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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First we initialize a Lie algebra.

 > $\mathrm{LD1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{alg1},\left[3\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: alg1}}$ (2.2)



Now define coordinates for the dual of the Lie algebra.

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

Calculate the infinitesimal generators for the co-adjoint action.

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

 N > $\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (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.

 N > $A≔\mathrm{Adjoint}\left(\mathrm{e3}\right)$
 ${A}{:=}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (2.6)
 alg1 > $\mathrm{convert}\left(\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right),\mathrm{DGvector},N\right)$
 ${-}{x}{}{\mathrm{D_x}}{+}\left({-}{x}{-}{y}\right){}{\mathrm{D_y}}$ (2.7)

Example 2.

First we initialize a 4-dimensional Lie algebra.

 N > $\mathrm{LD2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{alg2},\left[4\right]\right],\left[\left[\left[2,4,1\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right)$
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.8)
 N > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: alg2}}$ (2.9)

Now define coordinates for the dual of the Lie algebra.

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

Calculate the infinitesimal generators for the co-adjoint action.

 N2 > $\mathrm{Γ2}≔\mathrm{InfinitesimalCoadjointAction}\left(\mathrm{alg2},\mathrm{N2}\right)$
 ${\mathrm{Γ2}}{:=}\left[{w}{}{\mathrm{D_z}}{,}{y}{}{\mathrm{D_z}}{,}{-}{w}{}{\mathrm{D_x}}{-}{y}{}{\mathrm{D_y}}\right]$ (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 > $\mathrm{Center}\left(\mathrm{alg2}\right)$
 $\left[{\mathrm{e1}}\right]$ (2.12)
 alg2 > $\mathrm{QuotientAlgebra}\left(\left[\mathrm{e1}\right],\left[\mathrm{e2},\mathrm{e3},\mathrm{e4}\right]\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (2.13)
 alg2 > $\mathrm{LieAlgebraData}\left(\mathrm{Γ2}\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (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 > $\mathrm{LD3}≔\mathrm{Library}:-\mathrm{Retrieve}\left("Winternitz",1,\left[5,12\right],\mathrm{alg3}\right)$
 ${\mathrm{LD3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.15)
 alg2 > $\mathrm{DGsetup}\left(\mathrm{LD3}\right)$
 ${\mathrm{Lie algebra: alg3}}$ (2.16)

Calculate the infinitesimal generators for the co-adjoint action.

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

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

 N3 > $C≔\mathrm{expand}\left(\mathrm{GroupActions}:-\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ3},\left[1\right],\mathrm{output}="list"\right)\right)$
 ${C}{:=}\left[{-}{\mathrm{ln}}{}\left({\mathrm{x1}}\right){+}\frac{{\mathrm{x2}}}{{\mathrm{x1}}}{,}\frac{{1}}{{2}}{}{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{2}}{-}\frac{{\mathrm{ln}}{}\left({\mathrm{x1}}\right){}{\mathrm{x2}}}{{\mathrm{x1}}}{+}\frac{{\mathrm{x3}}}{{\mathrm{x1}}}{,}{-}\frac{{1}}{{6}}{}{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{3}}{+}\frac{{1}}{{2}}{}\frac{{{\mathrm{ln}}{}\left({\mathrm{x1}}\right)}^{{2}}{}{\mathrm{x2}}}{{\mathrm{x1}}}{-}\frac{{\mathrm{ln}}{}\left({\mathrm{x1}}\right){}{\mathrm{x3}}}{{\mathrm{x1}}}{+}\frac{{\mathrm{x4}}}{{\mathrm{x1}}}\right]$ (2.19)



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

 N3 > $\mathrm{expand}\left(\left[\mathrm{expand}\left(\mathrm{exp}\left(-C\left[1\right]\right),\mathrm{symbolic}\right),2C\left[2\right]-{C\left[1\right]}^{2},3C\left[3\right]+{C\left[1\right]}^{3}-3C\left[1\right]C\left[2\right]\right]\right)$
 $\left[\frac{{\mathrm{x1}}}{{{ⅇ}}^{\frac{{\mathrm{x2}}}{{\mathrm{x1}}}}}{,}\frac{{2}{}{\mathrm{x3}}}{{\mathrm{x1}}}{-}\frac{{{\mathrm{x2}}}^{{2}}}{{{\mathrm{x1}}}^{{2}}}{,}\frac{{3}{}{\mathrm{x4}}}{{\mathrm{x1}}}{+}\frac{{{\mathrm{x2}}}^{{3}}}{{{\mathrm{x1}}}^{{3}}}{-}\frac{{3}{}{\mathrm{x2}}{}{\mathrm{x3}}}{{{\mathrm{x1}}}^{{2}}}\right]$ (2.20)