Verma Modules

AUTHORS:

  • Travis Scrimshaw (2017-06-30): Initial version

Todo

Implement a sage.categories.pushout.ConstructionFunctor and return as the construction().

class sage.algebras.lie_algebras.verma_module.ModulePrinting(vector_name='v')[source]

Bases: object

Helper mixin class for printing the module vectors.

class sage.algebras.lie_algebras.verma_module.VermaModule(g, weight, basis_key=None, prefix='f', **kwds)[source]

Bases: ModulePrinting, CombinatorialFreeModule

A Verma module.

Let \(\lambda\) be a weight and \(\mathfrak{g}\) be a Kac–Moody Lie algebra with a fixed Borel subalgebra \(\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{g}^+\). The Verma module \(M_{\lambda}\) is a \(U(\mathfrak{g})\)-module given by

\[M_{\lambda} := U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} F_{\lambda},\]

where \(F_{\lambda}\) is the \(U(\mathfrak{b})\) module such that \(h \in U(\mathfrak{h})\) acts as multiplication by \(\langle \lambda, h \rangle\) and \(U(\mathfrak{g}^+) F_{\lambda} = 0\).

INPUT:

  • g – a Lie algebra

  • weight – a weight

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(2*La[1] + 3*La[2])
sage: pbw = M.pbw_basis()
sage: E1,E2,F1,F2,H1,H2 = [pbw(g) for g in L.gens()]
sage: v = M.highest_weight_vector()
sage: x = F2^3 * F1 * v
sage: x
f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]
sage: F1 * x
f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]
 + 3*f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]
sage: E1 * x
2*f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]
sage: H1 * x
3*f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]
sage: H2 * x
-2*f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(Integer(2)*La[Integer(1)] + Integer(3)*La[Integer(2)])
>>> pbw = M.pbw_basis()
>>> E1,E2,F1,F2,H1,H2 = [pbw(g) for g in L.gens()]
>>> v = M.highest_weight_vector()
>>> x = F2**Integer(3) * F1 * v
>>> x
f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]
>>> F1 * x
f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]
 + 3*f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]
>>> E1 * x
2*f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]
>>> H1 * x
3*f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]
>>> H2 * x
-2*f[-alpha[2]]^3*f[-alpha[1]]*v[2*Lambda[1] + 3*Lambda[2]]

REFERENCES:

class Element[source]

Bases: IndexedFreeModuleElement

contravariant_form(x, y)[source]

Return the contravariant form of x and y.

Let \(C(x, y)\) denote the (universal) contravariant form on \(U(\mathfrak{g})\). Then the contravariant form on \(M(\lambda)\) is given by evaluating \(C(x, y) \in U(\mathfrak{h})\) at \(\lambda\).

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['A', 1])
sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = g.verma_module(2*La[1])
sage: U = M.pbw_basis()
sage: v = M.highest_weight_vector()
sage: e, h, f = U.algebra_generators()
sage: elts = [f^k * v for k in range(8)]; elts
[v[2*Lambda[1]], f[-alpha[1]]*v[2*Lambda[1]],
 f[-alpha[1]]^2*v[2*Lambda[1]], f[-alpha[1]]^3*v[2*Lambda[1]],
 f[-alpha[1]]^4*v[2*Lambda[1]], f[-alpha[1]]^5*v[2*Lambda[1]],
 f[-alpha[1]]^6*v[2*Lambda[1]], f[-alpha[1]]^7*v[2*Lambda[1]]]
sage: matrix([[M.contravariant_form(x, y) for x in elts] for y in elts])
[1 0 0 0 0 0 0 0]
[0 2 0 0 0 0 0 0]
[0 0 4 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['A', Integer(1)])
>>> La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = g.verma_module(Integer(2)*La[Integer(1)])
>>> U = M.pbw_basis()
>>> v = M.highest_weight_vector()
>>> e, h, f = U.algebra_generators()
>>> elts = [f**k * v for k in range(Integer(8))]; elts
[v[2*Lambda[1]], f[-alpha[1]]*v[2*Lambda[1]],
 f[-alpha[1]]^2*v[2*Lambda[1]], f[-alpha[1]]^3*v[2*Lambda[1]],
 f[-alpha[1]]^4*v[2*Lambda[1]], f[-alpha[1]]^5*v[2*Lambda[1]],
 f[-alpha[1]]^6*v[2*Lambda[1]], f[-alpha[1]]^7*v[2*Lambda[1]]]
>>> matrix([[M.contravariant_form(x, y) for x in elts] for y in elts])
[1 0 0 0 0 0 0 0]
[0 2 0 0 0 0 0 0]
[0 0 4 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
degree_on_basis(m)[source]

Return the degree (or weight) of the basis element indexed by m.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(2*La[1] + 3*La[2])
sage: v = M.highest_weight_vector()
sage: M.degree_on_basis(v.leading_support())
2*Lambda[1] + 3*Lambda[2]

sage: pbw = M.pbw_basis()
sage: G = list(pbw.gens())
sage: f1, f2 = L.f()
sage: x = pbw(f1.bracket(f2)) * pbw(f1) * v
sage: x.degree()
-Lambda[1] + 3*Lambda[2]

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(Integer(2)*La[Integer(1)] + Integer(3)*La[Integer(2)])
>>> v = M.highest_weight_vector()
>>> M.degree_on_basis(v.leading_support())
2*Lambda[1] + 3*Lambda[2]

>>> pbw = M.pbw_basis()
>>> G = list(pbw.gens())
>>> f1, f2 = L.f()
>>> x = pbw(f1.bracket(f2)) * pbw(f1) * v
>>> x.degree()
-Lambda[1] + 3*Lambda[2]
dual()[source]

Return the dual module \(M(\lambda)^{\vee}\) in Category \(\mathcal{O}\).

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 2)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: M = L.verma_module(2*La[1])
sage: Mc = M.dual()

sage: Mp = L.verma_module(-2*La[1])
sage: Mp.dual() is Mp
True

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(2))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> M = L.verma_module(Integer(2)*La[Integer(1)])
>>> Mc = M.dual()

>>> Mp = L.verma_module(-Integer(2)*La[Integer(1)])
>>> Mp.dual() is Mp
True
gens()[source]

Return the generators of self as a \(U(\mathfrak{g})\)-module.

EXAMPLES:

sage: L = lie_algebras.sp(QQ, 6)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] - 3*La[2])
sage: M.gens()
(v[Lambda[1] - 3*Lambda[2]],)

>>> from sage.all import *
>>> L = lie_algebras.sp(QQ, Integer(6))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] - Integer(3)*La[Integer(2)])
>>> M.gens()
(v[Lambda[1] - 3*Lambda[2]],)
highest_weight()[source]

Return the highest weight of self.

EXAMPLES:

sage: L = lie_algebras.so(QQ, 7)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: M = L.verma_module(4*La[1] - 3/2*La[2])
sage: M.highest_weight()
4*Lambda[1] - 3/2*Lambda[2]

>>> from sage.all import *
>>> L = lie_algebras.so(QQ, Integer(7))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> M = L.verma_module(Integer(4)*La[Integer(1)] - Integer(3)/Integer(2)*La[Integer(2)])
>>> M.highest_weight()
4*Lambda[1] - 3/2*Lambda[2]
highest_weight_vector()[source]

Return the highest weight vector of self.

EXAMPLES:

sage: L = lie_algebras.sp(QQ, 6)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] - 3*La[2])
sage: M.highest_weight_vector()
v[Lambda[1] - 3*Lambda[2]]

>>> from sage.all import *
>>> L = lie_algebras.sp(QQ, Integer(6))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] - Integer(3)*La[Integer(2)])
>>> M.highest_weight_vector()
v[Lambda[1] - 3*Lambda[2]]
homogeneous_component_basis(d)[source]

Return a basis for the d-th homogeneous component of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: P = L.cartan_type().root_system().weight_lattice()
sage: La = P.fundamental_weights()
sage: al = P.simple_roots()
sage: mu = 2*La[1] + 3*La[2]
sage: M = L.verma_module(mu)
sage: M.homogeneous_component_basis(mu - al[2])
[f[-alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - 3*al[2])
[f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - 3*al[2] - 2*al[1])
[f[-alpha[2]]*f[-alpha[1] - alpha[2]]^2*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - La[1])
Family ()

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> P = L.cartan_type().root_system().weight_lattice()
>>> La = P.fundamental_weights()
>>> al = P.simple_roots()
>>> mu = Integer(2)*La[Integer(1)] + Integer(3)*La[Integer(2)]
>>> M = L.verma_module(mu)
>>> M.homogeneous_component_basis(mu - al[Integer(2)])
[f[-alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - Integer(3)*al[Integer(2)])
[f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - Integer(3)*al[Integer(2)] - Integer(2)*al[Integer(1)])
[f[-alpha[2]]*f[-alpha[1] - alpha[2]]^2*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - La[Integer(1)])
Family ()
is_projective()[source]

Return if self is a projective module in Category \(\mathcal{O}\).

A Verma module \(M_{\lambda}\) is projective (in Category \(\mathcal{O}\) if and only if \(\lambda\) is Verma dominant in the sense

\[\langle \lambda + \rho, \alpha^{\vee} \rangle \notin \ZZ_{<0}\]

for all positive roots \(\alpha\).

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: L.verma_module(La[1] + La[2]).is_projective()
True
sage: L.verma_module(-La[1] - La[2]).is_projective()
True
sage: L.verma_module(3/2*La[1] + 1/2*La[2]).is_projective()
True
sage: L.verma_module(3/2*La[1] + 1/3*La[2]).is_projective()
True
sage: L.verma_module(-3*La[1] + 2/3*La[2]).is_projective()
False

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> L.verma_module(La[Integer(1)] + La[Integer(2)]).is_projective()
True
>>> L.verma_module(-La[Integer(1)] - La[Integer(2)]).is_projective()
True
>>> L.verma_module(Integer(3)/Integer(2)*La[Integer(1)] + Integer(1)/Integer(2)*La[Integer(2)]).is_projective()
True
>>> L.verma_module(Integer(3)/Integer(2)*La[Integer(1)] + Integer(1)/Integer(3)*La[Integer(2)]).is_projective()
True
>>> L.verma_module(-Integer(3)*La[Integer(1)] + Integer(2)/Integer(3)*La[Integer(2)]).is_projective()
False
is_simple()[source]

Return if self is a simple module.

A Verma module \(M_{\lambda}\) is simple if and only if \(\lambda\) is Verma antidominant in the sense

\[\langle \lambda + \rho, \alpha^{\vee} \rangle \notin \ZZ_{>0}\]

for all positive roots \(\alpha\).

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: L.verma_module(La[1] + La[2]).is_simple()
False
sage: L.verma_module(-La[1] - La[2]).is_simple()
True
sage: L.verma_module(3/2*La[1] + 1/2*La[2]).is_simple()
False
sage: L.verma_module(3/2*La[1] + 1/3*La[2]).is_simple()
True
sage: L.verma_module(-3*La[1] + 2/3*La[2]).is_simple()
True

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> L.verma_module(La[Integer(1)] + La[Integer(2)]).is_simple()
False
>>> L.verma_module(-La[Integer(1)] - La[Integer(2)]).is_simple()
True
>>> L.verma_module(Integer(3)/Integer(2)*La[Integer(1)] + Integer(1)/Integer(2)*La[Integer(2)]).is_simple()
False
>>> L.verma_module(Integer(3)/Integer(2)*La[Integer(1)] + Integer(1)/Integer(3)*La[Integer(2)]).is_simple()
True
>>> L.verma_module(-Integer(3)*La[Integer(1)] + Integer(2)/Integer(3)*La[Integer(2)]).is_simple()
True
is_singular()[source]

Return if self is a singular Verma module.

A Verma module \(M_{\lambda}\) is singular if there does not exist a dominant weight \(\tilde{\lambda}\) that is in the dot orbit of \(\lambda\). We call a Verma module regular otherwise.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: M.is_singular()
False
sage: M = L.verma_module(La[1] - La[2])
sage: M.is_singular()
True
sage: M = L.verma_module(2*La[1] - 10*La[2])
sage: M.is_singular()
False
sage: M = L.verma_module(-2*La[1] - 2*La[2])
sage: M.is_singular()
False
sage: M = L.verma_module(-4*La[1] - La[2])
sage: M.is_singular()
True

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> M.is_singular()
False
>>> M = L.verma_module(La[Integer(1)] - La[Integer(2)])
>>> M.is_singular()
True
>>> M = L.verma_module(Integer(2)*La[Integer(1)] - Integer(10)*La[Integer(2)])
>>> M.is_singular()
False
>>> M = L.verma_module(-Integer(2)*La[Integer(1)] - Integer(2)*La[Integer(2)])
>>> M.is_singular()
False
>>> M = L.verma_module(-Integer(4)*La[Integer(1)] - La[Integer(2)])
>>> M.is_singular()
True
lie_algebra()[source]

Return the underlying Lie algebra of self.

EXAMPLES:

sage: L = lie_algebras.so(QQ, 9)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: M = L.verma_module(La[3] - 1/2*La[1])
sage: M.lie_algebra()
Lie algebra of ['B', 4] in the Chevalley basis

>>> from sage.all import *
>>> L = lie_algebras.so(QQ, Integer(9))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> M = L.verma_module(La[Integer(3)] - Integer(1)/Integer(2)*La[Integer(1)])
>>> M.lie_algebra()
Lie algebra of ['B', 4] in the Chevalley basis
pbw_basis()[source]

Return the PBW basis of the underlying Lie algebra used to define self.

EXAMPLES:

sage: L = lie_algebras.so(QQ, 8)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[2] - 2*La[3])
sage: M.pbw_basis()
Universal enveloping algebra of Lie algebra of ['D', 4] in the Chevalley basis
 in the Poincare-Birkhoff-Witt basis

>>> from sage.all import *
>>> L = lie_algebras.so(QQ, Integer(8))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(2)] - Integer(2)*La[Integer(3)])
>>> M.pbw_basis()
Universal enveloping algebra of Lie algebra of ['D', 4] in the Chevalley basis
 in the Poincare-Birkhoff-Witt basis
poincare_birkhoff_witt_basis()[source]

Return the PBW basis of the underlying Lie algebra used to define self.

EXAMPLES:

sage: L = lie_algebras.so(QQ, 8)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[2] - 2*La[3])
sage: M.pbw_basis()
Universal enveloping algebra of Lie algebra of ['D', 4] in the Chevalley basis
 in the Poincare-Birkhoff-Witt basis

>>> from sage.all import *
>>> L = lie_algebras.so(QQ, Integer(8))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(2)] - Integer(2)*La[Integer(3)])
>>> M.pbw_basis()
Universal enveloping algebra of Lie algebra of ['D', 4] in the Chevalley basis
 in the Poincare-Birkhoff-Witt basis
weight_space_basis(d)[source]

Return a basis for the d-th homogeneous component of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: P = L.cartan_type().root_system().weight_lattice()
sage: La = P.fundamental_weights()
sage: al = P.simple_roots()
sage: mu = 2*La[1] + 3*La[2]
sage: M = L.verma_module(mu)
sage: M.homogeneous_component_basis(mu - al[2])
[f[-alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - 3*al[2])
[f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - 3*al[2] - 2*al[1])
[f[-alpha[2]]*f[-alpha[1] - alpha[2]]^2*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]]
sage: M.homogeneous_component_basis(mu - La[1])
Family ()

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> P = L.cartan_type().root_system().weight_lattice()
>>> La = P.fundamental_weights()
>>> al = P.simple_roots()
>>> mu = Integer(2)*La[Integer(1)] + Integer(3)*La[Integer(2)]
>>> M = L.verma_module(mu)
>>> M.homogeneous_component_basis(mu - al[Integer(2)])
[f[-alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - Integer(3)*al[Integer(2)])
[f[-alpha[2]]^3*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - Integer(3)*al[Integer(2)] - Integer(2)*al[Integer(1)])
[f[-alpha[2]]*f[-alpha[1] - alpha[2]]^2*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^2*f[-alpha[1]]*f[-alpha[1] - alpha[2]]*v[2*Lambda[1] + 3*Lambda[2]],
 f[-alpha[2]]^3*f[-alpha[1]]^2*v[2*Lambda[1] + 3*Lambda[2]]]
>>> M.homogeneous_component_basis(mu - La[Integer(1)])
Family ()
class sage.algebras.lie_algebras.verma_module.VermaModuleHomset(X, Y, category=None, base=None, check=True)[source]

Bases: Homset

The set of morphisms from a Verma module to another module in Category \(\mathcal{O}\) considered as \(U(\mathfrak{g})\)-representations.

This currently assumes the codomain is a Verma module, its dual, or a simple module.

Let \(M_{w \cdot \lambda}\) and \(M_{w' \cdot \lambda'}\) be Verma modules, \(\cdot\) is the dot action, and \(\lambda + \rho\), \(\lambda' + \rho\) are dominant weights. Then we have

\[\dim \hom(M_{w \cdot \lambda}, M_{w' \cdot \lambda'}) = 1\]

if and only if \(\lambda = \lambda'\) and \(w' \leq w\) in Bruhat order. Otherwise the homset is 0 dimensional.

If the codomain is a dual Verma module \(M_{\mu}^{\vee}\), then the homset is \(\delta_{\lambda\mu}\) dimensional. When \(\mu = \lambda\), the image is the simple module \(L_{\lambda}\).

Element[source]

alias of VermaModuleMorphism

basis()[source]

Return a basis of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: Mp = L.verma_module(M.highest_weight().dot_action([2]))
sage: H = Hom(Mp, M)
sage: list(H.basis()) == [H.natural_map()]
True

sage: Mp = L.verma_module(La[1] + 2*La[2])
sage: H = Hom(Mp, M)
sage: H.basis()
Family ()

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> Mp = L.verma_module(M.highest_weight().dot_action([Integer(2)]))
>>> H = Hom(Mp, M)
>>> list(H.basis()) == [H.natural_map()]
True

>>> Mp = L.verma_module(La[Integer(1)] + Integer(2)*La[Integer(2)])
>>> H = Hom(Mp, M)
>>> H.basis()
Family ()
dimension()[source]

Return the dimension of self (as a vector space over the base ring).

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: Mp = L.verma_module(M.highest_weight().dot_action([2]))
sage: H = Hom(Mp, M)
sage: H.dimension()
1

sage: Mp = L.verma_module(La[1] + 2*La[2])
sage: H = Hom(Mp, M)
sage: H.dimension()
0

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> Mp = L.verma_module(M.highest_weight().dot_action([Integer(2)]))
>>> H = Hom(Mp, M)
>>> H.dimension()
1

>>> Mp = L.verma_module(La[Integer(1)] + Integer(2)*La[Integer(2)])
>>> H = Hom(Mp, M)
>>> H.dimension()
0
highest_weight_image()[source]

Return the image of the highest weight vector of the domain in the codomain.

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['C', 3])
sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = g.verma_module(La[1] + 2*La[3])
sage: Mc = M.dual()
sage: H = Hom(M, Mc)
sage: H.highest_weight_image()
v[Lambda[1] + 2*Lambda[3]]^*
sage: L = H.natural_map().image()
sage: Hp = Hom(M, L)
sage: Hp.highest_weight_image()
u[Lambda[1] + 2*Lambda[3]]

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['C', Integer(3)])
>>> La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = g.verma_module(La[Integer(1)] + Integer(2)*La[Integer(3)])
>>> Mc = M.dual()
>>> H = Hom(M, Mc)
>>> H.highest_weight_image()
v[Lambda[1] + 2*Lambda[3]]^*
>>> L = H.natural_map().image()
>>> Hp = Hom(M, L)
>>> Hp.highest_weight_image()
u[Lambda[1] + 2*Lambda[3]]
natural_map()[source]

Return the “natural map” of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: Mp = L.verma_module(M.highest_weight().dot_action([2]))
sage: H = Hom(Mp, M)
sage: H.natural_map()
Verma module morphism:
  From: Verma module with highest weight 3*Lambda[1] - 3*Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  Defn: v[3*Lambda[1] - 3*Lambda[2]] |-->
         f[-alpha[2]]^2*v[Lambda[1] + Lambda[2]]

sage: Mp = L.verma_module(La[1] + 2*La[2])
sage: H = Hom(Mp, M)
sage: H.natural_map()
Verma module morphism:
  From: Verma module with highest weight Lambda[1] + 2*Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  Defn: v[Lambda[1] + 2*Lambda[2]] |--> 0

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> Mp = L.verma_module(M.highest_weight().dot_action([Integer(2)]))
>>> H = Hom(Mp, M)
>>> H.natural_map()
Verma module morphism:
  From: Verma module with highest weight 3*Lambda[1] - 3*Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  Defn: v[3*Lambda[1] - 3*Lambda[2]] |-->
         f[-alpha[2]]^2*v[Lambda[1] + Lambda[2]]

>>> Mp = L.verma_module(La[Integer(1)] + Integer(2)*La[Integer(2)])
>>> H = Hom(Mp, M)
>>> H.natural_map()
Verma module morphism:
  From: Verma module with highest weight Lambda[1] + 2*Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + Lambda[2]
         of Lie algebra of ['A', 2] in the Chevalley basis
  Defn: v[Lambda[1] + 2*Lambda[2]] |--> 0
singular_vector()[source]

Return the singular vector in the codomain corresponding to the domain’s highest weight element or None if no such element exists.

ALGORITHM:

We essentially follow the algorithm laid out in [deG2005]. We split the main computation into two cases. If there exists an \(i\) such that \(\langle \lambda + \rho, \alpha_i^{\vee} \rangle = m > 0\) (i.e., the weight \(\lambda\) is \(i\)-dominant with respect to the dot action), then we use the \(\mathfrak{sl}_2\) relation on \(M_{s_i \cdot \lambda} \to M_{\lambda}\) to construct the singular vector \(f_i^m v_{\lambda}\). Otherwise we find the shortest root \(\alpha\) such that \(\langle \lambda + \rho, \alpha^{\vee} \rangle > 0\) and explicitly compute the kernel with respect to the weight basis elements. We iterate this until we reach \(\mu\).

EXAMPLES:

sage: L = lie_algebras.sp(QQ, 6)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: la = La[1] - La[3]
sage: mu = la.dot_action([1,2])
sage: M = L.verma_module(la)
sage: Mp = L.verma_module(mu)
sage: H = Hom(Mp, M)
sage: v = H.singular_vector(); v
f[-alpha[2]]*f[-alpha[1]]^3*v[Lambda[1] - Lambda[3]]
 + 3*f[-alpha[1]]^2*f[-alpha[1] - alpha[2]]*v[Lambda[1] - Lambda[3]]
sage: v.degree() == Mp.highest_weight()
True

>>> from sage.all import *
>>> L = lie_algebras.sp(QQ, Integer(6))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> la = La[Integer(1)] - La[Integer(3)]
>>> mu = la.dot_action([Integer(1),Integer(2)])
>>> M = L.verma_module(la)
>>> Mp = L.verma_module(mu)
>>> H = Hom(Mp, M)
>>> v = H.singular_vector(); v
f[-alpha[2]]*f[-alpha[1]]^3*v[Lambda[1] - Lambda[3]]
 + 3*f[-alpha[1]]^2*f[-alpha[1] - alpha[2]]*v[Lambda[1] - Lambda[3]]
>>> v.degree() == Mp.highest_weight()
True

sage: L = LieAlgebra(QQ, cartan_type=['F', 4])
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: la = La[1] + La[2] - La[3]
sage: mu = la.dot_action([1,2,3,2])
sage: M = L.verma_module(la)
sage: Mp = L.verma_module(mu)
sage: H = Hom(Mp, M)
sage: v = H.singular_vector()
sage: pbw = M.pbw_basis()
sage: E = [pbw(e) for e in L.e()]
sage: all(e * v == M.zero() for e in E)  # long time
True
sage: v.degree() == Mp.highest_weight()
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, cartan_type=['F', Integer(4)])
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> la = La[Integer(1)] + La[Integer(2)] - La[Integer(3)]
>>> mu = la.dot_action([Integer(1),Integer(2),Integer(3),Integer(2)])
>>> M = L.verma_module(la)
>>> Mp = L.verma_module(mu)
>>> H = Hom(Mp, M)
>>> v = H.singular_vector()
>>> pbw = M.pbw_basis()
>>> E = [pbw(e) for e in L.e()]
>>> all(e * v == M.zero() for e in E)  # long time
True
>>> v.degree() == Mp.highest_weight()
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, cartan_type=['F', Integer(4)])
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> la = La[Integer(1)] + La[Integer(2)] - La[Integer(3)]
>>> mu = la.dot_action([Integer(1),Integer(2),Integer(3),Integer(2)])
>>> M = L.verma_module(la)
>>> Mp = L.verma_module(mu)
>>> H = Hom(Mp, M)
>>> v = H.singular_vector()
>>> pbw = M.pbw_basis()
>>> E = [pbw(e) for e in L.e()]
>>> all(e * v == M.zero() for e in E)  # long time
True
>>> v.degree() == Mp.highest_weight()
True

When \(w \cdot \lambda \notin \lambda + Q^-\), there does not exist a singular vector:

sage: L = lie_algebras.sl(QQ, 4)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: la = 3/7*La[1] - 1/2*La[3]
sage: mu = la.dot_action([1,2])
sage: M = L.verma_module(la)
sage: Mp = L.verma_module(mu)
sage: H = Hom(Mp, M)
sage: H.singular_vector() is None
True

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(4))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> la = Integer(3)/Integer(7)*La[Integer(1)] - Integer(1)/Integer(2)*La[Integer(3)]
>>> mu = la.dot_action([Integer(1),Integer(2)])
>>> M = L.verma_module(la)
>>> Mp = L.verma_module(mu)
>>> H = Hom(Mp, M)
>>> H.singular_vector() is None
True

When we need to apply a non-simple reflection, we can compute the singular vector (see Issue #36793):

sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = g.verma_module((0*La[1]).dot_action([1]))
sage: Mp = g.verma_module((0*La[1]).dot_action([1,2]))
sage: H = Hom(Mp, M)
sage: v = H.singular_vector(); v
1/2*f[-alpha[2]]*f[-alpha[1]]*v[-2*Lambda[1] + Lambda[2]]
 + f[-alpha[1] - alpha[2]]*v[-2*Lambda[1] + Lambda[2]]
sage: pbw = M.pbw_basis()
sage: E = [pbw(e) for e in g.e()]
sage: all(e * v == M.zero() for e in E)
True
sage: v.degree() == Mp.highest_weight()
True

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['A', Integer(2)])
>>> La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = g.verma_module((Integer(0)*La[Integer(1)]).dot_action([Integer(1)]))
>>> Mp = g.verma_module((Integer(0)*La[Integer(1)]).dot_action([Integer(1),Integer(2)]))
>>> H = Hom(Mp, M)
>>> v = H.singular_vector(); v
1/2*f[-alpha[2]]*f[-alpha[1]]*v[-2*Lambda[1] + Lambda[2]]
 + f[-alpha[1] - alpha[2]]*v[-2*Lambda[1] + Lambda[2]]
>>> pbw = M.pbw_basis()
>>> E = [pbw(e) for e in g.e()]
>>> all(e * v == M.zero() for e in E)
True
>>> v.degree() == Mp.highest_weight()
True
zero()[source]

Return the zero morphism of self.

EXAMPLES:

sage: L = lie_algebras.sp(QQ, 6)
sage: La = L.cartan_type().root_system().weight_space().fundamental_weights()
sage: M = L.verma_module(La[1] + 2/3*La[2])
sage: Mp = L.verma_module(La[2] - La[3])
sage: H = Hom(Mp, M)
sage: H.zero()
Verma module morphism:
  From: Verma module with highest weight Lambda[2] - Lambda[3]
         of Lie algebra of ['C', 3] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + 2/3*Lambda[2]
         of Lie algebra of ['C', 3] in the Chevalley basis
  Defn: v[Lambda[2] - Lambda[3]] |--> 0

>>> from sage.all import *
>>> L = lie_algebras.sp(QQ, Integer(6))
>>> La = L.cartan_type().root_system().weight_space().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + Integer(2)/Integer(3)*La[Integer(2)])
>>> Mp = L.verma_module(La[Integer(2)] - La[Integer(3)])
>>> H = Hom(Mp, M)
>>> H.zero()
Verma module morphism:
  From: Verma module with highest weight Lambda[2] - Lambda[3]
         of Lie algebra of ['C', 3] in the Chevalley basis
  To:   Verma module with highest weight Lambda[1] + 2/3*Lambda[2]
         of Lie algebra of ['C', 3] in the Chevalley basis
  Defn: v[Lambda[2] - Lambda[3]] |--> 0
class sage.algebras.lie_algebras.verma_module.VermaModuleMorphism(parent, scalar)[source]

Bases: Morphism

A morphism of a Verma module to another module in Category \(\mathcal{O}\).

image()[source]

Return the image of self.

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['B', 2])
sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = g.verma_module(La[1] + 2*La[2])
sage: Mp = g.verma_module(La[1] + 3*La[2])
sage: phi = Hom(M, Mp).natural_map()
sage: phi.image()
Free module generated by {} over Rational Field
sage: Mc = M.dual()
sage: phi = Hom(M, Mc).natural_map()
sage: L = phi.image(); L
Simple module with highest weight Lambda[1] + 2*Lambda[2] of
 Lie algebra of ['B', 2] in the Chevalley basis
sage: psi = Hom(M, L).natural_map()
sage: psi.image()
Simple module with highest weight Lambda[1] + 2*Lambda[2] of
 Lie algebra of ['B', 2] in the Chevalley basis

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['B', Integer(2)])
>>> La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = g.verma_module(La[Integer(1)] + Integer(2)*La[Integer(2)])
>>> Mp = g.verma_module(La[Integer(1)] + Integer(3)*La[Integer(2)])
>>> phi = Hom(M, Mp).natural_map()
>>> phi.image()
Free module generated by {} over Rational Field
>>> Mc = M.dual()
>>> phi = Hom(M, Mc).natural_map()
>>> L = phi.image(); L
Simple module with highest weight Lambda[1] + 2*Lambda[2] of
 Lie algebra of ['B', 2] in the Chevalley basis
>>> psi = Hom(M, L).natural_map()
>>> psi.image()
Simple module with highest weight Lambda[1] + 2*Lambda[2] of
 Lie algebra of ['B', 2] in the Chevalley basis
is_injective()[source]

Return if self is injective or not.

A morphism \(\phi : M \to M'\) from a Verma module \(M\) to another Verma module \(M'\) is injective if and only if \(\dim \hom(M, M') = 1\) and \(\phi \neq 0\). If \(M'\) is a dual Verma or simple module, then the result is not injective.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: Mp = L.verma_module(M.highest_weight().dot_action([1,2]))
sage: Mpp = L.verma_module(M.highest_weight().dot_action([1,2]) + La[1])
sage: phi = Hom(Mp, M).natural_map()
sage: phi.is_injective()
True
sage: (0 * phi).is_injective()
False
sage: psi = Hom(Mpp, Mp).natural_map()
sage: psi.is_injective()
False

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> Mp = L.verma_module(M.highest_weight().dot_action([Integer(1),Integer(2)]))
>>> Mpp = L.verma_module(M.highest_weight().dot_action([Integer(1),Integer(2)]) + La[Integer(1)])
>>> phi = Hom(Mp, M).natural_map()
>>> phi.is_injective()
True
>>> (Integer(0) * phi).is_injective()
False
>>> psi = Hom(Mpp, Mp).natural_map()
>>> psi.is_injective()
False
is_surjective()[source]

Return if self is surjective or not.

A morphism \(\phi : M \to M'\) from a Verma module \(M\) to another Verma module \(M'\) is surjective if and only if the domain is equal to the codomain and it is not the zero morphism.

If \(M'\) is a simple module, then this surjective if and only if \(\dim \hom(M, M') = 1\) and \(\phi \neq 0\).

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 3)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] + La[2])
sage: Mp = L.verma_module(M.highest_weight().dot_action([1,2]))
sage: phi = Hom(M, M).natural_map()
sage: phi.is_surjective()
True
sage: (0 * phi).is_surjective()
False
sage: psi = Hom(Mp, M).natural_map()
sage: psi.is_surjective()
False

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(3))
>>> La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
>>> M = L.verma_module(La[Integer(1)] + La[Integer(2)])
>>> Mp = L.verma_module(M.highest_weight().dot_action([Integer(1),Integer(2)]))
>>> phi = Hom(M, M).natural_map()
>>> phi.is_surjective()
True
>>> (Integer(0) * phi).is_surjective()
False
>>> psi = Hom(Mp, M).natural_map()
>>> psi.is_surjective()
False