find the vector fields defining the infinitesimal co-adjoint action of a Lie group on its Lie algebra - Maple Programming Help

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

Calling Sequences

Parameters

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

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

Description

 • Let $G$ be an $n$-dimensional Lie group with Lie algebra $\mathrm{𝔤}$and let  be the structure equations for $\mathrm{𝔤}$. If are coordinates for the dual vector space $\mathrm{𝔤}$${}^{*}$, then the infinitesimal generators for the co-adjoint action of $G$ on${}^{*}$are the vector fields  .
 • The command InfinitesimalCoadjointAction(Algebra, Manifold) calculates the vector fields 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]]]);
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.1)
 > DGsetup(LD1);
 ${\mathrm{Lie algebra: alg1}}$ (2.2)



Now define coordinates for the dual of the Lie algebra.

 alg1 > DGsetup([x, y, z], N);
 ${\mathrm{frame name: N}}$ (2.3)

Calculate the infinitesimal generators for the co-adjoint action.

 N > Gamma := InfinitesimalCoadjointAction(alg1, N);
 ${\mathrm{Γ}}{:=}\left[{x}{}{\mathrm{D_z}}{,}\left({x}{+}{y}\right){}{\mathrm{D_z}}{,}{-}{x}{}{\mathrm{D_x}}{+}\left({-}{x}{-}{y}\right){}{\mathrm{D_y}}\right]$ (2.4)

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

 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.5)

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

 ${A}{:=}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (2.6)
 alg1 > convert(LinearAlgebra:-Transpose(A), DGvector, N);
 ${-}{x}{}{\mathrm{D_x}}{+}\left({-}{x}{-}{y}\right){}{\mathrm{D_y}}$ (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]]]);
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.8)
 N > DGsetup(LD2);
 ${\mathrm{Lie algebra: alg2}}$ (2.9)

Now define coordinates for the dual of the Lie algebra.

 alg2 > DGsetup([w, x, y, z], N2);
 ${\mathrm{frame name: N2}}$ (2.10)

Calculate the infinitesimal generators for the co-adjoint action.

 N2 > Gamma2 := InfinitesimalCoadjointAction(alg2, N2);
 ${\mathrm{Γ2}}{:=}\left[{w}{}{\mathrm{D_z}}{,}{y}{}{\mathrm{D_z}}{,}{-}{w}{}{\mathrm{D_x}}{-}{y}{}{\mathrm{D_y}}\right]$ (2.11)



In this example, the Lie algebra has a non-trivial center $\left[{e}_{1}\right]$ and now the structure equations for ${\mathrm{Γ}}_{2}$ are those for the quotient of by its center.

 N2 > Center(alg2);
 $\left[{\mathrm{e1}}\right]$ (2.12)
 alg2 > QuotientAlgebra([e1], [e2, e3, e4]);
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (2.13)
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (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);
 ${\mathrm{LD3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.15)
 alg2 > DGsetup(LD3);
 ${\mathrm{Lie algebra: alg3}}$ (2.16)

Calculate the infinitesimal generators for the co-adjoint action.

 alg2 > DGsetup([x1, x2, x3, x4, x5], N3);
 ${\mathrm{frame name: N3}}$ (2.17)
 N3 > Gamma3 := InfinitesimalCoadjointAction(alg3, N3);
 ${\mathrm{Γ3}}{:=}\left[{\mathrm{x1}}{}{\mathrm{D_x5}}{,}\left({\mathrm{x1}}{+}{\mathrm{x2}}\right){}{\mathrm{D_x5}}{,}\left({\mathrm{x2}}{+}{\mathrm{x3}}\right){}{\mathrm{D_x5}}{,}\left({\mathrm{x3}}{+}{\mathrm{x4}}\right){}{\mathrm{D_x5}}{,}{-}{\mathrm{x1}}{}{\mathrm{D_x1}}{+}\left({-}{\mathrm{x1}}{-}{\mathrm{x2}}\right){}{\mathrm{D_x2}}{+}\left({-}{\mathrm{x2}}{-}{\mathrm{x3}}\right){}{\mathrm{D_x3}}{+}\left({-}{\mathrm{x3}}{-}{\mathrm{x4}}\right){}{\mathrm{D_x4}}\right]$ (2.18)

We use the  InvariantGeometricObjectFields command to calculate the functions which invariant under the group generated by ${\mathrm{\Gamma }}_{3}$.

 N3 > C:= expand(GroupActions:-InvariantGeometricObjectFields(Gamma3, [1], output = "list"));
 ${C}{:=}\left[{-}{\mathrm{ln}}{}\left({\mathrm{x1}}\right){+}\frac{{\mathrm{x2}}}{{\mathrm{x1}}}{,}\frac{{1}}{{2}}{}{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{2}}{-}\frac{{\mathrm{ln}}{}\left({\mathrm{x1}}\right){}{\mathrm{x2}}}{{\mathrm{x1}}}{+}\frac{{\mathrm{x3}}}{{\mathrm{x1}}}{,}{-}\frac{{1}}{{6}}{}{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{3}}{+}\frac{{1}}{{2}}{}\frac{{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{2}}{}{\mathrm{x2}}}{{\mathrm{x1}}}{-}\frac{{\mathrm{ln}}{}\left({\mathrm{x1}}\right){}{\mathrm{x3}}}{{\mathrm{x1}}}{+}\frac{{\mathrm{x4}}}{{\mathrm{x1}}}\right]$ (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]]);
 $\left[\frac{{\mathrm{x1}}}{{{ⅇ}}^{\frac{{\mathrm{x2}}}{{\mathrm{x1}}}}}{,}\frac{{2}{}{\mathrm{x3}}}{{\mathrm{x1}}}{-}\frac{{{\mathrm{x2}}}^{{2}}}{{{\mathrm{x1}}}^{{2}}}{,}\frac{{3}{}{\mathrm{x4}}}{{\mathrm{x1}}}{+}\frac{{{\mathrm{x2}}}^{{3}}}{{{\mathrm{x1}}}^{{3}}}{-}\frac{{3}{}{\mathrm{x2}}{}{\mathrm{x3}}}{{{\mathrm{x1}}}^{{2}}}\right]$ (2.20)