diff --git a/src/sage/algebras/yangian.py b/src/sage/algebras/yangian.py index b63648bbb95..bfd69fbaf27 100644 --- a/src/sage/algebras/yangian.py +++ b/src/sage/algebras/yangian.py @@ -22,6 +22,7 @@ from sage.structure.unique_representation import UniqueRepresentation from sage.categories.hopf_algebras_with_basis import HopfAlgebrasWithBasis from sage.categories.graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis +from sage.categories.cartesian_product import cartesian_product from sage.rings.integer_ring import ZZ from sage.rings.infinity import infinity from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing @@ -34,155 +35,6 @@ import itertools -class GeneratorIndexingSet(UniqueRepresentation): - """ - Helper class for the indexing set of the generators. - """ - def __init__(self, index_set, level=None): - """ - Initialize ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - """ - self._index_set = index_set - self._level = level - - def __repr__(self): - """ - Return a string representation of ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: GeneratorIndexingSet((1,2)) - Cartesian product of Positive integers, (1, 2), (1, 2) - sage: GeneratorIndexingSet((1,2), 4) - Cartesian product of (1, 2, 3, 4), (1, 2), (1, 2) - """ - if self._level is None: - L = PositiveIntegers() - else: - L = tuple(range(1, self._level + 1)) - return "Cartesian product of {L}, {I}, {I}".format(L=L, I=self._index_set) - - def an_element(self): - """ - Initialize ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - sage: I.an_element() - (3, 1, 1) - sage: I = GeneratorIndexingSet((1,2), 5) - sage: I.an_element() - (3, 1, 1) - sage: I = GeneratorIndexingSet((1,2), 1) - sage: I.an_element() - (1, 1, 1) - """ - if self._level is not None and self._level < 3: - return (1, self._index_set[0], self._index_set[0]) - return (3, self._index_set[0], self._index_set[0]) - - def cardinality(self): - """ - Return the cardinality of ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - sage: I.cardinality() - +Infinity - sage: I = GeneratorIndexingSet((1,2), level=3) - sage: I.cardinality() == 3 * 2 * 2 - True - """ - if self._level is not None: - return self._level * len(self._index_set)**2 - return infinity - - __len__ = cardinality - - def __call__(self, x): - """ - Call ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - sage: I([1, 2]) - (1, 2) - """ - return tuple(x) - - def __contains__(self, x): - """ - Check containment of ``x`` in ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - sage: (4, 1, 2) in I - True - sage: [4, 2, 1] in I - True - sage: (-1, 1, 1) in I - False - sage: (1, 3, 1) in I - False - - :: - - sage: I3 = GeneratorIndexingSet((1,2), 3) - sage: (1, 1, 2) in I3 - True - sage: (3, 1, 1) in I3 - True - sage: (4, 1, 1) in I3 - False - """ - return (isinstance(x, (tuple, list)) and len(x) == 3 - and x[0] in ZZ and x[0] > 0 - and (self._level is None or x[0] <= self._level) - and x[1] in self._index_set - and x[2] in self._index_set) - - def __iter__(self): - """ - Iterate over ``self``. - - TESTS:: - - sage: from sage.algebras.yangian import GeneratorIndexingSet - sage: I = GeneratorIndexingSet((1,2)) - sage: it = iter(I) - sage: [next(it) for dummy in range(5)] - [(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1)] - - sage: I = GeneratorIndexingSet((1,2), 3) - sage: list(I) - [(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), - (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2), - (3, 1, 1), (3, 1, 2), (3, 2, 1), (3, 2, 2)] - """ - I = self._index_set - if self._level is not None: - for x in itertools.product(range(1, self._level + 1), I, I): - yield x - return - for i in PositiveIntegers(): - for x in itertools.product(I, I): - yield (i, x[0], x[1]) - - class Yangian(CombinatorialFreeModule): r""" The Yangian `Y(\mathfrak{gl}_n)`. @@ -392,7 +244,7 @@ def __init__(self, base_ring, n, variable_name, filtration): category = category.Connected() self._index_set = tuple(range(1, n + 1)) # The keys for the basis are tuples (l, i, j) - indices = GeneratorIndexingSet(self._index_set) + indices = cartesian_product([PositiveIntegers(), self._index_set, self._index_set]) # We note that the generators are non-commutative, but we always sort # them, so they are, in effect, indexed by the free abelian monoid basis_keys = IndexedFreeAbelianMonoid(indices, bracket=False, @@ -501,7 +353,7 @@ def _element_constructor_(self, x): True sage: Y6 = Yangian(QQ, 4, level=6, filtration='natural') sage: Y(Y6.an_element()) - t(1)[1,1]*t(1)[1,2]^2*t(1)[1,3]^3*t(3)[1,1] + t(1)[1,1]^2*t(1)[1,2]^2*t(1)[1,3]^3 + 2*t(1)[1,1] + 3*t(1)[1,2] + 1 """ if isinstance(x, CombinatorialFreeModule.Element): if isinstance(x.parent(), Yangian) and x.parent()._n <= self._n: @@ -543,8 +395,8 @@ def algebra_generators(self): sage: Y = Yangian(QQ, 4) sage: Y.algebra_generators() - Lazy family (generator(i))_{i in Cartesian product of - Positive integers, (1, 2, 3, 4), (1, 2, 3, 4)} + Lazy family (generator(i))_{i in The Cartesian product of + (Positive integers, {1, 2, 3, 4}, {1, 2, 3, 4})} """ return Family(self._indices._indices, self.gen, name="generator") @@ -816,7 +668,8 @@ def __init__(self, base_ring, n, level, variable_name, filtration): category = HopfAlgebrasWithBasis(base_ring).Filtered() self._index_set = tuple(range(1,n+1)) # The keys for the basis are tuples (l, i, j) - indices = GeneratorIndexingSet(self._index_set, level) + L = range(1, self._level + 1) + indices = cartesian_product([L, self._index_set, self._index_set]) # We note that the generators are non-commutative, but we always sort # them, so they are, in effect, indexed by the free abelian monoid basis_keys = IndexedFreeAbelianMonoid(indices, bracket=False, prefix=variable_name) @@ -1170,6 +1023,17 @@ def antipode_on_basis(self, m): -tbar(2)[1,1] sage: x = grY.an_element(); x + tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(42)[1,1] + sage: grY.antipode_on_basis(x.leading_support()) + -tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(42)[1,1] + - 2*tbar(1)[1,1]*tbar(1)[1,2]*tbar(1)[1,3]^3*tbar(42)[1,2] + - 3*tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^2*tbar(42)[1,3] + + 5*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(42)[1,1] + + 10*tbar(1)[1,2]*tbar(1)[1,3]^3*tbar(42)[1,2] + + 15*tbar(1)[1,2]^2*tbar(1)[1,3]^2*tbar(42)[1,3] + + sage: g = grY.indices().gens() + sage: x = grY(g[1,1,1] * g[1,1,2]^2 * g[1,1,3]^3 * g[3,1,1]); x tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(3)[1,1] sage: grY.antipode_on_basis(x.leading_support()) -tbar(1)[1,1]*tbar(1)[1,2]^2*tbar(1)[1,3]^3*tbar(3)[1,1]