#18230: Basic hierarchy of categories for representations of monoids, lie
algebras,
...
-------------------------------+------------------------
Reporter: nthiery | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.7
Component: algebra | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Description changed by nthiery:
Old description:
> Draft in finite-semigroup-nt.patch in the Sage-Combinat queue.
>
> Some notes:
> {{{
> class Semigroups.TransformationModules: # or .SetsWithAction
> """modules where the action is by discrete transformations"""
>
> class ParentMethods:
> def actor(): ?
> def semigroup(): ...
> # Design goal: be left/right agnostic whenever possible
> def action(self, x, s) or def action(self, s, x)
>
> class Semigroups.Finite:
> class ParentMethods:
> G.simple_modules(QQ): shorthand for
> G.algebras(QQ).simple_modules(QQ)?
>
> class Algebras.Modules:
>
> class ParentMethods:
> @abstract_method
> def simple_modules() -> return a family indexed by below:
> def simple_modules_index_set()
>
> def projective_indecomposable_modules()
> def cartan_matrix(self):
> pass
>
> def character(self):
>
> def brauer_character(self): # modular case; do we want to
> identify them?
>
> def character_value(self, s):
> return
>
> def class_function(self):
> return self.character_value
>
> # class_functions ? class_function_module? class_function_ring?
> # cyclic_homology?
> # trace_space Cyclic homology, second edition by Loday
> def character_ring(self):
>
> class FiniteDimensional:
>
> class ParentMethods:
>
> def representation():
> Return the morphism S -> Hom(self, self)
>
> @abstract_method # in the finite dimensional case
> def representation_matrix(self, s):
>
> @abstract_method
> def isotypic_components():
> """
> Return the isotypic components of ``self``.
>
> OUTPUT: a collection of submodules the internal direct
> sum of which is ``self``.
> """
> #
> induction / restriction
>
> M.induce(LargerAlgebra) / M.induced_module(LargerAlgebra)
>
> G.simple_modules(QQ): shorthand for
> G.algebras(QQ).simple_modules(QQ)
> }}}
New description:
Draft in finite-semigroup-nt.patch in the Sage-Combinat queue.
Some notes:
{{{
class Semigroups.TransformationModules: # or .SetsWithAction
"""modules where the action is by discrete transformations"""
class ParentMethods:
def actor(): ?
def semigroup(): ...
# Design goal: be left/right agnostic whenever possible
def action(self, x, s) or def action(self, s, x)
class Semigroups.Finite:
class ParentMethods:
G.simple_modules(QQ): shorthand for
G.algebras(QQ).simple_modules(QQ)?
class Algebras.Modules:
class ParentMethods:
@abstract_method
def simple_modules() -> return a family indexed by below:
def simple_modules_index_set()
def projective_indecomposable_modules()
def cartan_matrix(self):
pass
def character(self):
def brauer_character(self): # modular case; do we want to
identify them?
def character_value(self, s):
return
def class_function(self):
return self.character_value
# class_functions ? class_function_module? class_function_ring?
# cyclic_homology?
# trace_space Cyclic homology, second edition by Loday
def character_ring(self):
class FiniteDimensional:
class ParentMethods:
def representation():
Return the morphism S -> Hom(self, self)
@abstract_method # in the finite dimensional case
def representation_matrix(self, s):
@abstract_method
def isotypic_components():
"""
Return the isotypic components of ``self``.
OUTPUT: a collection of submodules the internal direct
sum of which is ``self``.
"""
#
induction / restriction
M.induce(LargerAlgebra) / M.induced_module(LargerAlgebra)
G.simple_modules(QQ): shorthand for
G.algebras(QQ).simple_modules(QQ)
}}}
Question: how to handle the case where the base ring of the module
does not match the base ring of the acting object? Example: complex
representations of real lie algebras
{{{
LieAlgebras(QQ).Modules(CC)
LieAlgebras(QQ).ComplexModules()
}}}
--
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Ticket URL: <http://trac.sagemath.org/ticket/18230#comment:3>
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