Homomorphisms of Lie Algebras

AUTHORS:

  • Travis Scrimshaw (07-15-2013): Initial implementation
class sage.algebras.lie_algebras.morphism.LieAlgebraHomomorphism_im_gens(parent, im_gens, check=True)

Bases: sage.categories.morphism.Morphism

A homomorphism of Lie algebras.

Let \(\mathfrak{g}\) and \(\mathfrak{g}^{\prime}\) be Lie algebras. A linear map \(f \colon \mathfrak{g} \to \mathfrak{g}^{\prime}\) is a homomorphism (of Lie algebras) if \(f([x, y]) = [f(x), f(y)]\) for all \(x, y \in \mathfrak{g}\). Thus homomorphisms are completely determined by the image of the generators of \(\mathfrak{g}\).

EXAMPLES:

sage: L = LieAlgebra(QQ, 'x,y,z')
sage: Lyn = L.Lyndon()
sage: H = L.Hall()
doctest:warning...:
FutureWarning: The Hall basis has not been fully proven correct, but currently no bugs are known
See http://trac.sagemath.org/16823 for details.
sage: phi = Lyn.coerce_map_from(H); phi
Lie algebra morphism:
  From: Free Lie algebra generated by (x, y, z) over Rational Field in the Hall basis
  To:   Free Lie algebra generated by (x, y, z) over Rational Field in the Lyndon basis
  Defn: x |--> x
        y |--> y
        z |--> z
im_gens()

Return the images of the generators of the domain.

OUTPUT:

  • list – a copy of the list of gens (it is safe to change this)

EXAMPLES:

sage: L = LieAlgebra(QQ, 'x,y,z')
sage: Lyn = L.Lyndon()
sage: H = L.Hall()
sage: f = Lyn.coerce_map_from(H)
sage: f.im_gens()
[x, y, z]
class sage.algebras.lie_algebras.morphism.LieAlgebraHomset(X, Y, category=None, base=None, check=True)

Bases: sage.categories.homset.Homset

Homset between two Lie algebras.

Todo

This is a very minimal implementation which does not have coercions of the morphisms.

zero()

Return the zero morphism.

EXAMPLES:

sage: L = LieAlgebra(QQ, 'x,y,z')
sage: Lyn = L.Lyndon()
sage: H = L.Hall()
sage: HS = Hom(Lyn, H)
sage: HS.zero()
Generic morphism:
  From: Free Lie algebra generated by (x, y, z) over Rational Field in the Lyndon basis
  To:   Free Lie algebra generated by (x, y, z) over Rational Field in the Hall basis