def __classcall_private__(cls,
R,
classification,
s_coeff,
names,
index_set=None,
category=None,
**kwds):
r"""
Normalize input to ensure a unique representation.
"""
names, index_set = standardize_names_index_set(names, index_set)
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
cat = LieAlgebras(R).FiniteDimensional().WithBasis().Nilpotent()
category = cat.or_subcategory(category)
return super(ClassifiedNilpotentLieAlgebra,
cls).__classcall__(cls,
R,
classification,
s_coeff,
names,
index_set,
category=category,
**kwds)
def __classcall_private__(cls, R, s_coeff, names=None, index_set=None,
category=None, **kwds):
"""
Normalize input to ensure a unique representation.
EXAMPLES:
If the variable order is specified, the order of structural
coefficients does not matter::
sage: from sage.algebras.lie_algebras.nilpotent_lie_algebra import NilpotentLieAlgebra_dense
sage: L1.<x,y,z> = NilpotentLieAlgebra_dense(QQ, {('x','y'): {'z': 1}})
sage: L2.<x,y,z> = NilpotentLieAlgebra_dense(QQ, {('y','x'): {'z': -1}})
sage: L1 is L2
True
If the variables are implicitly defined by the structural coefficients,
the ordering may be different and the Lie algebras will be considered
different::
sage: from sage.algebras.lie_algebras.nilpotent_lie_algebra import NilpotentLieAlgebra_dense
sage: L1 = NilpotentLieAlgebra_dense(QQ, {('x','y'): {'z': 1}})
sage: L2 = NilpotentLieAlgebra_dense(QQ, {('y','x'): {'z': -1}})
sage: L1
Nilpotent Lie algebra on 3 generators (x, y, z) over Rational Field
sage: L2
Nilpotent Lie algebra on 3 generators (y, x, z) over Rational Field
sage: L1 is L2
False
Constructed using two different methods from :class:`LieAlgebra`
yields the same Lie algebra::
sage: sc = {('X','Y'): {'Z': 1}}
sage: C = LieAlgebras(QQ).Nilpotent().FiniteDimensional().WithBasis()
sage: L1.<X,Y,Z> = LieAlgebra(QQ, sc, category=C)
sage: L2 = LieAlgebra(QQ, sc, nilpotent=True, names=['X','Y','Z'])
sage: L1 is L2
True
"""
if not names:
# extract names from structural coefficients
names = []
for (X, Y), d in s_coeff.items():
if X not in names: names.append(X)
if Y not in names: names.append(Y)
for k in d:
if k not in names: names.append(k)
from sage.structure.indexed_generators import standardize_names_index_set
names, index_set = standardize_names_index_set(names, index_set)
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
cat = LieAlgebras(R).FiniteDimensional().WithBasis().Nilpotent()
category = cat.or_subcategory(category)
return super(NilpotentLieAlgebra_dense, cls).__classcall__(
cls, R, s_coeff, names, index_set, category=category, **kwds)
def __init__(self, I, L, names, index_set, category=None):
r"""
Initialize ``self``.
TESTS::
sage: L.<x,y,z> = LieAlgebra(SR, {('x','y'): {'x':1}})
sage: K = L.quotient(y)
sage: K.dimension()
1
sage: TestSuite(K).run()
"""
B = L.basis()
sm = L.module().submodule_with_basis(
[I.reduce(B[i]).to_vector() for i in index_set])
SB = sm.basis()
# compute and normalize structural coefficients for the quotient
s_coeff = {}
for i, ind_i in enumerate(index_set):
for j in range(i + 1, len(index_set)):
ind_j = index_set[j]
brkt = I.reduce(L.bracket(SB[i], SB[j]))
brktvec = sm.coordinate_vector(brkt.to_vector())
s_coeff[(ind_i, ind_j)] = dict(zip(index_set, brktvec))
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
self._ambient = L
self._I = I
self._sm = sm
LieAlgebraWithStructureCoefficients.__init__(self,
L.base_ring(),
s_coeff,
names,
index_set,
category=category)
# register the quotient morphism as a conversion
H = Hom(L, self)
f = SetMorphism(H, self.retract)
self.register_conversion(f)
def __init__(self, I, L, names, index_set, category=None):
r"""
Initialize ``self``.
TESTS::
sage: L.<x,y,z> = LieAlgebra(SR, {('x','y'): {'x':1}})
sage: K = L.quotient(y)
sage: K.dimension()
1
sage: TestSuite(K).run()
"""
B = L.basis()
sm = L.module().submodule_with_basis([I.reduce(B[i]).to_vector()
for i in index_set])
SB = sm.basis()
# compute and normalize structural coefficients for the quotient
s_coeff = {}
for i, ind_i in enumerate(index_set):
for j in range(i + 1, len(index_set)):
ind_j = index_set[j]
brkt = I.reduce(L.bracket(SB[i], SB[j]))
brktvec = sm.coordinate_vector(brkt.to_vector())
s_coeff[(ind_i, ind_j)] = dict(zip(index_set, brktvec))
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
self._ambient = L
self._I = I
self._sm = sm
LieAlgebraWithStructureCoefficients.__init__(
self, L.base_ring(), s_coeff, names, index_set, category=category)
# register the quotient morphism as a conversion
H = Hom(L, self)
f = SetMorphism(H, self.retract)
self.register_conversion(f)
def __classcall_private__(cls,
R,
s_coeff,
names=None,
index_set=None,
category=None,
**kwds):
"""
Normalize input to ensure a unique representation.
EXAMPLES:
If the variable order is specified, the order of structural
coefficients does not matter::
sage: from sage.algebras.lie_algebras.nilpotent_lie_algebra import NilpotentLieAlgebra_dense
sage: L1.<x,y,z> = NilpotentLieAlgebra_dense(QQ, {('x','y'): {'z': 1}})
sage: L2.<x,y,z> = NilpotentLieAlgebra_dense(QQ, {('y','x'): {'z': -1}})
sage: L1 is L2
True
If the variables are implicitly defined by the structural coefficients,
the ordering may be different and the Lie algebras will be considered
different::
sage: from sage.algebras.lie_algebras.nilpotent_lie_algebra import NilpotentLieAlgebra_dense
sage: L1 = NilpotentLieAlgebra_dense(QQ, {('x','y'): {'z': 1}})
sage: L2 = NilpotentLieAlgebra_dense(QQ, {('y','x'): {'z': -1}})
sage: L1
Nilpotent Lie algebra on 3 generators (x, y, z) over Rational Field
sage: L2
Nilpotent Lie algebra on 3 generators (y, x, z) over Rational Field
sage: L1 is L2
False
Constructed using two different methods from :class:`LieAlgebra`
yields the same Lie algebra::
sage: sc = {('X','Y'): {'Z': 1}}
sage: C = LieAlgebras(QQ).Nilpotent().FiniteDimensional().WithBasis()
sage: L1.<X,Y,Z> = LieAlgebra(QQ, sc, category=C)
sage: L2 = LieAlgebra(QQ, sc, nilpotent=True, names=['X','Y','Z'])
sage: L1 is L2
True
"""
if not names:
# extract names from structural coefficients
names = []
for (X, Y), d in s_coeff.items():
if X not in names: names.append(X)
if Y not in names: names.append(Y)
for k in d:
if k not in names: names.append(k)
from sage.structure.indexed_generators import standardize_names_index_set
names, index_set = standardize_names_index_set(names, index_set)
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
cat = LieAlgebras(R).FiniteDimensional().WithBasis().Nilpotent()
category = cat.or_subcategory(category)
return super(NilpotentLieAlgebra_dense,
cls).__classcall__(cls,
R,
s_coeff,
names,
index_set,
category=category,
**kwds)
def __init__(self, R, r, s, names, naming, category, **kwds):
r"""
Initialize ``self``
EXAMPLES::
sage: L = LieAlgebra(ZZ, 2, step=2)
sage: TestSuite(L).run()
sage: L = LieAlgebra(QQ, 4, step=3)
sage: TestSuite(L).run() # long time
"""
if r not in ZZ or r <= 0:
raise ValueError("number of generators %s is not "
"a positive integer" % r)
if s not in ZZ or s <= 0:
raise ValueError("step %s is not a positive integer" % s)
# extract an index set from the Lyndon words of the corresponding
# free Lie algebra, and store the corresponding elements in a dict
from sage.algebras.lie_algebras.lie_algebra import LieAlgebra
free_gen_names = ['F%d' % k for k in range(r)]
free_gen_names_inv = {
val: i + 1
for i, val in enumerate(free_gen_names)
}
L = LieAlgebra(R, free_gen_names).Lyndon()
basis_by_deg = {d: [] for d in range(1, s + 1)}
for d in range(1, s + 1):
for X in L.graded_basis(d):
# convert brackets of form [X_1, [X_1, X_2]] to words (1,1,2)
w = tuple(free_gen_names_inv[s]
for s in X.leading_support().to_word())
basis_by_deg[d].append((w, X))
index_set = [ind for d in basis_by_deg for ind, val in basis_by_deg[d]]
if len(names) == 1 and len(index_set) > 1:
if not naming:
if r >= 10:
naming = 'linear'
else:
naming = 'index'
if naming == 'linear':
names = [
'%s_%d' % (names[0], k + 1) for k in range(len(index_set))
]
elif naming == 'index':
if r >= 10:
raise ValueError("'index' naming scheme not supported for "
"10 or more generators")
names = [
'%s_%s' % (names[0], "".join(str(s) for s in ind))
for ind in index_set
]
else:
raise ValueError("unknown naming scheme %s" % naming)
# extract structural coefficients from the free Lie algebra
s_coeff = {}
for dx in range(1, s + 1):
# Brackets are only computed when deg(X) + deg(Y) <= s
# We also require deg(Y) >= deg(X) by the ordering
for dy in range(dx, s + 1 - dx):
if dx == dy:
for i, val in enumerate(basis_by_deg[dx]):
X_ind, X = val
for Y_ind, Y in basis_by_deg[dy][i + 1:]:
Z = L[X, Y]
if not Z.is_zero():
s_coeff[(X_ind, Y_ind)] = {
W_ind: Z[W.leading_support()]
for W_ind, W in basis_by_deg[dx + dy]
}
else:
for X_ind, X in basis_by_deg[dx]:
for Y_ind, Y in basis_by_deg[dy]:
Z = L[X, Y]
if not Z.is_zero():
s_coeff[(X_ind, Y_ind)] = {
W_ind: Z[W.leading_support()]
for W_ind, W in basis_by_deg[dx + dy]
}
names, index_set = standardize_names_index_set(names, index_set)
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
NilpotentLieAlgebra_dense.__init__(self,
R,
s_coeff,
names,
index_set,
s,
category=category,
**kwds)
def __init__(self, R, r, s, names, naming, category, **kwds):
r"""
Initialize ``self``
EXAMPLES::
sage: L = LieAlgebra(ZZ, 2, step=2)
sage: TestSuite(L).run()
sage: L = LieAlgebra(QQ, 4, step=3)
sage: TestSuite(L).run() # long time
"""
if r not in ZZ or r <= 0:
raise ValueError("number of generators %s is not "
"a positive integer" % r)
if s not in ZZ or s <= 0:
raise ValueError("step %s is not a positive integer" % s)
# extract an index set from the Lyndon words of the corresponding
# free Lie algebra, and store the corresponding elements in a dict
from sage.algebras.lie_algebras.lie_algebra import LieAlgebra
free_gen_names = ['F%d' % k for k in range(r)]
free_gen_names_inv = {val: i + 1 for i, val in enumerate(free_gen_names)}
L = LieAlgebra(R, free_gen_names).Lyndon()
basis_by_deg = {d: [] for d in range(1, s + 1)}
for d in range(1, s + 1):
for X in L.graded_basis(d):
# convert brackets of form [X_1, [X_1, X_2]] to words (1,1,2)
w = tuple(free_gen_names_inv[s]
for s in X.leading_support().to_word())
basis_by_deg[d].append((w, X))
index_set = [ind for d in basis_by_deg for ind, val in basis_by_deg[d]]
if len(names) == 1 and len(index_set) > 1:
if not naming:
if r >= 10:
naming = 'linear'
else:
naming = 'index'
if naming == 'linear':
names = ['%s_%d' % (names[0], k + 1)
for k in range(len(index_set))]
elif naming == 'index':
if r >= 10:
raise ValueError("'index' naming scheme not supported for "
"10 or more generators")
names = ['%s_%s' % (names[0], "".join(str(s) for s in w))
for w in index_set]
else:
raise ValueError("unknown naming scheme %s" % naming)
# extract structural coefficients from the free Lie algebra
s_coeff = {}
for dx in range(1, s + 1):
# Brackets are only computed when deg(X) + deg(Y) <= s
# We also require deg(Y) >= deg(X) by the ordering
for dy in range(dx, s + 1 - dx):
if dx == dy:
for i, val in enumerate(basis_by_deg[dx]):
X_ind, X = val
for Y_ind, Y in basis_by_deg[dy][i + 1:]:
Z = L[X, Y]
if not Z.is_zero():
s_coeff[(X_ind, Y_ind)] = {W_ind: Z[W.leading_support()]
for W_ind, W in basis_by_deg[dx + dy]}
else:
for X_ind, X in basis_by_deg[dx]:
for Y_ind, Y in basis_by_deg[dy]:
Z = L[X, Y]
if not Z.is_zero():
s_coeff[(X_ind, Y_ind)] = {W_ind: Z[W.leading_support()]
for W_ind, W in basis_by_deg[dx + dy]}
names, index_set = standardize_names_index_set(names, index_set)
s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
s_coeff, index_set)
NilpotentLieAlgebra_dense.__init__(self, R, s_coeff, names,
index_set, s,
category=category, **kwds)
