Homomorphisms of Lie Algebras¶
AUTHORS:
Travis Scrimshaw (07-15-2013): Initial implementation
Eero Hakavuori (08-09-2018): Morphisms defined by a generating subset
- class sage.algebras.lie_algebras.morphism.LieAlgebraHomomorphism_im_gens(parent, im_gens, base_map=None, 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 \(\mathfrak{g}\).
INPUT:
parent
– a homset between two Lie algebrasim_gens
– the image of the generators of the domainbase_map
– a homomorphism to apply to the coefficients. It should be a map from the base ring of the domain to the base ring of the codomain. Note that if base_map is nontrivial then the result will not be a morphism in the category of lie algebras over the base ring.check
– whether to run checks on the validity of the defining data
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
You can provide a base map, creating a semilinear map that (sometimes) preserves the Lie bracket:
sage: R.<x> = ZZ[] sage: K.<i> = NumberField(x^2 + 1) sage: cc = K.hom([-i]) sage: L.<X,Y,Z,W> = LieAlgebra(K, {('X','Y'): {'Z':1}, ('X','Z'): {'W':1}}) sage: M.<A,B,C,D> = LieAlgebra(K, {('A','B'): {'C':1}, ('A','C'): {'D':1}}) sage: phi = L.morphism({X:A, Y:B, Z:C, W:D}, base_map=cc) sage: phi(X) A sage: phi(i*X) -i*A sage: all(phi(x.bracket(y)) == phi(x).bracket(phi(y)) for x,y in cartesian_product_iterator([[X,Y,Z,W],[X,Y,Z,W]])) True
Note that the Lie bracket should still be preserved, even though the map is no longer linear over the base ring:
sage: L.<X,Y,Z,W> = LieAlgebra(K, {('X','Y'): {'Z':i}, ('X','Z'): {'W':1}}) sage: M.<A,B,C,D> = LieAlgebra(K, {('A','B'): {'C':-i}, ('A','C'): {'D':1}}) sage: phi = L.morphism({X:A, Y:B, Z:C, W:D}, base_map=cc) sage: phi(X.bracket(Y)) -i*C sage: phi(X).bracket(phi(Y)) -i*C
- base_map()¶
Return the map on the base ring that is part of the defining data for this morphism. May return
None
if a coercion is used.EXAMPLES:
sage: R.<x> = ZZ[] sage: K.<i> = NumberField(x^2 + 1) sage: cc = K.hom([-i]) sage: L.<X,Y,Z,W> = LieAlgebra(K, {('X','Y'): {'Z':1}, ('X','Z'): {'W':1}}) sage: M.<A,B> = LieAlgebra(K, abelian=True) sage: phi = L.morphism({X: A, Y: B}, base_map=cc) sage: phi(X) A sage: phi(i*X) -i*A sage: phi.base_map() Ring endomorphism of Number Field in i with defining polynomial x^2 + 1 Defn: i |--> -i
- 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, base_map=None, category=None)¶
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 givenbase_map
– a homomorphism to apply to the coefficients. It should be a map from the base ring of the domain to the base ring of the codomain. Note that if base_map is nontrivial then the result will not be a morphism in the category of lie algebras over the base ring.check
– (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 non-fields, 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, X-Y: 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