xyD_x+y2x2D_y

(2.16)
M > 

IsoAlg,IsoRepIsotropySubalgebraGamma,x=1,y=2,output=Vector,Representation

IsoAlg,IsoRep:=825xy325x2+325y21D_x625xy425x2+425y2D_y,425y2+425x2+625xyD_x825xy325x2+325y21D_y,15y2+15x2+45xyxD_x25y225x2+25xyyD_y,25y225x2+25xyyD_x45xy15x2+15y2+xD_y,25454525,45252545,1001,0110

(2.17)

 

M > 

L3LieAlgebraDataIsoAlg,Alg3

L3:=e1,e3=e125e345e4,e1,e4=e2+45e325e4,e2,e3=e245e3+25e4,e2,e4=e125e345e4

(2.18)
M > 

DGsetupL3:

Alg3 > 

ρ3RepresentationAlg3,M,IsoRep

ρ3:=e1,25454525,e2,45252545,e3,1001,e4,0110

(2.19)
Alg3 > 

Queryρ3,Representation

true

(2.20)

 

Example 4.

Here is 4 dimensional faithful representation of an indecomposable 4-dimensional Lie algebra with a center.

Alg3 > 

L4LieAlgebraDatax2,x4=x1,x3,x4=x3,x1,x2,x3,x4,Alg4

L4:=e2,e4=e1,e3,e4=e3

(2.21)
M > 

DGsetupL4

Lie algebra: Alg4

(2.22)
Alg4 > 

M4Matrix0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,Matrix0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,Matrix0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,Matrix1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0

M4:=0000000100000000,0000000000010000,0001000000000000,1000001000000000

(2.23)
Alg4 > 

DGsetupx1,x2,x3,x4,V:

V > 

ρ4RepresentationAlg4,V,M4

ρ4:=e1,0000000100000000,e2,0000000000010000,e3,0001000000000000,e4,1000001000000000

(2.24)
Alg4 > 

Queryρ4,Representation

true

(2.25)

See Also

DifferentialGeometry

GroupActions

Library

Adjoint

Center

Derivations

IsotropySubalgebra

Query

Retrieve