gh-37693: Implement the hypercenter and upper central series for fini…

…te dimensional Lie algebras

    
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We implement the upper central series and hypercenter of a finite
dimensional Lie algebra.

As part of the testing, we found improvements could be made to the
generic example (needed so the example would work with these tests): We
moved the `reduce()` method up to the category. Implemented methods
expected for the subalgebras. Better input verification. Elements are
now hashable. We had to work around the fact that `lift()` is serving
multiple roles as to the UEA and to the ambient space for a Lie
subalgebra. This functionality needs to be divided (I take full
responsibility for this bad design) by renaming `lift` to something for
the UEA map, but this is a significant change that should be done on a
followup PR.

We also fixed some bugs and improved efficiency with changing vectors
into Lie algebra elements.

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- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: #37693
Reported by: Travis Scrimshaw
Reviewer(s): Matthias Köppe, Travis Scrimshaw

src/sage/algebras/lie_algebras/quotient.py

@@ -218,8 +218,10 @@ def __classcall_private__(cls, I, ambient=None, names=None,
         index_set = [i for i in sorted_indices if i not in I_supp]
 
         if names is None:
-            amb_names = dict(zip(sorted_indices, ambient.variable_names()))
-            names = [amb_names[i] for i in index_set]
+            if ambient._names is not None:
+                # ambient has assigned variable names, so use those
+                amb_names = dict(zip(sorted_indices, ambient.variable_names()))
+                names = [amb_names[i] for i in index_set]
         elif isinstance(names, str):
             if len(index_set) == 1:
                 names = [names]
@@ -252,15 +254,15 @@ def __init__(self, I, L, names, index_set, category=None):
         B = L.basis()
         sm = L.module().submodule_with_basis([I.reduce(B[i]).to_vector()
                                               for i in index_set])
-        SB = sm.basis()
+        SB = [L.from_vector(b) for b in sm.basis()]
 
         # compute and normalize structural coefficients for the quotient
         s_coeff = {}
         for i, ind_i in enumerate(index_set):
             for j in range(i + 1, len(index_set)):
                 ind_j = index_set[j]
 
-                brkt = I.reduce(L.bracket(SB[i], SB[j]))
+                brkt = I.reduce(SB[i].bracket(SB[j]))
                 brktvec = sm.coordinate_vector(brkt.to_vector())
                 s_coeff[(ind_i, ind_j)] = dict(zip(index_set, brktvec))
         s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
@@ -291,11 +293,26 @@ def _repr_(self):
             L: General linear Lie algebra of rank 2 over Rational Field
             I: Ideal ([0 0]
             [0 1])
+
+            sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+            sage: a,b,c = L.gens()
+            sage: I = L.ideal([a + 2*b, b + 3*c])
+            sage: Q = L.quotient(I)
+            sage: Q
+            Lie algebra quotient L/I of dimension 1 over Rational Field where
+            L: An example of a finite dimensional Lie algebra with basis:
+                the 3-dimensional abelian Lie algebra over Rational Field
+            I: Ideal ((1, 0, -6), (0, 1, 3))
         """
+        try:
+            ideal_repr = self._I._repr_short()
+        except AttributeError:
+            ideal_repr = repr(tuple(self._I.gens()))
+
         return ("Lie algebra quotient L/I of dimension %s"
                 " over %s where\nL: %s\nI: Ideal %s" % (
                     self.dimension(), self.base_ring(),
-                    self.ambient(), self._I._repr_short()))
+                    self.ambient(), ideal_repr))
 
     def _repr_generator(self, i):
         r"""

src/sage/algebras/lie_algebras/subalgebra.py

@@ -888,71 +888,6 @@ def is_ideal(self, A):
             return True
         return super().is_ideal(A)
 
-    def reduce(self, X):
-        r"""
-        Reduce an element of the ambient Lie algebra modulo the
-        ideal ``self``.
-
-        INPUT:
-
-        - ``X`` -- an element of the ambient Lie algebra
-
-        OUTPUT:
-
-        An element `Y` of the ambient Lie algebra that is contained in a fixed
-        complementary submodule `V` to ``self`` such that `X = Y` mod ``self``.
-
-        When the base ring of ``self`` is a field, the complementary submodule
-        `V` is spanned by the elements of the basis that are not the leading
-        supports of the basis of ``self``.
-
-        EXAMPLES:
-
-        An example reduction in a 6 dimensional Lie algebra::
-
-            sage: sc = {('a','b'): {'d': 1}, ('a','c'): {'e': 1},
-            ....:       ('b','c'): {'f': 1}}
-            sage: L.<a,b,c,d,e,f> = LieAlgebra(QQ, sc)
-            sage: I =  L.ideal(c)
-            sage: I.reduce(a + b + c + d + e + f)
-            a + b + d
-
-        The reduction of an element is zero if and only if the
-        element belongs to the subalgebra::
-
-            sage: I.reduce(c + e)
-            0
-            sage: c + e in I
-            True
-
-        Over non-fields, the complementary submodule may not be spanned by
-        a subset of the basis of the ambient Lie algebra::
-
-            sage: L.<X,Y,Z> = LieAlgebra(ZZ, {('X','Y'): {'Z': 3}})
-            sage: I = L.ideal(Y)
-            sage: I.basis()
-            Family (Y, 3*Z)
-            sage: I.reduce(3*Z)
-            0
-            sage: I.reduce(Y + 14*Z)
-            2*Z
-        """
-        R = self.base_ring()
-        for Y in self.basis():
-            Y = self.lift(Y)
-            k, c = Y.leading_item(key=self._order)
-
-            if R.is_field():
-                X = X - X[k] / c * Y
-            else:
-                try:
-                    q, _ = X[k].quo_rem(c)
-                    X = X - q * Y
-                except AttributeError:
-                    pass
-
-        return X
-
     class Element(LieSubalgebraElementWrapper):
         def adjoint_matrix(self, sparse=False):
             """

src/sage/categories/examples/finite_dimensional_lie_algebras_with_basis.py

@@ -95,6 +95,10 @@ def __init__(self, R, n=None, M=None, ambient=None):
         self._ambient = ambient
         Parent.__init__(self, base=R, category=cat)
 
+        from sage.categories.lie_algebras import LiftMorphism
+        self._lift_uea = LiftMorphism(self, self._construct_UEA())
+        self._lift_uea.register_as_coercion()
+
     def _repr_(self):
         """
         EXAMPLES::
@@ -135,6 +139,57 @@ def _element_constructor_(self, x):
             x = x.value
         return self.element_class(self, self._M(x))
 
+    def lift(self, x):
+        r"""
+        Return the lift of ``self``.
+
+        EXAMPLES::
+
+            sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+            sage: a, b, c = L.gens()
+            sage: L.lift(a)
+            b0
+            sage: L.lift(b).parent() is L.universal_enveloping_algebra()
+            True
+
+            sage: I = L.ideal([a + 2*b, b + 3*c])
+            sage: I.lift(I.basis()[0])
+            (1, 0, -6)
+        """
+        # FIXME: This method can likely be simplified or removed once we
+        #   disentangle the UEA lift from the generic lift
+        A = self._ambient
+        if A is self:
+            return self._lift_uea(x)
+        return A.element_class(A, A._M(x.value))
+
+    def universal_enveloping_algebra(self):
+        r"""
+        Return the universal enveloping algebra of ``self``.
+
+        EXAMPLES::
+
+            sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+            sage: L.universal_enveloping_algebra()
+            Noncommutative Multivariate Polynomial Ring in b0, b1, b2
+             over Rational Field, nc-relations: {}
+        """
+        # FIXME: This method can likely be removed once we
+        #   disentangle the UEA lift from the generic lift
+        return self._lift_uea.codomain()
+
+    def _order(self, x):
+        r"""
+        Return a key for sorting for the index ``x``.
+
+        TESTS::
+
+            sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+            sage: L._order(2)
+            2
+        """
+        return x
+
     @cached_method
     def zero(self):
         """
@@ -194,6 +249,8 @@ def subalgebra(self, gens):
             [   1    0 -1/2]
             [   0    1    1]
         """
+        if isinstance(gens, AbelianLieAlgebra):
+            gens = [self(g) for g in gens.gens()]
         N = self._M.subspace([g.value for g in gens])
         return AbelianLieAlgebra(self.base_ring(), M=N, ambient=self._ambient)
 
@@ -252,7 +309,7 @@ def gens(self) -> tuple:
             sage: L.gens()
             ((1, 0, 0), (0, 1, 0), (0, 0, 1))
         """
-        return tuple(self._M.basis())
+        return tuple([self.element_class(self, b) for b in self._M.basis()])
 
     def module(self):
         """
@@ -302,7 +359,36 @@ def from_vector(self, v, order=None):
         """
         return self.element_class(self, self._M(v))
 
+    def leading_monomials(self):
+        r"""
+        Return the set of leading monomials of the basis of ``self``.
+
+        EXAMPLES::
+
+            sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+            sage: a, b, c = L.lie_algebra_generators()
+            sage: I = L.ideal([2*a + b, b + c])
+            sage: I.leading_monomials()
+            ((1, 0, 0), (0, 1, 0))
+        """
+        # for free modules, the leading monomial is actually the trailing monomial
+        return tuple([self._ambient._M(b.value).trailing_monomial()
+                      for b in self.basis()])
+
     class Element(BaseExample.Element):
+        def __init__(self, parent, value):
+            """
+            Initialize ``self``.
+
+            EXAMPLES::
+
+                sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
+                sage: a, b, c = L.lie_algebra_generators()
+                sage: TestSuite(a).run()
+            """
+            value.set_immutable()
+            super().__init__(parent, value)
+
         def __iter__(self):
             """
             Iterate over ``self`` by returning pairs ``(i, c)`` where ``i``

src/sage/categories/examples/lie_algebras.py

@@ -198,6 +198,20 @@ def __ne__(self, rhs):
             """
             return not self.__eq__(rhs)
 
+        def __hash__(self):
+            r"""
+            Return the hash of ``self``.
+
+            EXAMPLES::
+
+                sage: # needs sage.combinat sage.groups
+                sage: L = LieAlgebras(QQ).example()
+                sage: x, y = L.lie_algebra_generators()
+                sage: hash(x) == hash(x.value)
+                True
+            """
+            return hash(self.value)
+
         def __bool__(self) -> bool:
             """
             Check non-zero.