def __init__(self, R, p):
"""
Initialize ``self``.
EXAMPLES::
sage: L = lie_algebras.pwitt(GF(5), 5); L
The 5-Witt Lie algebra over Finite Field of size 5
sage: TestSuite(L).run()
We skip the grading test as we need to be able to echelonize a
matrix over the base ring as part of the test::
sage: L = lie_algebras.pwitt(Zmod(6), 6)
sage: TestSuite(L).run(skip="_test_grading")
"""
if R(p) != 0:
raise ValueError("{} is not 0 in {}".format(p, R))
cat = LieAlgebras(R).FiniteDimensional().WithBasis().Graded()
FinitelyGeneratedLieAlgebra.__init__(self,
R,
index_set=list(range(p)),
category=cat)
IndexedGenerators.__init__(self,
list(range(p)),
prefix='d',
bracket='[')
self._p = p
def __init__(self, g, kac_moody):
"""
Initalize ``self``.
EXAMPLES::
sage: asl = lie_algebras.Affine(QQ, ['A',4,1])
sage: TestSuite(asl).run()
"""
self._g = g
self._cartan_type = g.cartan_type().affine()
R = g.base_ring()
names = list(g.variable_names()) + ['e0', 'f0', 'c']
if kac_moody:
names += ['d']
self._kac_moody = kac_moody
names = tuple(names)
self._ordered_indices = names
cat = LieAlgebras(R).WithBasis()
FinitelyGeneratedLieAlgebra.__init__(self,
R,
names,
names,
category=cat)
def __init__(self, R, p):
"""
Initialize ``self``.
EXAMPLES::
sage: L = lie_algebras.pwitt(GF(5), 5); L
The 5-Witt Lie algebra over Finite Field of size 5
sage: TestSuite(L).run()
sage: L = lie_algebras.pwitt(Zmod(6), 6)
sage: TestSuite(L).run() # not tested -- universal envelope doesn't work
sage: L._test_jacobi_identity()
"""
if R(p) != 0:
raise ValueError("{} is not 0 in {}".format(p, R))
cat = LieAlgebras(R).FiniteDimensional().WithBasis()
FinitelyGeneratedLieAlgebra.__init__(self,
R,
index_set=list(range(p)),
category=cat)
IndexedGenerators.__init__(self,
list(range(p)),
prefix='d',
bracket='[')
self._p = p
def __init__(self, R, s_coeff, names, index_set, category=None, prefix=None,
bracket=None, latex_bracket=None, string_quotes=None, **kwds):
"""
Initialize ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x,y', {('x','y'): {'x':1}})
sage: TestSuite(L).run()
"""
default = (names != tuple(index_set))
if prefix is None:
if default:
prefix = 'L'
else:
prefix = ''
if bracket is None:
bracket = default
if latex_bracket is None:
latex_bracket = default
if string_quotes is None:
string_quotes = default
#self._pos_to_index = dict(enumerate(index_set))
self._index_to_pos = {k: i for i,k in enumerate(index_set)}
if "sorting_key" not in kwds:
kwds["sorting_key"] = self._index_to_pos.__getitem__
cat = LieAlgebras(R).WithBasis().FiniteDimensional().or_subcategory(category)
FinitelyGeneratedLieAlgebra.__init__(self, R, names, index_set, cat)
IndexedGenerators.__init__(self, self._indices, prefix=prefix,
bracket=bracket, latex_bracket=latex_bracket,
string_quotes=string_quotes, **kwds)
self._M = FreeModule(R, len(index_set))
# Transform the values in the structure coefficients to elements
def to_vector(tuples):
vec = [R.zero()]*len(index_set)
for k,c in tuples:
vec[self._index_to_pos[k]] = c
vec = self._M(vec)
vec.set_immutable()
return vec
self._s_coeff = {(self._index_to_pos[k[0]], self._index_to_pos[k[1]]):
to_vector(s_coeff[k])
for k in s_coeff.keys()}
def __init__(self, R, s_coeff, names, index_set, category=None, prefix=None,
bracket=None, latex_bracket=None, string_quotes=None, **kwds):
"""
Initialize ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x,y', {('x','y'): {'x':1}})
sage: TestSuite(L).run()
"""
default = (names != tuple(index_set))
if prefix is None:
if default:
prefix = 'L'
else:
prefix = ''
if bracket is None:
bracket = default
if latex_bracket is None:
latex_bracket = default
if string_quotes is None:
string_quotes = default
#self._pos_to_index = dict(enumerate(index_set))
self._index_to_pos = {k: i for i,k in enumerate(index_set)}
if "sorting_key" not in kwds:
kwds["sorting_key"] = self._index_to_pos.__getitem__
cat = LieAlgebras(R).WithBasis().FiniteDimensional().or_subcategory(category)
FinitelyGeneratedLieAlgebra.__init__(self, R, names, index_set, cat)
IndexedGenerators.__init__(self, self._indices, prefix=prefix,
bracket=bracket, latex_bracket=latex_bracket,
string_quotes=string_quotes, **kwds)
self._M = FreeModule(R, len(index_set))
# Transform the values in the structure coefficients to elements
def to_vector(tuples):
vec = [R.zero()]*len(index_set)
for k,c in tuples:
vec[self._index_to_pos[k]] = c
vec = self._M(vec)
vec.set_immutable()
return vec
self._s_coeff = {(self._index_to_pos[k[0]], self._index_to_pos[k[1]]):
to_vector(s_coeff[k])
for k in s_coeff.keys()}
def __init__(self, R, p):
"""
Initialize ``self``.
EXAMPLES::
sage: L = lie_algebras.pwitt(GF(5), 5); L
The 5-Witt Lie algebra over Finite Field of size 5
sage: TestSuite(L).run()
sage: L = lie_algebras.pwitt(Zmod(6), 6)
sage: TestSuite(L).run() # not tested -- universal envelope doesn't work
sage: L._test_jacobi_identity()
"""
cat = LieAlgebras(R).FiniteDimensional().WithBasis()
FinitelyGeneratedLieAlgebra.__init__(self, R, index_set=range(p), category=cat)
IndexedGenerators.__init__(self, range(p), prefix='d', bracket='[')
self._p = p
def __init__(self, lie, basis_name):
"""
Initialize ``self``.
EXAMPLES::
sage: L.<x, y> = LieAlgebra(QQ)
sage: 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.
Free Lie algebra generated by (x, y) over Rational Field in the Hall basis
"""
self._basis_name = basis_name
IndexedGenerators.__init__(self, lie._indices, prefix='', bracket=False)
FinitelyGeneratedLieAlgebra.__init__(self, lie.base_ring(),
names=lie._names, index_set=lie._indices,
category=FreeLieAlgebraBases(lie))
def __init__(self, lie, basis_name):
"""
Initialize ``self``.
EXAMPLES::
sage: L.<x, y> = LieAlgebra(QQ)
sage: 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.
Free Lie algebra generated by (x, y) over Rational Field in the Hall basis
"""
self._basis_name = basis_name
IndexedGenerators.__init__(self, lie._indices, prefix='', bracket=False)
FinitelyGeneratedLieAlgebra.__init__(self, lie.base_ring(),
names=lie._names, index_set=lie._indices,
category=FreeLieAlgebraBases(lie))
def __init__(self, g, kac_moody):
"""
Initalize ``self``.
EXAMPLES::
sage: asl = lie_algebras.Affine(QQ, ['A',4,1])
sage: TestSuite(asl).run()
"""
self._g = g
self._cartan_type = g.cartan_type().affine()
R = g.base_ring()
names = list(g.variable_names()) + ['e0', 'f0', 'c']
if kac_moody:
names += ['d']
self._kac_moody = kac_moody
names = tuple(names)
self._ordered_indices = names
cat = LieAlgebras(R).WithBasis()
FinitelyGeneratedLieAlgebra.__init__(self, R, names, names, category=cat)