60.9 The Adjoint Representation

is the matrix of the adjoint representation of the element x w.r.t. the basis B. The adjoint map is the left multiplication by x. The i-th column of the resulting matrix represents the image of the the i-th basis vector of B under left multiplication by x.

gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> AdjointMatrix( Basis( L ), Basis( L )[1] );
[ [ 0, 0, -2 ], [ 0, 0, 0 ], [ 0, 1, 0 ] ]

is the associative matrix algebra (with 1) generated by the matrices of the adjoint representation of the subalgebra K on the Lie algebra L.

gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> AdjointAssociativeAlgebra( L, L );
<algebra of dimension 9 over Rationals>
gap> AdjointAssociativeAlgebra( L, CartanSubalgebra( L ) );
<algebra of dimension 3 over Rationals>

is the matrix of the Killing form k with respect to the basis B, i.e., the matrix (k(b_i,b_j)) where b₁,b₂¼ are the basis vectors of B.

gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> KillingMatrix( Basis( L ) );
[ [ 0, 4, 0 ], [ 4, 0, 0 ], [ 0, 0, 8 ] ]

is the orthogonal complement of the subspace U of the Lie algebra L with respect to the Killing form k, that is, the set U^{^} = { x Î L; k(x,y) = 0 for all y Î L }.

U^{^} is a subspace of L, and if U is an ideal of L then U^{^} is a subalgebra of L.

gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> b:= BasisVectors( Basis( L ) );;
gap> V:= VectorSpace( Rationals, [b[1],b[2]] );;
gap> KappaPerp( L, V );
<vector space of dimension 1 over Rationals>

x is nilpotent in L if its adjoint matrix is a nilpotent matrix.

gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
gap> IsNilpotentElement( L, Basis( L )[1] );
true

A non-nilpotent element of a Lie algebra L is an element x such that ad x is not nilpotent. If L is not nilpotent, then by Engel's theorem non nilpotent elements exist in L. In this case this function returns a non nilpotent element of L, otherwise (if L is nilpotent) fail is returned.

gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
gap> NonNilpotentElement( L );  
v.13
gap> IsNilpotentElement( L, last );
false

This function tries to find a subalgebra S of the Lie algebra L with S isomorphic to sl₂ and such that the nilpotent element x of L is contained in S. If such an algebra exists then it is returned, otherwise fail is returned.

gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
gap> b:= BasisVectors( Basis( L ) );;
gap> IsNilpotentElement( L, b[1] );
true
gap> FindSl2( L, b[1] );
<Lie algebra of dimension 3 over Rationals>

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