Homomorphisms of Lie Algebras¶
AUTHORS:
 Travis Scrimshaw (07152013): Initial implementation
 Eero Hakavuori (08092018): Morphisms defined by a generating subset

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


class
sage.algebras.lie_algebras.morphism.
LieAlgebraMorphism_from_generators
(on_generators, domain=None, codomain=None, check=True)¶ Bases:
sage.algebras.lie_algebras.morphism.LieAlgebraHomomorphism_im_gens
A morphism between two Lie algebras defined by images of a generating set as a Lie algebra.
This is the Lie algebra morphism \(\phi \colon L \to K\) defined on the chosen basis of \(L\) to that of \(K\) be using the image of some generating set (as a Lie algebra) of \(L\).
INPUT:
on_generators
– dictionary{X: Y}
of the images \(Y\) incodomain
of elements \(X\) ofdomain
codomain
– a Lie algebra (optional); this is inferred from the values ofon_generators
if not givencheck
– (default:True
) boolean; ifFalse
the values on the Lie brackets implied byon_generators
will not be checked for contradictory values
EXAMPLES:
A reflection of one horizontal vector in the Heisenberg algebra:
sage: L.<X,Y,Z> = LieAlgebra(QQ, {('X','Y'): {'Z':1}}) sage: phi = L.morphism({X:X, Y:Y}); phi Lie algebra endomorphism of Lie algebra on 3 generators (X, Y, Z) over Rational Field Defn: X > X Y > Y Z > Z
There is no Lie algebra morphism that reflects one horizontal vector, but not the vertical one:
sage: L.morphism({X:X, Y:Y, Z:Z}) Traceback (most recent call last): ... ValueError: this does not define a Lie algebra morphism; contradictory values for brackets of length 2
Checking for mistakes can be disabled, which can produce invalid results:
sage: phi = L.morphism({X:X, Y:Y, Z:Z}, check=False); phi Lie algebra endomorphism of Lie algebra on 3 generators (X, Y, Z) over Rational Field Defn: X > X Y > Y Z > Z sage: L[phi(X), phi(Y)] == phi(L[X,Y]) False
The set of keys must generate the Lie algebra:
sage: L.morphism({X: X}) Traceback (most recent call last): ... ValueError: [X] is not a generating set of Lie algebra on 3 generators (X, Y, Z) over Rational Field
Over nonfields, generating subsets are more restricted:
sage: L.<X,Y,Z> = LieAlgebra(ZZ, {('X','Y'): {'Z':2}}) sage: L.morphism({X: X, Y: Y}) Traceback (most recent call last): ... ValueError: [X, Y] is not a generating set of Lie algebra on 3 generators (X, Y, Z) over Integer Ring
The generators do not have to correspond to the defined generating set of the domain:
sage: L.<X,Y,Z,W> = LieAlgebra(QQ, {('X','Y'): {'Z':1}, ('X','Z'): {'W':1}}) sage: K.<A,B,C> = LieAlgebra(QQ, {('A','B'): {'C':2}}) sage: phi = L.morphism({X+2*Y: A, XY: B}); phi Lie algebra morphism: From: Lie algebra on 4 generators (X, Y, Z, W) over Rational Field To: Lie algebra on 3 generators (A, B, C) over Rational Field Defn: X > 1/3*A + 2/3*B Y > 1/3*A  1/3*B Z > 2/3*C W > 0 sage: phi(X+2*Y) A sage: phi(X) 1/3*A + 2/3*B sage: phi(W) 0 sage: phi(Z) 2/3*C sage: all(K[phi(p), phi(q)] == phi(L[p,q]) ....: for p in L.basis() for q in L.basis()) True
A quotient type Lie algebra morphism:
sage: K.<A,B> = LieAlgebra(SR, abelian=True) sage: L.morphism({X: A, Y: B}) Lie algebra morphism: From: Lie algebra on 4 generators (X, Y, Z, W) over Rational Field To: Abelian Lie algebra on 2 generators (A, B) over Symbolic Ring Defn: X > A Y > B Z > 0 W > 0