Homomorphisms of Lie Algebras¶
AUTHORS:
- Travis Scrimshaw (07-15-2013): Initial implementation
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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
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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]
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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.
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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
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