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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 $\mathrm{𝔤}$

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 $\mathrm{𝔤}$

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

Description

 • Let be a Lie algebra and  Then the adjoint transformation defined by $x$ is the linear transformation addefined by adfor all . The transformation $\mathrm{ad}\left(x\right)$always defines a derivation on that is, ad. The mapping $x\to$addefines a representation of $\mathrm{𝔤}\mathit{.}$ The exponential of usually denoted by Ad($x$), is a Lie algebra isomorphism.
 • Adjoint(x) returns the matrix representing the linear transformation ${\mathrm{ad}}\left({x}\right)$.
 • AdjointExp(x) returns the matrix representing the linear transformation Ad(x) = exp($\mathrm{ad}\left(x\right))$.
 • Adjoint() returns the list of adjoint matrices for the basis vectors of the current algebra $\mathrm{𝔤}$.
 • 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 $\mathrm{𝔤}$, with representation space V.
 • Adjoint(x, h) calculates the restriction of ad(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

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

Example 1.

First initialize a Lie algebra.

 > $\mathrm{L1}â‰”\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,4,1\right],1\right],\left[\left[1,4,2\right],-1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1 > $\mathrm{Adjoint}\left(t\mathrm{e4}\right)$
 $\left[\begin{array}{cccc}{0}& {-}{t}& {0}& {0}\\ {t}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.2)

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

 Alg1 > $\mathrm{AdjointExp}\left(t\mathrm{e4}\right)$
 $\left[\begin{array}{cccc}{\mathrm{cos}}{}\left({t}\right)& {-}{\mathrm{sin}}{}\left({t}\right)& {0}& {0}\\ {\mathrm{sin}}{}\left({t}\right)& {\mathrm{cos}}{}\left({t}\right)& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (2.3)
 Alg1 > $\mathrm{Adjoint}\left(\mathrm{e1}+2\mathrm{e3}\right)$
 $\left[\begin{array}{rrrr}{-}{2}& {0}& {1}& {0}\\ {0}& {-}{2}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.4)

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

 Alg1 > $\mathrm{Adjoint}\left(\mathrm{e3}\right),\mathrm{Adjoint}\left(\mathrm{e3},\left[\mathrm{e1},\mathrm{e2}\right]\right)$
 $\left[\begin{array}{rrrr}{-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{-}{1}& {0}\\ {0}& {-}{1}\end{array}\right]$ (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 > $\mathrm{Adjoint}\left(\mathrm{e4}+2\mathrm{e3}\right),\mathrm{Adjoint}\left(\mathrm{e4}+2\mathrm{e3},\left[\mathrm{e1},\mathrm{e2}\right],\left[\mathrm{e3},\mathrm{e4}\right]\right)$
 $\left[\begin{array}{rrrr}{-}{2}& {-}{1}& {0}& {0}\\ {1}& {-}{2}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{-}{2}& {-}{1}\\ {1}& {-}{2}\end{array}\right]$ (2.6)



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

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.7)
 > $\mathrm{ρ}â‰”\mathrm{Adjoint}\left(\mathrm{Alg1},\mathrm{representationspace}=V\right)$
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.8)
 Alg1 > $\mathrm{Query}\left(\mathrm{ρ},"Representation"\right)$
 ${\mathrm{true}}$ (2.9)

 See Also

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