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 𝔤

     Alg      - (optional) the name of a Lie algebra

     h        - a list of vectors defining a basis for a subspace of a Lie algebra 𝔤

 

Description

Examples

Description

• 

The Killing form on a ndimensional Lie algebra 𝔤  is the symmetric quadratic form B defined by  Bx, y =traceadxady for any x, y  𝔤 . Here adx and ady are the adjoint matrices for the vectors x and y. In terms of the structure constants Cijk with respect to the basis {ei }for 𝔤,one has bij = Bei, ej =k,ℓ =1nCiℓk Ckjℓ . If 𝔥  𝔤 is a subspace with basis x1, x2, ... ,xp, then the restriction of the Killing form to 𝔥 is given by the p ×p matrix b rs = Bxr, xs.

• 

 Killing() returns the n ×n symmetric matrix bij for the Lie algebra defined by the current frame. Killing(Alg) returns the n ×n symmetric matrix bij for the Lie algebra Alg. Alg.Killing(h) returns the Killing Matrix b rs restricted to the subalgebra 𝔥.

• 

KillingForm(Alg) returns the symmetric rank 2-tensor bij θi  θj, where the {θi} are the dual 1-forms to the basis {ei }.

• 

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");

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

(2.1)

 

Compute the Killing form on the vectors x = e1 + e2 and y = e1  e2 +e3.

Alg1 > 

X := evalDG(e1 + e3);

X:=e1+e3

(2.2)
Alg1 > 

Y := evalDG(e1 - e2 + e3);

Y:=e1e2+e3

(2.3)
Alg1 > 

Killing(X, Y);

4

(2.4)

 

Compute the Killing form for the current Lie algebra.

Alg1 > 

K := Killing();

K:=200020002

(2.5)

 

Compute the Killing form restricted to the subspace S = spane2, e3.

Alg1 > 

S := [e2, e3]:

Alg1 > 

Killing(S);

2002

(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);

2θ1θ12θ2θ22θ3θ3

(2.7)

 

 

 

See Also

DifferentialGeometry

LieAlgebras

Adjoint

MultiplicationTable

Query[Semisimple]