find the Killing form (matrix) of a Lie algebra, evaluate the Killing form on a pair of vectors, evaluate the Killing form on a subspace - Maple Programming Help

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LieAlgebras[Killing] - find the Killing form (matrix) of a Lie algebra, evaluate the Killing form on a pair of vectors, evaluate the Killing form on a subspace

LieAlgebras[KillingForm] - find the Killing form (symmetric tensor) of a Lie algebra

Calling Sequences

Killing(x, y)

Killing(Alg)

Killing(h)

KilllingForm(Alg)

Parameters

x,y      - a pair of vectors in a Lie algebra $\mathrm{𝔤}$

Alg      - (optional) the name of a Lie algebra

h        - a list of vectors defining a basis for a subspace of a Lie algebra $\mathrm{𝔤}$

Description

 • The Killing form on a $n-$dimensional Lie algebra  is the symmetric quadratic form defined by tracefor any . Here $\mathrm{ad}\left(x\right)$ and $\mathrm{ad}\left(y\right)$ are the adjoint matrices for the vectors $x$ and In terms of the structure constants with respect to the basis {}for $\mathrm{𝔤}\mathit{,}$one has . If $\mathrm{𝔥}$is a subspace with basis , then the restriction of the Killing form to $𝔥$ is given by the matrix
 • Killing() returns the symmetric matrix $\left[{b}_{\mathrm{ij}}\right]$ for the Lie algebra defined by the current frame. Killing(Alg) returns the symmetric matrix $\left[{b}_{\mathrm{ij}}\right]$ for the Lie algebra Alg. Alg.Killing(h) returns the Killing Matrix restricted to the subalgebra $\mathrm{𝔥}\mathit{.}$
 • KillingForm(Alg) returns the symmetric rank 2-tensor , where the {are the dual 1-forms to the basis {.
 • The command Killing is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Killing(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Killing(...).

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.

 > L1 := _DG([["LieAlgebra", Alg1, [3]], [[[2, 3, 1], 1], [[1, 3, 2], -1], [[1, 2, 3], 1]]]):
 Alg1   > DGsetup(L1):
 > MultiplicationTable("LieBracket");
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.1)

Compute the Killing form on the vectors and y = .

 Alg1 > X := evalDG(e1 + e3);
 ${X}{:=}{\mathrm{e1}}{+}{\mathrm{e3}}$ (2.2)
 Alg1 > Y := evalDG(e1 - e2 + e3);
 ${Y}{:=}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}$ (2.3)
 Alg1 > Killing(X, Y);
 ${-}{4}$ (2.4)

Compute the Killing form for the current Lie algebra.

 Alg1 > K := Killing();
 ${K}{:=}\left[\begin{array}{rrr}{-}{2}& {0}& {0}\\ {0}& {-}{2}& {0}\\ {0}& {0}& {-}{2}\end{array}\right]$ (2.5)

Compute the Killing form restricted to the subspace span.

 Alg1 > S := [e2, e3]:
 Alg1 > Killing(S);
 $\left[\begin{array}{rr}{-}{2}& {0}\\ {0}& {-}{2}\end{array}\right]$ (2.6)

Example 2.

Here is the Killing form for the Lie algebra from Example 1, given as a symmetric, covariant tensor on the Lie algebra.

 > KillingForm(Alg1);
 ${-}{2}{}{\mathrm{θ1}}{}{\mathrm{θ1}}{-}{2}{}{\mathrm{θ2}}{}{\mathrm{θ2}}{-}{2}{}{\mathrm{θ3}}{}{\mathrm{θ3}}$ (2.7)