Finite Dimensional Lie Algebras With Basis

AUTHORS:

• Travis Scrimshaw (07-15-2013): Initial implementation

class sage.categories.finite_dimensional_lie_algebras_with_basis.FiniteDimensionalLieAlgebrasWithBasis(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

Category of finite dimensional Lie algebras with a basis.

Todo

Many of these tests should use non-abelian Lie algebras and need to be added after trac ticket #16820.

class ElementMethods

adjoint_matrix()

Return the matrix of the adjoint action of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.an_element().adjoint_matrix()
[0 0 0]
[0 0 0]
[0 0 0]

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: x.adjoint_matrix()
[0 0]
[1 0]
sage: y.adjoint_matrix()
[-1  0]
[ 0  0]

to_vector()

Return the vector in g.module() corresponding to the element self of g (where g is the parent of self).

Implement this if you implement g.module(). See sage.categories.lie_algebras.LieAlgebras.module() for how this is to be done.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.an_element().to_vector()
(0, 0, 0)

sage: D = DescentAlgebra(QQ, 4).D()
sage: L = LieAlgebra(associative=D)
sage: L.an_element().to_vector()
(1, 1, 1, 1, 1, 1, 1, 1)

class ParentMethods

as_finite_dimensional_algebra()

Return self as a FiniteDimensionalAlgebra.

EXAMPLES:

sage: L = lie_algebras.cross_product(QQ)
sage: x,y,z = L.basis()
sage: F = L.as_finite_dimensional_algebra()
sage: X,Y,Z = F.basis()
sage: x.bracket(y)
Z
sage: X * Y
Z

center()

Return the center of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: Z = L.center(); Z
An example of a finite dimensional Lie algebra with basis: the
 3-dimensional abelian Lie algebra over Rational Field
sage: Z.basis_matrix()
[1 0 0]
[0 1 0]
[0 0 1]

centralizer(S)

Return the centralizer of S in self.

INPUT:

• S – a subalgebra of self or a list of elements that represent generators for a subalgebra

See also

centralizer_basis()

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a,b,c = L.lie_algebra_generators()
sage: S = L.centralizer([a + b, 2*a + c]); S
An example of a finite dimensional Lie algebra with basis:
 the 3-dimensional abelian Lie algebra over Rational Field
sage: S.basis_matrix()
[1 0 0]
[0 1 0]
[0 0 1]

centralizer_basis(S)

Return a basis of the centralizer of S in self.

INPUT:

• S – a subalgebra of self or a list of elements that represent generators for a subalgebra

See also

centralizer()

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a,b,c = L.lie_algebra_generators()
sage: L.centralizer_basis([a + b, 2*a + c])
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

sage: H = lie_algebras.Heisenberg(QQ, 2)
sage: H.centralizer_basis(H)
[z]


sage: D = DescentAlgebra(QQ, 4).D()
sage: L = LieAlgebra(associative=D)
sage: L.centralizer_basis(L)
[D{},
 D{1} + D{1, 2} + D{2, 3} + D{3},
 D{1, 2, 3} + D{1, 3} + D{2}]
sage: D.center_basis()
(D{},
 D{1} + D{1, 2} + D{2, 3} + D{3},
 D{1, 2, 3} + D{1, 3} + D{2})

derived_series()

Return the derived series \((\mathfrak{g}^{(i)})_i\) of self where the rightmost \(\mathfrak{g}^{(k)} = \mathfrak{g}^{(k+1)} = \cdots\).

We define the derived series of a Lie algebra \(\mathfrak{g}\) recursively by \(\mathfrak{g}^{(0)} := \mathfrak{g}\) and

\[\mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}]\]

and recall that \(\mathfrak{g}^{(k)} \supseteq \mathfrak{g}^{(k+1)}\). Alternatively we can express this as

\[\mathfrak{g} \supseteq [\mathfrak{g}, \mathfrak{g}] \supseteq \bigl[ [\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}] \bigr] \supseteq \biggl[ \bigl[ [\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}] \bigr], \bigl[ [\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}] \bigr] \biggr] \supseteq \cdots.\]

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.derived_series()
(An example of a finite dimensional Lie algebra with basis:
    the 3-dimensional abelian Lie algebra over Rational Field,
 An example of a finite dimensional Lie algebra with basis:
    the 0-dimensional abelian Lie algebra over Rational Field
    with basis matrix:
    [])

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: L.derived_series() # todo: not implemented - #17416
(Lie algebra on 2 generators (x, y) over Rational Field,
 Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
(x,),
 Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
())

derived_subalgebra()

Return the derived subalgebra of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.derived_subalgebra()
An example of a finite dimensional Lie algebra with basis:
 the 0-dimensional abelian Lie algebra over Rational Field
 with basis matrix:
[]

from_vector(v)

Return the element of self corresponding to the vector v in self.module().

Implement this if you implement module(); see the documentation of sage.categories.lie_algebras.LieAlgebras.module() for how this is to be done.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: u = L.from_vector(vector(QQ, (1, 0, 0))); u
(1, 0, 0)
sage: parent(u) is L
True

is_abelian()

Return if self is an abelian Lie algebra.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.is_abelian()
True

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'): {'x':1}})
sage: L.is_abelian()
False

is_ideal(A)

Return if self is an ideal of A.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: I = L.ideal([2*a - c, b + c])
sage: I.is_ideal(L)
True

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: L.is_ideal(L)
True

sage: F = LieAlgebra(QQ, 'F', representation='polynomial')
sage: L.is_ideal(F)
Traceback (most recent call last):
...
NotImplementedError: A must be a finite dimensional Lie algebra
 with basis

is_nilpotent()

Return if self is a nilpotent Lie algebra.

A Lie algebra is nilpotent if the lower central series eventually becomes \(0\).

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.is_nilpotent()
True

is_semisimple()

Return if self if a semisimple Lie algebra.

A Lie algebra is semisimple if the solvable radical is zero. In characteristic 0, this is equivalent to saying the Killing form is non-degenerate.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.is_semisimple()
False

is_solvable()

Return if self is a solvable Lie algebra.

A Lie algebra is solvable if the derived series eventually becomes \(0\).

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.is_solvable()
True

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: L.is_solvable() # todo: not implemented - #17416
False

killing_form(x, y)

Return the Killing form on x and y, where x and y are two elements of self.

The Killing form is defined as

\[\langle x \mid y \rangle = \operatorname{tr}\left( \operatorname{ad}_x \circ \operatorname{ad}_y \right).\]

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a,b,c = L.lie_algebra_generators()
sage: L.killing_form(a, b)
0

killing_form_matrix()

Return the matrix of the Killing form of self.

The rows and the columns of this matrix are indexed by the elements of the basis of self (in the order provided by basis()).

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.killing_form_matrix()
[0 0 0]
[0 0 0]
[0 0 0]

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example(0)
sage: m = L.killing_form_matrix(); m
[]
sage: parent(m)
Full MatrixSpace of 0 by 0 dense matrices over Rational Field

killing_matrix(x, y)

Return the Killing matrix of x and y, where x and y are two elements of self.

The Killing matrix is defined as the matrix corresponding to the action of \(\operatorname{ad}_x \circ \operatorname{ad}_y\) in the basis of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a,b,c = L.lie_algebra_generators()
sage: L.killing_matrix(a, b)
[0 0 0]
[0 0 0]
[0 0 0]

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: L.killing_matrix(x, y)
[ 0  0]
[-1  0]

lower_central_series()

Return the lower central series \((\mathfrak{g}_{i})_i\) of self where the rightmost \(\mathfrak{g}_k = \mathfrak{g}_{k+1} = \cdots\).

We define the lower central series of a Lie algebra \(\mathfrak{g}\) recursively by \(\mathfrak{g}_0 := \mathfrak{g}\) and

\[\mathfrak{g}_{k+1} = [\mathfrak{g}, \mathfrak{g}_{k}]\]

and recall that \(\mathfrak{g}_{k} \supseteq \mathfrak{g}_{k+1}\). Alternatively we can express this as

\[\mathfrak{g} \supseteq [\mathfrak{g}, \mathfrak{g}] \supseteq \bigl[ [\mathfrak{g}, \mathfrak{g}], \mathfrak{g} \bigr] \supseteq\biggl[\bigl[ [\mathfrak{g}, \mathfrak{g}], \mathfrak{g} \bigr], \mathfrak{g}\biggr] \supseteq \cdots.\]

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.derived_series()
(An example of a finite dimensional Lie algebra with basis:
    the 3-dimensional abelian Lie algebra over Rational Field,
 An example of a finite dimensional Lie algebra with basis:
    the 0-dimensional abelian Lie algebra over Rational Field
    with basis matrix:
    [])

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: L.lower_central_series() # todo: not implemented - #17416
(Lie algebra on 2 generators (x, y) over Rational Field,
 Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
(x,))

module(R=None)

Return a dense free module associated to self over R.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L._dense_free_module()
Vector space of dimension 3 over Rational Field

product_space(L, submodule=False)

Return the product space [self, L].

INPUT:

• L – a Lie subalgebra of self
• submodule – (default: False) if True, then the result is forced to be a submodule of self

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a,b,c = L.lie_algebra_generators()
sage: X = L.subalgebra([a, b+c])
sage: L.product_space(X)
An example of a finite dimensional Lie algebra with basis:
 the 0-dimensional abelian Lie algebra over Rational Field
 with basis matrix:
[]
sage: Y = L.subalgebra([a, 2*b-c])
sage: X.product_space(Y)
An example of a finite dimensional Lie algebra with basis:
 the 0-dimensional abelian Lie algebra over Rational
 Field with basis matrix:
[]

sage: H = lie_algebras.Heisenberg(ZZ, 4)
sage: Hp = H.product_space(H, submodule=True).basis()
sage: [H.from_vector(v) for v in Hp]
[z]

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'):{'x':1}})
sage: Lp = L.product_space(L) # todo: not implemented - #17416
sage: Lp # todo: not implemented - #17416
Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
(x,)
sage: Lp.product_space(L) # todo: not implemented - #17416
Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
(x,)
sage: L.product_space(Lp) # todo: not implemented - #17416
Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
(x,)
sage: Lp.product_space(Lp) # todo: not implemented - #17416
Subalgebra generated of Lie algebra on 2 generators (x, y) over Rational Field with basis:
()

structure_coefficients(include_zeros=False)

Return the structure coefficients of self.

INPUT:

• include_zeros – (default: False) if True, then include the \([x, y] = 0\) pairs in the output

OUTPUT:

A dictionary whose keys are pairs of basis indices \((i, j)\) with \(i < j\), and whose values are the corresponding elements \([b_i, b_j]\) in the Lie algebra.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: L.structure_coefficients()
Finite family {}
sage: L.structure_coefficients(True)
Finite family {(0, 1): (0, 0, 0), (1, 2): (0, 0, 0), (0, 2): (0, 0, 0)}

sage: G = SymmetricGroup(3)
sage: S = GroupAlgebra(G, QQ)
sage: L = LieAlgebra(associative=S)
sage: L.structure_coefficients()
Finite family {((1,3,2), (1,3)): (2,3) - (1,2),
               ((1,2), (1,2,3)): -(2,3) + (1,3),
               ((1,2,3), (1,3)): -(2,3) + (1,2),
               ((2,3), (1,3,2)): -(1,2) + (1,3),
               ((2,3), (1,3)): -(1,2,3) + (1,3,2),
               ((2,3), (1,2)): (1,2,3) - (1,3,2),
               ((2,3), (1,2,3)): (1,2) - (1,3),
               ((1,2), (1,3,2)): (2,3) - (1,3),
               ((1,2), (1,3)): (1,2,3) - (1,3,2)}

class Subobjects(category, *args)

Bases: sage.categories.subobjects.SubobjectsCategory

A category for subalgebras of a finite dimensional Lie algebra with basis.

class ParentMethods

ambient()

Return the ambient Lie algebra of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: S = L.subalgebra([2*a+b, b + c])
sage: S.ambient() == L
True

basis_matrix()

Return the basis matrix of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: S = L.subalgebra([2*a+b, b + c])
sage: S.basis_matrix()
[   1    0 -1/2]
[   0    1    1]

example(n=3)

Return an example of a finite dimensional Lie algebra with basis as per Category.example.

EXAMPLES:

sage: C = LieAlgebras(QQ).FiniteDimensional().WithBasis()
sage: C.example()
An example of a finite dimensional Lie algebra with basis:
 the 3-dimensional abelian Lie algebra over Rational Field

Other dimensions can be specified as an optional argument:

sage: C.example(5)
An example of a finite dimensional Lie algebra with basis:
 the 5-dimensional abelian Lie algebra over Rational Field