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tensor_free_module.py
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tensor_free_module.py
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r"""
Tensor products of free modules
The class :class:`TensorFreeModule` implements tensor products of the type
.. MATH::
T^{(k,l)}(M) = \underbrace{M\otimes\cdots\otimes M}_{k\ \; \mbox{times}}
\otimes \underbrace{M^*\otimes\cdots\otimes M^*}_{l\ \; \mbox{times}},
where `M` is a free module of finite rank over a commutative ring `R` and
`M^*=\mathrm{Hom}_R(M,R)` is the dual of `M`.
Note that `T^{(1,0)}(M) = M` and `T^{(0,1)}(M) = M^*`.
Thanks to the canonical isomorphism `M^{**} \simeq M` (which holds since `M`
is a free module of finite rank), `T^{(k,l)}(M)` can be identified with the
set of tensors of type `(k,l)` defined as multilinear maps
.. MATH::
\underbrace{M^*\times\cdots\times M^*}_{k\ \; \mbox{times}}
\times \underbrace{M\times\cdots\times M}_{l\ \; \mbox{times}}
\longrightarrow R
Accordingly, :class:`TensorFreeModule` is a Sage *parent* class, whose
*element* class is
:class:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor`.
`T^{(k,l)}(M)` is itself a free module over `R`, of rank `n^{k+l}`, `n`
being the rank of `M`. Accordingly the class :class:`TensorFreeModule`
inherits from the class
:class:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule`.
.. TODO::
implement more general tensor products, i.e. tensor product of the type
`M_1\otimes\cdots\otimes M_n`, where the `M_i`'s are `n` free modules of
finite rank over the same ring `R`.
AUTHORS:
- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
REFERENCES:
- \K. Conrad: *Tensor products* [Con2015]_
- Chap. 21 (Exer. 4) of R. Godement: *Algebra* [God1968]_
- Chap. 16 of S. Lang: *Algebra* [Lan2002]_
"""
#******************************************************************************
# Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr>
# Copyright (C) 2015 Michal Bejger <bejger@camk.edu.pl>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#******************************************************************************
from sage.categories.modules import Modules
from sage.misc.cachefunc import cached_method
from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
from sage.tensor.modules.free_module_tensor import FreeModuleTensor
from sage.tensor.modules.alternating_contr_tensor import AlternatingContrTensor
from sage.tensor.modules.free_module_alt_form import FreeModuleAltForm
from sage.tensor.modules.free_module_morphism import \
FiniteRankFreeModuleMorphism
from sage.tensor.modules.free_module_automorphism import FreeModuleAutomorphism
from .tensor_free_submodule_basis import TensorFreeSubmoduleBasis_sym
class TensorFreeModule(FiniteRankFreeModule_abstract):
r"""
Class for the free modules over a commutative ring `R` that are
tensor products of a given free module `M` over `R` with itself and its
dual `M^*`:
.. MATH::
T^{(k,l)}(M) = \underbrace{M\otimes\cdots\otimes M}_{k\ \; \mbox{times}}
\otimes \underbrace{M^*\otimes\cdots\otimes M^*}_{l\ \; \mbox{times}}
As recalled above, `T^{(k,l)}(M)` can be canonically identified with the
set of tensors of type `(k,l)` on `M`.
This is a Sage *parent* class, whose *element* class is
:class:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor`.
INPUT:
- ``fmodule`` -- free module `M` of finite rank over a commutative ring
`R`, as an instance of
:class:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule`
- ``tensor_type`` -- pair ``(k, l)`` with ``k`` being the contravariant
rank and ``l`` the covariant rank
- ``name`` -- (default: ``None``) string; name given to the tensor module
- ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote the
tensor module; if none is provided, it is set to ``name``
EXAMPLES:
Set of tensors of type `(1,2)` on a free `\ZZ`-module of rank 3::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: from sage.tensor.modules.tensor_free_module import TensorFreeModule
sage: T = TensorFreeModule(M, (1,2)) ; T
Free module of type-(1,2) tensors on the
Rank-3 free module M over the Integer Ring
Instead of importing TensorFreeModule in the global name space, it is
recommended to use the module's method
:meth:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule.tensor_module`::
sage: T = M.tensor_module(1,2) ; T
Free module of type-(1,2) tensors on the
Rank-3 free module M over the Integer Ring
sage: latex(T)
T^{(1, 2)}\left(M\right)
The module ``M`` itself is considered as the set of tensors of
type `(1,0)`::
sage: M is M.tensor_module(1,0)
True
``T`` is a module (actually a free module) over `\ZZ`::
sage: T.category()
Category of tensor products of finite dimensional modules over Integer Ring
sage: T in Modules(ZZ)
True
sage: T.rank()
27
sage: T.base_ring()
Integer Ring
sage: T.base_module()
Rank-3 free module M over the Integer Ring
``T`` is a *parent* object, whose elements are instances of
:class:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor`::
sage: t = T.an_element() ; t
Type-(1,2) tensor on the Rank-3 free module M over the Integer Ring
sage: from sage.tensor.modules.free_module_tensor import FreeModuleTensor
sage: isinstance(t, FreeModuleTensor)
True
sage: t in T
True
sage: T.is_parent_of(t)
True
Elements can be constructed from ``T``. In particular, 0 yields
the zero element of ``T``::
sage: T(0)
Type-(1,2) tensor zero on the Rank-3 free module M over the Integer Ring
sage: T(0) is T.zero()
True
while non-zero elements are constructed by providing their components in
a given basis::
sage: e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: comp = [[[i-j+k for k in range(3)] for j in range(3)] for i in range(3)]
sage: t = T(comp, basis=e, name='t') ; t
Type-(1,2) tensor t on the Rank-3 free module M over the Integer Ring
sage: t.comp(e)[:]
[[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]],
[[1, 2, 3], [0, 1, 2], [-1, 0, 1]],
[[2, 3, 4], [1, 2, 3], [0, 1, 2]]]
sage: t.display(e)
t = e_0⊗e^0⊗e^1 + 2 e_0⊗e^0⊗e^2 - e_0⊗e^1⊗e^0 + e_0⊗e^1⊗e^2
- 2 e_0⊗e^2⊗e^0 - e_0⊗e^2⊗e^1 + e_1⊗e^0⊗e^0 + 2 e_1⊗e^0⊗e^1
+ 3 e_1⊗e^0⊗e^2 + e_1⊗e^1⊗e^1 + 2 e_1⊗e^1⊗e^2 - e_1⊗e^2⊗e^0
+ e_1⊗e^2⊗e^2 + 2 e_2⊗e^0⊗e^0 + 3 e_2⊗e^0⊗e^1 + 4 e_2⊗e^0⊗e^2
+ e_2⊗e^1⊗e^0 + 2 e_2⊗e^1⊗e^1 + 3 e_2⊗e^1⊗e^2 + e_2⊗e^2⊗e^1
+ 2 e_2⊗e^2⊗e^2
An alternative is to construct the tensor from an empty list of components
and to set the nonzero components afterwards::
sage: t = T([], name='t')
sage: t.set_comp(e)[0,1,1] = -3
sage: t.set_comp(e)[2,0,1] = 4
sage: t.display(e)
t = -3 e_0⊗e^1⊗e^1 + 4 e_2⊗e^0⊗e^1
See the documentation of
:class:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor`
for the full list of arguments that can be provided to the __call__
operator. For instance, to construct a tensor symmetric with respect to the
last two indices::
sage: t = T([], name='t', sym=(1,2))
sage: t.set_comp(e)[0,1,1] = -3
sage: t.set_comp(e)[2,0,1] = 4
sage: t.display(e) # notice that t^2_{10} has be set equal to t^2_{01} by symmetry
t = -3 e_0⊗e^1⊗e^1 + 4 e_2⊗e^0⊗e^1 + 4 e_2⊗e^1⊗e^0
The tensor modules over a given module `M` are unique::
sage: T is M.tensor_module(1,2)
True
There is a coercion map from `\Lambda^p(M^*)`, the set of alternating
forms of degree `p`, to `T^{(0,p)}(M)`::
sage: L2 = M.dual_exterior_power(2) ; L2
2nd exterior power of the dual of the Rank-3 free module M over the
Integer Ring
sage: T02 = M.tensor_module(0,2) ; T02
Free module of type-(0,2) tensors on the Rank-3 free module M over the
Integer Ring
sage: T02.has_coerce_map_from(L2)
True
Of course, for `p\geq 2`, there is no coercion in the reverse direction,
since not every tensor of type `(0,p)` is alternating::
sage: L2.has_coerce_map_from(T02)
False
The coercion map `\Lambda^2(M^*)\rightarrow T^{(0,2)}(M)` in action::
sage: a = M.alternating_form(2, name='a') ; a
Alternating form a of degree 2 on the Rank-3 free module M over the
Integer Ring
sage: a[0,1], a[1,2] = 4, -3
sage: a.display(e)
a = 4 e^0∧e^1 - 3 e^1∧e^2
sage: a.parent() is L2
True
sage: ta = T02(a) ; ta
Type-(0,2) tensor a on the Rank-3 free module M over the Integer Ring
sage: ta.display(e)
a = 4 e^0⊗e^1 - 4 e^1⊗e^0 - 3 e^1⊗e^2 + 3 e^2⊗e^1
sage: ta.symmetries() # the antisymmetry is of course preserved
no symmetry; antisymmetry: (0, 1)
For the degree `p=1`, we have the identity `\Lambda^1(M^*) = T^{(0,1)}(M) = M^*`::
sage: M.dual_exterior_power(1) is M.tensor_module(0,1)
True
sage: M.tensor_module(0,1) is M.dual()
True
There is a canonical identification between tensors of type `(1,1)` and
endomorphisms of module `M`. Accordingly, coercion maps have been
implemented between `T^{(1,1)}(M)` and `\mathrm{End}(M)` (the module of
all endomorphisms of `M`, see
:class:`~sage.tensor.modules.free_module_homset.FreeModuleHomset`)::
sage: T11 = M.tensor_module(1,1) ; T11
Free module of type-(1,1) tensors on the Rank-3 free module M over the
Integer Ring
sage: End(M)
Set of Morphisms from Rank-3 free module M over the Integer Ring
to Rank-3 free module M over the Integer Ring
in Category of finite dimensional modules over Integer Ring
sage: T11.has_coerce_map_from(End(M))
True
sage: End(M).has_coerce_map_from(T11)
True
The coercion map `\mathrm{End}(M)\rightarrow T^{(1,1)}(M)` in action::
sage: phi = End(M).an_element() ; phi
Generic endomorphism of Rank-3 free module M over the Integer Ring
sage: phi.matrix(e)
[1 1 1]
[1 1 1]
[1 1 1]
sage: tphi = T11(phi) ; tphi # image of phi by the coercion map
Type-(1,1) tensor on the Rank-3 free module M over the Integer Ring
sage: tphi[:]
[1 1 1]
[1 1 1]
[1 1 1]
sage: t = M.tensor((1,1))
sage: t[0,0], t[1,1], t[2,2] = -1,-2,-3
sage: t[:]
[-1 0 0]
[ 0 -2 0]
[ 0 0 -3]
sage: s = t + phi ; s # phi is coerced to a type-(1,1) tensor prior to the addition
Type-(1,1) tensor on the Rank-3 free module M over the Integer Ring
sage: s[:]
[ 0 1 1]
[ 1 -1 1]
[ 1 1 -2]
The coercion map `T^{(1,1)}(M) \rightarrow \mathrm{End}(M)` in action::
sage: phi1 = End(M)(tphi) ; phi1
Generic endomorphism of Rank-3 free module M over the Integer Ring
sage: phi1 == phi
True
sage: s = phi + t ; s # t is coerced to an endomorphism prior to the addition
Generic endomorphism of Rank-3 free module M over the Integer Ring
sage: s.matrix(e)
[ 0 1 1]
[ 1 -1 1]
[ 1 1 -2]
There is a coercion `\mathrm{GL}(M)\rightarrow T^{(1,1)}(M)`, i.e. from
automorphisms of `M` to type-`(1,1)` tensors on `M`::
sage: GL = M.general_linear_group() ; GL
General linear group of the Rank-3 free module M over the Integer Ring
sage: T11.has_coerce_map_from(GL)
True
The coercion map `\mathrm{GL}(M)\rightarrow T^{(1,1)}(M)` in action::
sage: a = GL.an_element() ; a
Automorphism of the Rank-3 free module M over the Integer Ring
sage: a.matrix(e)
[ 1 0 0]
[ 0 -1 0]
[ 0 0 1]
sage: ta = T11(a) ; ta
Type-(1,1) tensor on the Rank-3 free module M over the Integer Ring
sage: ta.display(e)
e_0⊗e^0 - e_1⊗e^1 + e_2⊗e^2
sage: a.display(e)
e_0⊗e^0 - e_1⊗e^1 + e_2⊗e^2
Of course, there is no coercion in the reverse direction, since not
every type-`(1,1)` tensor is invertible::
sage: GL.has_coerce_map_from(T11)
False
"""
Element = FreeModuleTensor
def __init__(self, fmodule, tensor_type, name=None, latex_name=None, category=None):
r"""
TESTS::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(2, 3)
sage: TestSuite(T).run()
"""
self._fmodule = fmodule
self._tensor_type = tuple(tensor_type)
ring = fmodule._ring
rank = pow(fmodule._rank, tensor_type[0] + tensor_type[1])
if self._tensor_type == (0,1): # case of the dual
category = Modules(ring).FiniteDimensional().or_subcategory(category)
if name is None and fmodule._name is not None:
name = fmodule._name + '*'
if latex_name is None and fmodule._latex_name is not None:
latex_name = fmodule._latex_name + r'^*'
else:
category = Modules(ring).FiniteDimensional().TensorProducts().or_subcategory(category)
if name is None and fmodule._name is not None:
name = 'T^' + str(self._tensor_type) + '(' + fmodule._name + \
')'
if latex_name is None and fmodule._latex_name is not None:
latex_name = r'T^{' + str(self._tensor_type) + r'}\left(' + \
fmodule._latex_name + r'\right)'
super().__init__(fmodule._ring, rank, name=name, latex_name=latex_name, category=category)
fmodule._all_modules.add(self)
def tensor_factors(self):
r"""
Return the tensor factors of this tensor module.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(2, 3)
sage: T.tensor_factors()
[Rank-3 free module M over the Integer Ring,
Rank-3 free module M over the Integer Ring,
Dual of the Rank-3 free module M over the Integer Ring,
Dual of the Rank-3 free module M over the Integer Ring,
Dual of the Rank-3 free module M over the Integer Ring]
"""
tensor_type = self.tensor_type()
if tensor_type == (0,1): # case of the dual
raise NotImplementedError
bmodule = self.base_module()
factors = [bmodule] * tensor_type[0]
dmodule = bmodule.dual()
if tensor_type[1]:
factors += [dmodule] * tensor_type[1]
return factors
#### Parent Methods
def _element_constructor_(self, comp=[], basis=None, name=None,
latex_name=None, sym=None, antisym=None):
r"""
Construct a tensor.
EXAMPLES::
sage: M = FiniteRankFreeModule(QQ, 2, name='M')
sage: T = M.tensor_module(1,1)
sage: T._element_constructor_(0) is T.zero()
True
sage: e = M.basis('e')
sage: t = T._element_constructor_(comp=[[2,0],[1/2,-3]], basis=e,
....: name='t') ; t
Type-(1,1) tensor t on the 2-dimensional vector space M over the
Rational Field
sage: t.display()
t = 2 e_0⊗e^0 + 1/2 e_1⊗e^0 - 3 e_1⊗e^1
sage: t.parent()
Free module of type-(1,1) tensors on the 2-dimensional vector
space M over the Rational Field
sage: t.parent() is T
True
"""
from sage.rings.integer import Integer
if isinstance(comp, (int, Integer)) and comp == 0:
return self.zero()
if isinstance(comp, FiniteRankFreeModuleMorphism):
# coercion of an endomorphism to a type-(1,1) tensor:
endo = comp # for readability
if self._tensor_type == (1,1) and endo.is_endomorphism() and \
self._fmodule is endo.domain():
resu = self.element_class(self._fmodule, (1,1),
name=endo._name,
latex_name=endo._latex_name,
parent=self)
for basis, mat in endo._matrices.items():
resu.add_comp(basis[0])[:] = mat
else:
raise TypeError("cannot coerce the {}".format(endo) +
" to an element of {}".format(self))
elif isinstance(comp, AlternatingContrTensor):
# coercion of an alternating contravariant tensor of degree
# p to a type-(p,0) tensor:
tensor = comp # for readability
p = tensor.degree()
if self._tensor_type != (p,0) or \
self._fmodule != tensor.base_module():
raise TypeError("cannot coerce the {}".format(tensor) +
" to an element of {}".format(self))
if p == 1:
asym = None
else:
asym = range(p)
resu = self.element_class(self._fmodule, (p,0),
name=tensor._name,
latex_name=tensor._latex_name,
antisym=asym,
parent=self)
for basis, comp in tensor._components.items():
resu._components[basis] = comp.copy()
elif isinstance(comp, FreeModuleAltForm):
# coercion of an alternating form to a type-(0,p) tensor:
form = comp # for readability
p = form.degree()
if self._tensor_type != (0,p) or \
self._fmodule != form.base_module():
raise TypeError("cannot coerce the {}".format(form) +
" to an element of {}".format(self))
if p == 1:
asym = None
else:
asym = range(p)
resu = self.element_class(self._fmodule, (0,p), name=form._name,
latex_name=form._latex_name,
antisym=asym,
parent=self)
for basis, comp in form._components.items():
resu._components[basis] = comp.copy()
elif isinstance(comp, FreeModuleAutomorphism):
# coercion of an automorphism to a type-(1,1) tensor:
autom = comp # for readability
if self._tensor_type != (1,1) or \
self._fmodule != autom.base_module():
raise TypeError("cannot coerce the {}".format(autom) +
" to an element of {}".format(self))
resu = self.element_class(self._fmodule, (1,1), name=autom._name,
latex_name=autom._latex_name,
parent=self)
for basis, comp in autom._components.items():
resu._components[basis] = comp.copy()
elif isinstance(comp, FreeModuleTensor):
tensor = comp
if self._tensor_type != tensor._tensor_type or \
self._fmodule != tensor.base_module():
raise TypeError("cannot coerce the {}".format(tensor) +
" to an element of {}".format(self))
resu = self.element_class(self._fmodule, self._tensor_type,
name=name, latex_name=latex_name,
sym=sym, antisym=antisym,
parent=self)
for basis, comp in tensor._components.items():
resu._components[basis] = comp.copy()
else:
# Standard construction:
resu = self.element_class(self._fmodule, self._tensor_type,
name=name, latex_name=latex_name,
sym=sym, antisym=antisym, parent=self)
if comp:
resu.set_comp(basis)[:] = comp
return resu
@cached_method
def zero(self):
r"""
Return the zero of ``self``.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: T11 = M.tensor_module(1,1)
sage: T11.zero()
Type-(1,1) tensor zero on the Rank-3 free module M over the Integer
Ring
The zero element is cached::
sage: T11.zero() is T11(0)
True
"""
resu = self._element_constructor_(name='zero', latex_name='0')
for basis in self._fmodule._known_bases:
resu._add_comp_unsafe(basis)
# (since new components are initialized to zero)
resu._is_zero = True # This element is certainly zero
resu.set_immutable()
return resu
def _an_element_(self):
r"""
Construct some (unnamed) element of ``self``.
EXAMPLES::
sage: M = FiniteRankFreeModule(QQ, 2, name='M')
sage: T = M.tensor_module(1,1)
sage: t = T._an_element_() ; t
Type-(1,1) tensor on the 2-dimensional vector space M over the
Rational Field
sage: t.display()
1/2 e_0⊗e^0
sage: t.parent() is T
True
sage: M.tensor_module(2,3)._an_element_().display()
1/2 e_0⊗e_0⊗e^0⊗e^0⊗e^0
TESTS::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: T60M = M.tensor_module(6, 0)
sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
sage: t = Sym0123x45M._an_element_()
sage: t.parent() is Sym0123x45M
True
"""
resu = self([])
# Make sure that the base module has a default basis
self._fmodule.an_element()
sindex = self._fmodule._sindex
ind = [sindex for i in range(resu._tensor_rank)]
resu.set_comp()[ind] = self._fmodule._ring.an_element()
return resu
def _coerce_map_from_(self, other):
r"""
Determine whether coercion to ``self`` exists from other parent.
EXAMPLES:
Sets of module endomorphisms coerce to type-`(1,1)` tensor modules::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: M.tensor_module(1,1)._coerce_map_from_(End(M))
True
but not to tensor modules of other types::
sage: M.tensor_module(0,2)._coerce_map_from_(End(M))
False
and not to type-`(1,1)` tensor modules defined on another free module::
sage: N = FiniteRankFreeModule(ZZ, 3, name='N')
sage: f = N.basis('f')
sage: M.tensor_module(1,1)._coerce_map_from_(End(N))
False
There is no coercion if the module morphisms are not endomorphisms::
sage: M.tensor_module(1,1)._coerce_map_from_(Hom(M,N))
False
Coercion from alternating contravariant tensors::
sage: M.tensor_module(2,0)._coerce_map_from_(M.exterior_power(2))
True
sage: M.tensor_module(2,0)._coerce_map_from_(M.exterior_power(3))
False
sage: M.tensor_module(2,0)._coerce_map_from_(N.exterior_power(2))
False
Coercion from alternating forms::
sage: M.tensor_module(0,2)._coerce_map_from_(M.dual_exterior_power(2))
True
sage: M.tensor_module(0,2)._coerce_map_from_(M.dual_exterior_power(3))
False
sage: M.tensor_module(0,2)._coerce_map_from_(N.dual_exterior_power(2))
False
Coercion from submodules::
sage: Sym01M = M.tensor_module(2, 0, sym=((0, 1)))
sage: M.tensor_module(2,0)._coerce_map_from_(Sym01M)
True
"""
from .free_module_homset import FreeModuleHomset
from .ext_pow_free_module import (ExtPowerFreeModule,
ExtPowerDualFreeModule)
from .free_module_linear_group import FreeModuleLinearGroup
if isinstance(other, FreeModuleHomset):
# Coercion of an endomorphism to a type-(1,1) tensor:
if self._tensor_type == (1,1):
return other.is_endomorphism_set() and \
self._fmodule is other.domain()
else:
return False
if isinstance(other, ExtPowerFreeModule):
# Coercion of an alternating contravariant tensor to a
# type-(p,0) tensor:
return self._tensor_type == (other.degree(), 0) and \
self._fmodule is other.base_module()
if isinstance(other, ExtPowerDualFreeModule):
# Coercion of an alternating form to a type-(0,p) tensor:
return self._tensor_type == (0, other.degree()) and \
self._fmodule is other.base_module()
if isinstance(other, FreeModuleLinearGroup):
# Coercion of an automorphism to a type-(1,1) tensor:
return self._tensor_type == (1,1) and \
self._fmodule is other.base_module()
try:
if other.is_submodule(self):
return True
except AttributeError:
pass
return False
#### End of parent methods
def _repr_(self):
r"""
Return a string representation of ``self``.
EXAMPLES::
sage: M = FiniteRankFreeModule(QQ, 2, name='M')
sage: M.tensor_module(1,1)
Free module of type-(1,1) tensors on the 2-dimensional vector space
M over the Rational Field
"""
description = "Free module of type-({},{}) tensors on the {}".format(
self._tensor_type[0], self._tensor_type[1], self._fmodule)
return description
def base_module(self):
r"""
Return the free module on which ``self`` is constructed.
OUTPUT:
- instance of :class:`FiniteRankFreeModule` representing the free
module on which the tensor module is defined.
EXAMPLES:
Base module of a type-`(1,2)` tensor module::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(1,2)
sage: T.base_module()
Rank-3 free module M over the Integer Ring
sage: T.base_module() is M
True
"""
return self._fmodule
def tensor_type(self):
r"""
Return the tensor type of ``self``.
OUTPUT:
- pair `(k,l)` such that ``self`` is the module tensor product
`T^{(k,l)}(M)`
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3)
sage: T = M.tensor_module(1,2)
sage: T.tensor_type()
(1, 2)
"""
return self._tensor_type
@cached_method
def basis(self, symbol, latex_symbol=None, from_family=None,
indices=None, latex_indices=None, symbol_dual=None,
latex_symbol_dual=None):
r"""
Return the standard basis of ``self`` corresponding to a basis of the base module.
INPUT:
- ``symbol``, ``indices`` -- passed to the base module's method
:meth:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule.basis`
to select a basis of the :meth:`base_module` of ``self``,
or to create it.
- other parameters -- passed to
:meth:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule.basis`; when
the basis does not exist yet, it will be created using these parameters.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(1,1)
sage: e_T = T.basis('e'); e_T
Standard basis on the
Free module of type-(1,1) tensors on the Rank-3 free module M over the Integer Ring
induced by Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: for a in e_T: a.display()
e_0⊗e^0
e_0⊗e^1
e_0⊗e^2
e_1⊗e^0
e_1⊗e^1
e_1⊗e^2
e_2⊗e^0
e_2⊗e^1
e_2⊗e^2
sage: Sym2M = M.tensor_module(2, 0, sym=range(2))
sage: e_Sym2M = Sym2M.basis('e'); e_Sym2M
Standard basis on the
Free module of fully symmetric type-(2,0) tensors on the Rank-3 free module M over the Integer Ring
induced by Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: for a in e_Sym2M: a.display()
e_0⊗e_0
e_0⊗e_1 + e_1⊗e_0
e_0⊗e_2 + e_2⊗e_0
e_1⊗e_1
e_1⊗e_2 + e_2⊗e_1
e_2⊗e_2
sage: M = FiniteRankFreeModule(ZZ, 2)
sage: e = M.basis('e')
sage: f = M.basis('f', from_family=(-e[1], e[0]))
sage: for b in f: b.display()
f_0 = -e_1
f_1 = e_0
sage: S = M.tensor_module(2, 0, sym=(0,1))
sage: fS = S.basis('f')
sage: for b in fS: b.display()
e_1⊗e_1
-e_0⊗e_1 - e_1⊗e_0
e_0⊗e_0
sage: for b in fS: b.display(f)
f_0⊗f_0
f_0⊗f_1 + f_1⊗f_0
f_1⊗f_1
"""
return TensorFreeSubmoduleBasis_sym(self, symbol=symbol, latex_symbol=latex_symbol,
indices=indices, latex_indices=latex_indices,
symbol_dual=symbol_dual, latex_symbol_dual=latex_symbol_dual)
@cached_method
def _basis_sym(self):
r"""
Return an instance of :class:`~sage.tensor.modules.comp.Components`.
This implementation returns an instance without symmetry.
The subclass :class:`~sage.tensor.modules.tensor_free_submodule.TensorFreeSubmodule_sym`
overrides this method to encode the prescribed symmetry of the submodule.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(1,1)
sage: c = T._basis_sym(); c
2-indices components w.r.t. (0, 1, 2)
"""
frame = tuple(self.base_module().irange())
tensor = self.ambient()()
return tensor._new_comp(frame)