Group, ring, etc. actions on objects#
The terminology and notation used is suggestive of groups acting on sets, but this framework can be used for modules, algebras, etc.
A group action \(G \times S \rightarrow S\) is a functor from \(G\) to Sets.
Warning
An Action
object only keeps a weak reference to the underlying set
which is acted upon. This decision was made in Issue #715 in order to
allow garbage collection within the coercion framework (this is where
actions are mainly used) and avoid memory leaks.
sage: from sage.categories.action import Action
sage: class P: pass
sage: A = Action(P(),P())
sage: import gc
sage: _ = gc.collect()
sage: A
<repr(<sage.categories.action.Action at 0x...>) failed:
RuntimeError: This action acted on a set that became garbage collected>
>>> from sage.all import *
>>> from sage.categories.action import Action
>>> class P: pass
>>> A = Action(P(),P())
>>> import gc
>>> _ = gc.collect()
>>> A
<repr(<sage.categories.action.Action at 0x...>) failed:
RuntimeError: This action acted on a set that became garbage collected>
To avoid garbage collection of the underlying set, it is sufficient to create a strong reference to it before the action is created.
sage: _ = gc.collect()
sage: from sage.categories.action import Action
sage: class P: pass
sage: q = P()
sage: A = Action(P(),q)
sage: gc.collect()
0
sage: A
Left action by <__main__.P ... at ...> on <__main__.P ... at ...>
>>> from sage.all import *
>>> _ = gc.collect()
>>> from sage.categories.action import Action
>>> class P: pass
>>> q = P()
>>> A = Action(P(),q)
>>> gc.collect()
0
>>> A
Left action by <__main__.P ... at ...> on <__main__.P ... at ...>
AUTHOR:
Robert Bradshaw: initial version
- class sage.categories.action.Action[source]#
Bases:
Functor
The action of
G
onS
.INPUT:
G
– a parent or Python typeS
– a parent or Python typeis_left
– (boolean, default:True
) whether elements ofG
are on the leftop
– (default:None
) operation. This is not used byAction
itself, but other classes may use it
- act(g, x)[source]#
This is a consistent interface for acting on
x
byg
, regardless of whether it’s a left or right action.If needed,
g
andx
are converted to the correct parent.EXAMPLES:
sage: R.<x> = ZZ [] sage: from sage.structure.coerce_actions import IntegerMulAction sage: A = IntegerMulAction(ZZ, R, True) # Left action sage: A.act(5, x) 5*x sage: A.act(int(5), x) 5*x sage: A = IntegerMulAction(ZZ, R, False) # Right action sage: A.act(5, x) 5*x sage: A.act(int(5), x) 5*x
>>> from sage.all import * >>> R = ZZ ['x']; (x,) = R._first_ngens(1) >>> from sage.structure.coerce_actions import IntegerMulAction >>> A = IntegerMulAction(ZZ, R, True) # Left action >>> A.act(Integer(5), x) 5*x >>> A.act(int(Integer(5)), x) 5*x >>> A = IntegerMulAction(ZZ, R, False) # Right action >>> A.act(Integer(5), x) 5*x >>> A.act(int(Integer(5)), x) 5*x
- class sage.categories.action.ActionEndomorphism[source]#
Bases:
Morphism
The endomorphism defined by the action of one element.
EXAMPLES:
sage: A = ZZ['x'].get_action(QQ, self_on_left=False, op=operator.mul) sage: A Left scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Integer Ring sage: A(1/2) Action of 1/2 on Univariate Polynomial Ring in x over Integer Ring under Left scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Integer Ring.
>>> from sage.all import * >>> A = ZZ['x'].get_action(QQ, self_on_left=False, op=operator.mul) >>> A Left scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Integer Ring >>> A(Integer(1)/Integer(2)) Action of 1/2 on Univariate Polynomial Ring in x over Integer Ring under Left scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Integer Ring.
- class sage.categories.action.InverseAction[source]#
Bases:
Action
An action that acts as the inverse of the given action.
EXAMPLES:
sage: V = QQ^3 # needs sage.modules sage: v = V((1, 2, 3)) # needs sage.modules sage: cm = get_coercion_model() sage: # needs sage.modules sage: a = cm.get_action(V, QQ, operator.mul) sage: a Right scalar multiplication by Rational Field on Vector space of dimension 3 over Rational Field sage: ~a Right inverse action by Rational Field on Vector space of dimension 3 over Rational Field sage: (~a)(v, 1/3) (3, 6, 9) sage: # needs sage.modules sage: b = cm.get_action(QQ, V, operator.mul) sage: b Left scalar multiplication by Rational Field on Vector space of dimension 3 over Rational Field sage: ~b Left inverse action by Rational Field on Vector space of dimension 3 over Rational Field sage: (~b)(1/3, v) (3, 6, 9) sage: c = cm.get_action(ZZ, list, operator.mul) sage: c Left action by Integer Ring on <... 'list'> sage: ~c Traceback (most recent call last): ... TypeError: no inverse defined for Left action by Integer Ring on <... 'list'>
>>> from sage.all import * >>> V = QQ**Integer(3) # needs sage.modules >>> v = V((Integer(1), Integer(2), Integer(3))) # needs sage.modules >>> cm = get_coercion_model() >>> # needs sage.modules >>> a = cm.get_action(V, QQ, operator.mul) >>> a Right scalar multiplication by Rational Field on Vector space of dimension 3 over Rational Field >>> ~a Right inverse action by Rational Field on Vector space of dimension 3 over Rational Field >>> (~a)(v, Integer(1)/Integer(3)) (3, 6, 9) >>> # needs sage.modules >>> b = cm.get_action(QQ, V, operator.mul) >>> b Left scalar multiplication by Rational Field on Vector space of dimension 3 over Rational Field >>> ~b Left inverse action by Rational Field on Vector space of dimension 3 over Rational Field >>> (~b)(Integer(1)/Integer(3), v) (3, 6, 9) >>> c = cm.get_action(ZZ, list, operator.mul) >>> c Left action by Integer Ring on <... 'list'> >>> ~c Traceback (most recent call last): ... TypeError: no inverse defined for Left action by Integer Ring on <... 'list'>
- class sage.categories.action.PrecomposedAction[source]#
Bases:
Action
A precomposed action first applies given maps, and then applying an action to the return values of the maps.
EXAMPLES:
We demonstrate that an example discussed on Issue #14711 did not become a problem:
sage: # needs sage.libs.flint sage.modular sage: E = ModularSymbols(11).2 sage: s = E.modular_symbol_rep() sage: del E,s sage: import gc sage: _ = gc.collect() sage: E = ModularSymbols(11).2 sage: v = E.manin_symbol_rep() sage: c,x = v[0] sage: y = x.modular_symbol_rep() sage: coercion_model.get_action(QQ, parent(y), op=operator.mul) Left scalar multiplication by Rational Field on Abelian Group of all Formal Finite Sums over Rational Field with precomposition on right by Coercion map: From: Abelian Group of all Formal Finite Sums over Integer Ring To: Abelian Group of all Formal Finite Sums over Rational Field
>>> from sage.all import * >>> # needs sage.libs.flint sage.modular >>> E = ModularSymbols(Integer(11)).gen(2) >>> s = E.modular_symbol_rep() >>> del E,s >>> import gc >>> _ = gc.collect() >>> E = ModularSymbols(Integer(11)).gen(2) >>> v = E.manin_symbol_rep() >>> c,x = v[Integer(0)] >>> y = x.modular_symbol_rep() >>> coercion_model.get_action(QQ, parent(y), op=operator.mul) Left scalar multiplication by Rational Field on Abelian Group of all Formal Finite Sums over Rational Field with precomposition on right by Coercion map: From: Abelian Group of all Formal Finite Sums over Integer Ring To: Abelian Group of all Formal Finite Sums over Rational Field
- left_precomposition#
The left map to precompose with, or
None
if there is no left precomposition map.
- right_precomposition#
The right map to precompose with, or
None
if there is no right precomposition map.