Merge branch 'u/gh-ehaka/free_nilpotent_lie_algebras-26076' of git://trac.sagemath.org/sage into public/lie_algebras/free_nilpotent-26076

Author: tscrim

src/sage/algebras/lie_algebras/lie_algebra.py

@@ -319,6 +319,17 @@ class LieAlgebra(Parent, UniqueRepresentation): # IndexedGenerators):
         sage: L.<X,Y,Z> = LieAlgebra(QQ, {('X','Y'): {'Z': 1}}, category=C); L
         Nilpotent Lie algebra on 3 generators (X, Y, Z) over Rational Field
 
+    Free nilpotent Lie algebras are the truncated versions of the free Lie
+    algebras. That is, the only relations other than anticommutativity and the
+    Jacobi identity among the Lie brackets are that brackets of length higher
+    than the nilpotency step vanish. They can be created by using the
+    ``step`` keyword::
+
+        sage: L = LieAlgebra(ZZ, 2, step=3); L
+        Free Nilpotent Lie algebra on 5 generators (X_1, X_2, X_12, X_112, X_122) over Integer Ring
+        sage: L.step()
+        3
+
     REFERENCES:
 
     - [deG2000]_ Willem A. de Graaf. *Lie Algebras: Theory and Algorithms*.
@@ -422,6 +433,13 @@ def __classcall_private__(cls, R=None, arg0=None, arg1=None, names=None,
         # Otherwise it must be either a free or abelian Lie algebra
 
         if arg1 in ZZ:
+            step = kwds.get("step", None)
+            if step:
+                # Parse input as a free nilpotent Lie algebra
+                from sage.algebras.lie_algebras.nilpotent_lie_algebra import FreeNilpotentLieAlgebra
+                del kwds["step"]
+                return FreeNilpotentLieAlgebra(R, arg1, step, names=names, **kwds)
+
             if isinstance(arg0, str):
                 names = arg0
             if names is None:

src/sage/algebras/lie_algebras/nilpotent_lie_algebra.py

@@ -153,3 +153,268 @@ def _repr_(self):
             Nilpotent Lie algebra on 4 generators (X, Y, Z, W) over Rational Field
         """
         return "Nilpotent %s" % (super(NilpotentLieAlgebra_dense, self)._repr_())
+
+class FreeNilpotentLieAlgebra(NilpotentLieAlgebra_dense):
+    r"""
+    Returns the free nilpotent Lie algebra of step ``s`` with ``r`` generators.
+
+    The free nilpotent Lie algebra `L` of step `s` with `r` generators is the
+    quotient of the free Lie algebra on `r` generators by the `s+1`th term of
+    the lower central series. That is, the only relations in the Lie algebra `L`
+    are anticommutativity, the Jacobi identity, and the vanishing of all
+    brackets of length more than `s`.
+
+    INPUT:
+
+    - ``R`` -- the base ring
+    - ``r`` -- an integer; the number of generators
+    - ``s`` -- an integer; the nilpotency step of the algebra
+    - ``names`` -- (optional) a string or a list of strings used to name the
+      basis elements; If ``names`` is a string, then names for the basis will be
+      autogenerated as determined by the ``naming`` parameter.
+    - ``naming`` -- (optional) a string; the naming scheme to use for the basis.
+      Valid values are:
+          'index' : The basis elements are ``names_w``, where `w` are Lyndon
+                    words indexing the basis.
+                    This naming scheme is not supported if `r > 10`, since it
+                    leads to ambiguous names. When `r \leq 10`, this is the
+                    default naming scheme.
+          'linear' : the basis is indexed ``names_1``,...,``names_n`` in the
+                     ordering of the Lyndon basis. When `r > 10`, this is the
+                     default naming scheme.
+
+    EXAMPLES:
+
+    We compute the free step 4 Lie algebra on 2 generators and verify the only
+    non-trivial relation [[1,[1,2]],2] = [1,[[1,2],2]]::
+
+        sage: L = LieAlgebra(QQ, 2, step=4)
+        sage: L.basis().list()
+        [X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222]
+        sage: X_1, X_2 = L.basis().list()[:2]
+        sage: L[[X_1, [X_1, X_2]], X_2]
+        X_1122
+        sage: L[[X_1, [X_1, X_2]], X_2] == L[X_1, [[X_1, X_2], X_2]]
+        True
+
+    The linear naming scheme on the same Lie algebra::
+
+        sage: K = LieAlgebra(QQ, 2, step=4, names='Y', naming='linear')
+        sage: K.basis().list()
+        [Y_1, Y_2, Y_3, Y_4, Y_5, Y_6, Y_7, Y_8]
+        sage: K.inject_variables()
+        Defining Y_1, Y_2, Y_3, Y_4, Y_5, Y_6, Y_7, Y_8
+        sage: Y_2.bracket(Y_3)
+        -Y_5
+        sage: Y_5.bracket(Y_1)
+        -Y_7
+        sage: Y_3.bracket(Y_4)
+        0
+
+    A fully custom naming scheme on the Heisenberg algebra::
+
+        sage: L = LieAlgebra(ZZ, 2, step=2, names = ('X', 'Y', 'Z'))
+        sage: a, b, c = L.basis()
+        sage: L.basis().list()
+        [X, Y, Z]
+        sage: a.bracket(b)
+        Z
+
+    An equivalent way to define custom names for the basis elements and
+    bind them as local variables simultaneously::
+
+        sage: L.<X,Y,Z> = LieAlgebra(ZZ, 2, step=2)
+        sage: L.basis().list()
+        [X, Y, Z]
+        sage: X.bracket(Y)
+        Z
+
+    A free nilpotent Lie algebra is a stratified nilpotent Lie algebra::
+
+        sage: L = LieAlgebra(QQ, 3, step=3)
+        sage: L.category()
+        Category of finite dimensional stratified lie algebras with basis over Rational Field
+        sage: L in LieAlgebras(QQ).Nilpotent()
+        True
+
+    Being graded means that each basis element has a degree::
+
+        sage: L in LieAlgebras(QQ).Graded()
+        True
+        sage: L.homogeneous_component_basis(1).list()
+        [X_1, X_2, X_3]
+        sage: L.homogeneous_component_basis(2).list()
+        [X_12, X_13, X_23]
+        sage: L.homogeneous_component_basis(3).list()
+        [X_112, X_113, X_122, X_123, X_132, X_133, X_223, X_233]
+
+    TESTS:
+
+    Verify all bracket relations in the free nilpotent Lie algebra of step 5
+    with 2 generators::
+
+        sage: L = LieAlgebra(QQ, 2, step=5)
+        sage: L.inject_variables()
+        Defining X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222,
+        X_11112, X_11122, X_11212, X_11222, X_12122, X_12222
+        sage: [X_1.bracket(Xk) for Xk in L.basis()]
+        [0, X_12, X_112, X_1112, X_1122,
+        X_11112, X_11122, X_11222, 0, 0, 0, 0, 0, 0]
+        sage: [X_2.bracket(Xk) for Xk in L.basis()]
+        [-X_12, 0, -X_122, -X_1122, -X_1222,
+        -X_11122 + X_11212, -X_11222 - X_12122, -X_12222, 0, 0, 0, 0, 0, 0]
+        sage: [X_12.bracket(Xk) for Xk in L.basis()]
+        [-X_112, X_122, 0, -X_11212, X_12122, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_112.bracket(Xk) for Xk in L.basis()]
+        [-X_1112, X_1122, X_11212, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_122.bracket(Xk) for Xk in L.basis()]
+        [-X_1122, X_1222, -X_12122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_1112.bracket(Xk) for Xk in L.basis()]
+        [-X_11112, X_11122 - X_11212, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_1122.bracket(Xk) for Xk in L.basis()]
+        [-X_11122, X_11222 + X_12122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_1222.bracket(Xk) for Xk in L.basis()]
+        [-X_11222, X_12222, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_11112.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_11122.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_11212.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_11222.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_12122.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+        sage: [X_12222.bracket(Xk) for Xk in L.basis()]
+        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+
+    The dimensions of the smallest free nilpotent Lie algebras on 2 or 3
+    generators::
+
+        sage: l = [LieAlgebra(QQ, 2, step=k) for k in range(1, 7)]
+        sage: [L.dimension() for L in l]
+        [2, 3, 5, 8, 14, 23]
+        sage: l = [LieAlgebra(QQ, 3, step=k) for k in range(1, 4)]
+        sage: [L.dimension() for L in l]
+        [3, 6, 14]
+
+    Test suite for a free nilpotent Lie algebra::
+
+        sage: L = LieAlgebra(QQ, 4, step=3)
+        sage: TestSuite(L).run()
+    """
+
+    def __init__(self, R, r, s, names=None, naming=None, category=None, **kwds):
+        r"""
+        Initialize ``self``
+
+        EXAMPLES::
+
+            sage: LieAlgebra(ZZ, 2, step=2)
+            Free Nilpotent Lie algebra on 3 generators (X_1, X_2, X_12) over Integer Ring
+        """
+        from sage.rings.integer_ring import ZZ
+
+        if r not in ZZ or r <= 0:
+            raise ValueError("number of generators %s is not "
+                             "a positive integer" % r)
+        if s not in ZZ or s <= 0:
+            raise ValueError("step %s is not a positive integer" % s)
+
+        # extract an index set from the Lyndon words of the corresponding
+        # free Lie algebra, and store the corresponding elements in a dict
+        from sage.algebras.lie_algebras.lie_algebra import LieAlgebra
+
+        free_gen_names = ['F%d' % k for k in range(r)]
+        L = LieAlgebra(R, free_gen_names).Lyndon()
+        from collections import OrderedDict
+        basis_dict = OrderedDict()
+        for d in range(1, s + 1):
+            for X in L.graded_basis(d):
+                # convert brackets of form [X_1, [X_1, X_2]] to words (1,1,2)
+                w = tuple(free_gen_names.index(s) + 1
+                          for s in X.leading_support().to_word())
+                basis_dict[w] = X
+
+        # define names for basis elements
+        if not names:
+            names = 'X'
+        if isinstance(names, str):
+            names = tuple(names)
+        if len(names) == 1 and len(basis_dict)>1:
+            if not naming:
+                if r > 10:
+                    naming = 'linear'
+                else:
+                    naming = 'index'
+            if naming == 'linear':
+                names = ['%s_%d' % (names[0], k + 1)
+                         for k in range(len(basis_dict))]
+            elif naming == 'index':
+                if r > 10:
+                    raise ValueError("'index' naming scheme not supported for "
+                                     "over 10 generators")
+                names = ['%s_%s' % (names[0], "".join(str(s) for s in w))
+                         for w in basis_dict]
+            else:
+                raise ValueError("unknown naming scheme %s" % naming)
+
+        # extract structural coefficients from the free Lie algebra
+        s_coeff = {}
+        for k,X_ind in enumerate(basis_dict):
+            X = basis_dict[X_ind]
+            degX = len(X_ind)
+            for Y_ind in basis_dict.keys()[k+1:]:
+                # brackets are only computed when deg(X) + deg(Y) <= s
+                degY = len(Y_ind)
+                if degX + degY > s:
+                    continue
+
+                Y = basis_dict[Y_ind]
+                Z = L[X, Y]
+                if not Z.is_zero():
+                    s_coeff[(X_ind, Y_ind)] = {W: Z[basis_dict[W].leading_support()]
+                                               for W in basis_dict}
+
+        from sage.structure.indexed_generators import standardize_names_index_set
+        names, index_set = standardize_names_index_set(names, basis_dict.keys())
+        s_coeff = LieAlgebraWithStructureCoefficients._standardize_s_coeff(
+            s_coeff, index_set)
+
+        category = LieAlgebras(R).FiniteDimensional().WithBasis() \
+                                 .Graded().Stratified().or_subcategory(category)
+        NilpotentLieAlgebra_dense.__init__(self, R, s_coeff, names,
+                                           index_set, s,
+                                           category=category, **kwds)
+
+    def _repr_generator(self, w):
+        r"""
+        Returns the string representation of the basis element
+        indexed by the word ``w`` in ``self``.
+
+        EXAMPLES::
+
+            sage: L = LieAlgebra(QQ, 2, step=4)
+            sage: L._repr_generator((1, 1, 2, 2))
+            'X_1122'
+            sage: L = LieAlgebra(QQ, 2, step=4, naming='linear')
+            sage: L._repr_generator((1, 1, 2, 2))
+            'X_7'
+            sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
+            sage: L._repr_generator((1, 2))
+            'Z'
+        """
+        i = self.indices().index(w)
+        return self.variable_names()[i]
+
+    def _repr_(self):
+        """
+        Return a string representation of ``self``.
+
+        EXAMPLES::
+
+            sage: L = LieAlgebra(QQ, 2, step=3)
+            sage: L
+            Free Nilpotent Lie algebra on 5 generators (X_1, X_2, X_12, X_112, X_122) over Rational Field
+        """
+        return "Free %s" % (super(FreeNilpotentLieAlgebra, self)._repr_())