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LieAlgebras[InfinitesimalCoadjointAction] - find the vector fields defining the infinitesimal co-adjoint action of a Lie group on its Lie algebra

Calling Sequences

     InfinitesimalCoadjointAction(Alg,  M)

Parameters

     Alg       - name or string, the name of an initialized Lie algebra

     M         - name or string, the name of an initialized manifold

 

Description

Examples

Description

• 

Let G be an n-dimensional Lie group with Lie algebra 𝔤and let ei, ej = Cijk ek be the structure equations for 𝔤. If xi are coordinates for the dual vector space 𝔤*, then the infinitesimal generators for the co-adjoint action of G on 𝔤*are the vector fields Xi= Cijk xj   xk .

• 

The command InfinitesimalCoadjointAction(Algebra, Manifold) calculates the vector fields Xi for the Lie algebra Algebra using the coordinates for the dual space provide by M.

• 

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

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

First we initialize a Lie algebra.

LD1 := _DG([["LieAlgebra", alg1, [3]], [[[1, 3, 1], 1], [[2, 3, 1], 1], [[2, 3, 2], 1]]]);

LD1:=e1,e3=e1,e2,e3=e1+e2

(2.1)

DGsetup(LD1);

Lie algebra: alg1

(2.2)

Now define coordinates for the dual of the Lie algebra.

alg1 > 

DGsetup([x, y, z], N);

frame name: N

(2.3)

 

Calculate the infinitesimal generators for the co-adjoint action.

N > 

Gamma := InfinitesimalCoadjointAction(alg1, N);

Γ:=xD_z,x+yD_z,xD_x+xyD_y

(2.4)

 

The center of the Lie algebra alg1 is trivial and therefore the structure equations for the Lie algebra Γ are the same as those for alg1.

N > 

LieAlgebraData(Gamma);

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

(2.5)

 

The vector fields Γ may be calculated directly using the Adjoint and convert/DGvector commands. For example, we obtain the last vector in Γ as follows.

N > 

A := Adjoint(e3);

A:=110010000

(2.6)
alg1 > 

convert(LinearAlgebra:-Transpose(A), DGvector, N);

xD_x+xyD_y

(2.7)

 

Example 2.

First we initialize a 4-dimensional Lie algebra.

N > 

LD2 := _DG([["LieAlgebra", alg2, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

LD2:=e2,e4=e1,e3,e4=e3

(2.8)
N > 

DGsetup(LD2);

Lie algebra: alg2

(2.9)

 

Now define coordinates for the dual of the Lie algebra.

alg2 > 

DGsetup([w, x, y, z], N2);

frame name: N2

(2.10)

 

Calculate the infinitesimal generators for the co-adjoint action.

N2 > 

Gamma2 := InfinitesimalCoadjointAction(alg2, N2);

Γ2:=wD_z,yD_z,wD_xyD_y

(2.11)

In this example, the Lie algebra has a non-trivial center e1 and now the structure equations for Γ2 are those for the quotient of alg2 by its center.

N2 > 

Center(alg2);

e1

(2.12)
alg2 > 

QuotientAlgebra([e1], [e2, e3, e4]);

e2,e3=e2

(2.13)
alg2 > 

LieAlgebraData(Gamma2);

e2,e3=e2

(2.14)

 

Example 3.

The invariants for the co-adjoint action are called generalized Casimir operators (See J. Patera, R. T. Sharp , P. Winternitz and H. Zassenhaus, Invariants of real low dimensional Lie algebras, J. Math. Phys. 17, No 6, June 1976, 966--994).

 

We calculate the generalized Casimir operators for the Lie algebra [5,12] from this article. First use the Retrieve command to obtain the structure equations for this algebra and initialize the Lie algebra.

alg2 > 

LD3 := Library:-Retrieve("Winternitz", 1, [5, 12], alg3);

LD3:=e1,e5=e1,e2,e5=e1+e2,e3,e5=e2+e3,e4,e5=e3+e4

(2.15)
alg2 > 

DGsetup(LD3);

Lie algebra: alg3

(2.16)

 

Calculate the infinitesimal generators for the co-adjoint action.

alg2 > 

DGsetup([x1, x2, x3, x4, x5], N3);

frame name: N3

(2.17)
N3 > 

Gamma3 := InfinitesimalCoadjointAction(alg3, N3);

Γ3:=x1D_x5,x1+x2D_x5,x2+x3D_x5,x3+x4D_x5,x1D_x1+x1x2D_x2+x2x3D_x3+x3x4D_x4

(2.18)

 

We use the  InvariantGeometricObjectFields command to calculate the functions which invariant under the group generated by Γ3.

N3 > 

C:= expand(GroupActions:-InvariantGeometricObjectFields(Gamma3, [1], output = "list"));

C:=lnx1+x2x1,12lnx12lnx1x2x1+x3x1,16lnx13+12lnx12x2x1lnx1x3x1+x4x1

(2.19)

Functional combinations of these invariants give the formulas for the generalized Casimir operators in the Patera, Sharp, et al. paper.

N3 > 

expand([expand(exp(-C[1]), symbolic), 2*C[2] -C[1]^2, 3*C[3]  + C[1]^3 -3*C[1]*C[2]]);

x1ⅇx2x1,2x3x1x22x12,3x4x1+x23x133x2x3x12

(2.20)

See Also

DifferentialGeometry

GroupActions

Library

LieAlgebras

convert/DGvector

LieAlgebraData

Adjoint

Retrieve

InvariantGeometricObjectFields