Nilpotent Lie groups¶

AUTHORS:

Eero Hakavuori (2018-09-25): initial version of nilpotent Lie groups

class sage.groups.lie_gps.nilpotent_lie_group.NilpotentLieGroup(L, name, **kwds)[source]¶

Bases: Group, DifferentiableManifold

A nilpotent Lie group.

INPUT:

L – the Lie algebra of the Lie group; must be a finite dimensional nilpotent Lie algebra with basis over a topological field, e.g. \(\QQ\) or \(\RR\)
name – string; name (symbol) given to the Lie group

Two types of exponential coordinates are defined on any nilpotent Lie group using the basis of the Lie algebra, see chart_exp1() and chart_exp2().

EXAMPLES:

Creation of a nilpotent Lie group:

Sage

sage: L = lie_algebras.Heisenberg(QQ, 1)
sage: G = L.lie_group(); G
Lie group G of Heisenberg algebra of rank 1 over Rational Field

Python

>>> from sage.all import *
>>> L = lie_algebras.Heisenberg(QQ, Integer(1))
>>> G = L.lie_group(); G
Lie group G of Heisenberg algebra of rank 1 over Rational Field

Giving a different name to the group:

Sage

sage: L.lie_group('H')
Lie group H of Heisenberg algebra of rank 1 over Rational Field

Python

>>> from sage.all import *
>>> L.lie_group('H')
Lie group H of Heisenberg algebra of rank 1 over Rational Field

Elements can be created using the exponential map:

Sage

sage: p,q,z = L.basis()
sage: g = G.exp(p); g
exp(p1)
sage: h = G.exp(q); h
exp(q1)

Python

>>> from sage.all import *
>>> p,q,z = L.basis()
>>> g = G.exp(p); g
exp(p1)
>>> h = G.exp(q); h
exp(q1)

Lie group multiplication has the usual product syntax:

Sage

sage: k = g*h; k
exp(p1 + q1 + 1/2*z)

Python

>>> from sage.all import *
>>> k = g*h; k
exp(p1 + q1 + 1/2*z)

The identity element is given by one():

Sage

sage: e = G.one(); e
exp(0)
sage: e*k == k and k*e == k
True

Python

>>> from sage.all import *
>>> e = G.one(); e
exp(0)
>>> e*k == k and k*e == k
True

The default coordinate system is exponential coordinates of the first kind:

Sage

sage: G.default_chart() == G.chart_exp1()
True
sage: G.chart_exp1()
Chart (G, (x_0, x_1, x_2))

Python

>>> from sage.all import *
>>> G.default_chart() == G.chart_exp1()
True
>>> G.chart_exp1()
Chart (G, (x_0, x_1, x_2))

Changing the default coordinates to exponential coordinates of the second kind will change how elements are printed:

Sage

sage: G.set_default_chart(G.chart_exp2())
sage: k
exp(z)exp(q1)exp(p1)
sage: G.set_default_chart(G.chart_exp1())
sage: k
exp(p1 + q1 + 1/2*z)

Python

>>> from sage.all import *
>>> G.set_default_chart(G.chart_exp2())
>>> k
exp(z)exp(q1)exp(p1)
>>> G.set_default_chart(G.chart_exp1())
>>> k
exp(p1 + q1 + 1/2*z)

The frames of left- or right-invariant vector fields are created using left_invariant_frame() and right_invariant_frame():

Sage

sage: X = G.left_invariant_frame(); X
Vector frame (G, (X_0,X_1,X_2))
sage: X[0]
Vector field X_0 on the Lie group G of Heisenberg algebra of rank 1 over Rational Field

Python

>>> from sage.all import *
>>> X = G.left_invariant_frame(); X
Vector frame (G, (X_0,X_1,X_2))
>>> X[Integer(0)]
Vector field X_0 on the Lie group G of Heisenberg algebra of rank 1 over Rational Field

A vector field can be displayed with respect to a coordinate frame:

Sage

sage: exp1_frame = G.chart_exp1().frame()
sage: exp2_frame = G.chart_exp2().frame()
sage: X[0].display(exp1_frame)
X_0 = ∂/∂x_0 - 1/2*x_1 ∂/∂x_2
sage: X[0].display(exp2_frame)
X_0 = ∂/∂y_0
sage: X[1].display(exp1_frame)
X_1 = ∂/∂x_1 + 1/2*x_0 ∂/∂x_2
sage: X[1].display(exp2_frame)
X_1 = ∂/∂y_1 + x_0 ∂/∂y_2

Python

>>> from sage.all import *
>>> exp1_frame = G.chart_exp1().frame()
>>> exp2_frame = G.chart_exp2().frame()
>>> X[Integer(0)].display(exp1_frame)
X_0 = ∂/∂x_0 - 1/2*x_1 ∂/∂x_2
>>> X[Integer(0)].display(exp2_frame)
X_0 = ∂/∂y_0
>>> X[Integer(1)].display(exp1_frame)
X_1 = ∂/∂x_1 + 1/2*x_0 ∂/∂x_2
>>> X[Integer(1)].display(exp2_frame)
X_1 = ∂/∂y_1 + x_0 ∂/∂y_2

Defining a left translation by a generic point:

Sage

sage: g = G.point([var('a'), var('b'), var('c')]); g
exp(a*p1 + b*q1 + c*z)
sage: L_g = G.left_translation(g); L_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
sage: L_g.display()
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, -1/2*b*x_0 + 1/2*a*x_1 + c + x_2)
   (x_0, x_1, x_2) ↦ (y_0, y_1, y_2) = (a + x_0, b + x_1,
                                        1/2*a*b + 1/2*(2*a + x_0)*x_1 + c + x_2)
   (y_0, y_1, y_2) ↦ (x_0, x_1, x_2) = (a + y_0, b + y_1,
                                        -1/2*b*y_0 + 1/2*(a - y_0)*y_1 + c + y_2)
   (y_0, y_1, y_2) ↦ (a + y_0, b + y_1, 1/2*a*b + a*y_1 + c + y_2)

Python

>>> from sage.all import *
>>> g = G.point([var('a'), var('b'), var('c')]); g
exp(a*p1 + b*q1 + c*z)
>>> L_g = G.left_translation(g); L_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
>>> L_g.display()
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, -1/2*b*x_0 + 1/2*a*x_1 + c + x_2)
   (x_0, x_1, x_2) ↦ (y_0, y_1, y_2) = (a + x_0, b + x_1,
                                        1/2*a*b + 1/2*(2*a + x_0)*x_1 + c + x_2)
   (y_0, y_1, y_2) ↦ (x_0, x_1, x_2) = (a + y_0, b + y_1,
                                        -1/2*b*y_0 + 1/2*(a - y_0)*y_1 + c + y_2)
   (y_0, y_1, y_2) ↦ (a + y_0, b + y_1, 1/2*a*b + a*y_1 + c + y_2)

Verifying the left-invariance of the left-invariant frame:

Sage

sage: x = G(G.chart_exp1()[:])
sage: L_g.differential(x)(X[0].at(x)) == X[0].at(L_g(x))
True
sage: L_g.differential(x)(X[1].at(x)) == X[1].at(L_g(x))
True
sage: L_g.differential(x)(X[2].at(x)) == X[2].at(L_g(x))
True

Python

>>> from sage.all import *
>>> x = G(G.chart_exp1()[:])
>>> L_g.differential(x)(X[Integer(0)].at(x)) == X[Integer(0)].at(L_g(x))
True
>>> L_g.differential(x)(X[Integer(1)].at(x)) == X[Integer(1)].at(L_g(x))
True
>>> L_g.differential(x)(X[Integer(2)].at(x)) == X[Integer(2)].at(L_g(x))
True

An element of the Lie algebra can be extended to a left or right invariant vector field:

Sage

sage: X_L = G.left_invariant_extension(p + 3*q); X_L
Vector field p1 + 3*q1 on the Lie group G of Heisenberg algebra of rank 1 over Rational Field
sage: X_L.display(exp1_frame)
p1 + 3*q1 = ∂/∂x_0 + 3 ∂/∂x_1 + (3/2*x_0 - 1/2*x_1) ∂/∂x_2
sage: X_R = G.right_invariant_extension(p + 3*q)
sage: X_R.display(exp1_frame)
p1 + 3*q1 = ∂/∂x_0 + 3 ∂/∂x_1 + (-3/2*x_0 + 1/2*x_1) ∂/∂x_2

Python

>>> from sage.all import *
>>> X_L = G.left_invariant_extension(p + Integer(3)*q); X_L
Vector field p1 + 3*q1 on the Lie group G of Heisenberg algebra of rank 1 over Rational Field
>>> X_L.display(exp1_frame)
p1 + 3*q1 = ∂/∂x_0 + 3 ∂/∂x_1 + (3/2*x_0 - 1/2*x_1) ∂/∂x_2
>>> X_R = G.right_invariant_extension(p + Integer(3)*q)
>>> X_R.display(exp1_frame)
p1 + 3*q1 = ∂/∂x_0 + 3 ∂/∂x_1 + (-3/2*x_0 + 1/2*x_1) ∂/∂x_2

The nilpotency step of the Lie group is the nilpotency step of its algebra. Nilpotency for Lie groups means that group commutators that are longer than the nilpotency step vanish:

Sage

sage: G.step()
2
sage: g = G.exp(p); h = G.exp(q)
sage: c = g*h*~g*~h; c
exp(z)
sage: g*c*~g*~c
exp(0)

Python

>>> from sage.all import *
>>> G.step()
2
>>> g = G.exp(p); h = G.exp(q)
>>> c = g*h*~g*~h; c
exp(z)
>>> g*c*~g*~c
exp(0)

class Element(parent, **kwds)[source]¶

Bases: ManifoldPoint, MultiplicativeGroupElement

A base class for an element of a Lie group.

EXAMPLES:

Elements of the group are printed in the default exponential coordinates:

Sage

sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: g = G.exp(2*X + 3*Z); g
exp(2*X + 3*Z)
sage: h = G.point([ var('a'), var('b'), 0]); h
exp(a*X + b*Y)
sage: G.set_default_chart(G.chart_exp2())
sage: g
exp(3*Z)exp(2*X)
sage: h
exp(1/2*a*b*Z)exp(b*Y)exp(a*X)

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2), names=('X', 'Y', 'Z',)); (X, Y, Z,) = L._first_ngens(3)
>>> G = L.lie_group()
>>> g = G.exp(Integer(2)*X + Integer(3)*Z); g
exp(2*X + 3*Z)
>>> h = G.point([ var('a'), var('b'), Integer(0)]); h
exp(a*X + b*Y)
>>> G.set_default_chart(G.chart_exp2())
>>> g
exp(3*Z)exp(2*X)
>>> h
exp(1/2*a*b*Z)exp(b*Y)exp(a*X)

Multiplication of two elements uses the usual product syntax:

Sage

sage: G.exp(Y)*G.exp(X)
exp(Y)exp(X)
sage: G.exp(X)*G.exp(Y)
exp(Z)exp(Y)exp(X)
sage: G.set_default_chart(G.chart_exp1())
sage: G.exp(X)*G.exp(Y)
exp(X + Y + 1/2*Z)

Python

>>> from sage.all import *
>>> G.exp(Y)*G.exp(X)
exp(Y)exp(X)
>>> G.exp(X)*G.exp(Y)
exp(Z)exp(Y)exp(X)
>>> G.set_default_chart(G.chart_exp1())
>>> G.exp(X)*G.exp(Y)
exp(X + Y + 1/2*Z)

adjoint(g)[source]¶

Return the adjoint map as an automorphism of the Lie algebra of self.

INPUT:

g – an element of self

For a Lie group element \(g\), the adjoint map \(\operatorname{Ad}_g\) is the map on the Lie algebra \(\mathfrak{g}\) given by the differential of the conjugation by \(g\) at the identity.

If the Lie algebra of self does not admit symbolic coefficients, the adjoint is not in general defined for abstract points.

EXAMPLES:

An example of an adjoint map:

Sage

sage: L = LieAlgebra(QQ, 2, step=3)
sage: G = L.lie_group()
sage: g = G.exp(L.basis().list()[0]); g
exp(X_1)
sage: Ad_g = G.adjoint(g); Ad_g
Lie algebra endomorphism of Free Nilpotent Lie algebra on 5
generators (X_1, X_2, X_12, X_112, X_122) over Rational Field
  Defn: X_1 |--> X_1
        X_2 |--> X_2 + X_12 + 1/2*X_112
        X_12 |--> X_12 + X_112
        X_112 |--> X_112
        X_122 |--> X_122

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(3))
>>> G = L.lie_group()
>>> g = G.exp(L.basis().list()[Integer(0)]); g
exp(X_1)
>>> Ad_g = G.adjoint(g); Ad_g
Lie algebra endomorphism of Free Nilpotent Lie algebra on 5
generators (X_1, X_2, X_12, X_112, X_122) over Rational Field
  Defn: X_1 |--> X_1
        X_2 |--> X_2 + X_12 + 1/2*X_112
        X_12 |--> X_12 + X_112
        X_112 |--> X_112
        X_122 |--> X_122

Usually the adjoint map of a symbolic point is not defined:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: g = G.point([var('a'), var('b'), var('c')]); g
exp(a*X_1 + b*X_2 + c*X_12)
sage: G.adjoint(g)
Traceback (most recent call last):
...
TypeError: unable to convert -b to a rational

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> g = G.point([var('a'), var('b'), var('c')]); g
exp(a*X_1 + b*X_2 + c*X_12)
>>> G.adjoint(g)
Traceback (most recent call last):
...
TypeError: unable to convert -b to a rational

However, if the adjoint map is independent from the symbolic terms, the map is still well defined:

Sage

sage: g = G.point([0, 0, var('a')]); g
exp(a*X_12)
sage: G.adjoint(g)
Lie algebra endomorphism of Free Nilpotent Lie algebra on 3 generators (X_1, X_2, X_12) over Rational Field
  Defn: X_1 |--> X_1
        X_2 |--> X_2
        X_12 |--> X_12

Python

>>> from sage.all import *
>>> g = G.point([Integer(0), Integer(0), var('a')]); g
exp(a*X_12)
>>> G.adjoint(g)
Lie algebra endomorphism of Free Nilpotent Lie algebra on 3 generators (X_1, X_2, X_12) over Rational Field
  Defn: X_1 |--> X_1
        X_2 |--> X_2
        X_12 |--> X_12

chart_exp1()[source]¶

Return the chart of exponential coordinates of the first kind.

Exponential coordinates of the first kind are

\[\exp(x_1X_1 + \cdots + x_nX_n) \mapsto (x_1, \ldots, x_n).\]

EXAMPLES:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: G.chart_exp1()
Chart (G, (x_1, x_2, x_12))

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> G.chart_exp1()
Chart (G, (x_1, x_2, x_12))

chart_exp2()[source]¶

Return the chart of exponential coordinates of the second kind.

Exponential coordinates of the second kind are

\[\exp(x_nX_n) \cdots \exp(x_1X_1) \mapsto (x_1, \ldots, x_n).\]

EXAMPLES:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: G.chart_exp2()
Chart (G, (y_1, y_2, y_12))

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> G.chart_exp2()
Chart (G, (y_1, y_2, y_12))

conjugation(g)[source]¶

Return the conjugation by g as an automorphism of self.

The conjugation by \(g\) on a Lie group \(G\) is the map

\[G \to G, \qquad h \mapsto ghg^{-1}.\]

INPUT:

g – an element of self

EXAMPLES:

A generic conjugation in the Heisenberg group:

Sage

sage: H = lie_algebras.Heisenberg(QQ, 1)
sage: p,q,z = H.basis()
sage: G = H.lie_group()
sage: g = G.point([var('a'), var('b'), var('c')])
sage: C_g = G.conjugation(g); C_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
sage: C_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (x_0, x_1, -b*x_0 + a*x_1 + x_2)

Python

>>> from sage.all import *
>>> H = lie_algebras.Heisenberg(QQ, Integer(1))
>>> p,q,z = H.basis()
>>> G = H.lie_group()
>>> g = G.point([var('a'), var('b'), var('c')])
>>> C_g = G.conjugation(g); C_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
>>> C_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (x_0, x_1, -b*x_0 + a*x_1 + x_2)

exp(X)[source]¶

Return the group element \(exp(X)\).

INPUT:

X – an element of the Lie algebra of self

EXAMPLES:

Sage

sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: G.exp(X)
exp(X)
sage: G.exp(Y)
exp(Y)
sage: G.exp(X + Y)
exp(X + Y)

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2), names=('X', 'Y', 'Z',)); (X, Y, Z,) = L._first_ngens(3)
>>> G = L.lie_group()
>>> G.exp(X)
exp(X)
>>> G.exp(Y)
exp(Y)
>>> G.exp(X + Y)
exp(X + Y)

gens()[source]¶

Return a tuple of elements whose one-parameter subgroups generate the Lie group.

EXAMPLES:

Sage

sage: L = lie_algebras.Heisenberg(QQ, 1)
sage: G = L.lie_group()
sage: G.gens()
(exp(p1), exp(q1), exp(z))

Python

>>> from sage.all import *
>>> L = lie_algebras.Heisenberg(QQ, Integer(1))
>>> G = L.lie_group()
>>> G.gens()
(exp(p1), exp(q1), exp(z))

left_invariant_extension(X, name=None)[source]¶

Return the left-invariant vector field that has the value X at the identity.

INPUT:

X – an element of the Lie algebra of self
name – (optional) a string to use as a name for the vector field; if nothing is given, the name of the vector X is used

EXAMPLES:

A left-invariant extension in the Heisenberg group:

Sage

sage: L = lie_algebras.Heisenberg(QQ, 1)
sage: p, q, z = L.basis()
sage: H = L.lie_group('H')
sage: X = H.left_invariant_extension(p); X
Vector field p1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
sage: X.display(H.chart_exp1().frame())
p1 = ∂/∂x_0 - 1/2*x_1 ∂/∂x_2

Python

>>> from sage.all import *
>>> L = lie_algebras.Heisenberg(QQ, Integer(1))
>>> p, q, z = L.basis()
>>> H = L.lie_group('H')
>>> X = H.left_invariant_extension(p); X
Vector field p1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
>>> X.display(H.chart_exp1().frame())
p1 = ∂/∂x_0 - 1/2*x_1 ∂/∂x_2

Default vs. custom naming for the invariant vector field:

Sage

sage: Y = H.left_invariant_extension(p + q); Y
Vector field p1 + q1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
sage: Z = H.left_invariant_extension(p + q, 'Z'); Z
Vector field Z on the Lie group H of Heisenberg algebra of rank 1 over Rational Field

Python

>>> from sage.all import *
>>> Y = H.left_invariant_extension(p + q); Y
Vector field p1 + q1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
>>> Z = H.left_invariant_extension(p + q, 'Z'); Z
Vector field Z on the Lie group H of Heisenberg algebra of rank 1 over Rational Field

left_invariant_frame(**kwds)[source]¶

Return the frame of left-invariant vector fields of self.

The labeling of the frame and the dual frame can be customized using keyword parameters as described in sage.manifolds.differentiable.manifold.DifferentiableManifold.vector_frame().

EXAMPLES:

The default left-invariant frame:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: livf = G.left_invariant_frame(); livf
Vector frame (G, (X_1,X_2,X_12))
sage: coord_frame = G.chart_exp1().frame()
sage: livf[0].display(coord_frame)
X_1 = ∂/∂x_1 - 1/2*x_2 ∂/∂x_12
sage: livf[1].display(coord_frame)
X_2 = ∂/∂x_2 + 1/2*x_1 ∂/∂x_12
sage: livf[2].display(coord_frame)
X_12 = ∂/∂x_12

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> livf = G.left_invariant_frame(); livf
Vector frame (G, (X_1,X_2,X_12))
>>> coord_frame = G.chart_exp1().frame()
>>> livf[Integer(0)].display(coord_frame)
X_1 = ∂/∂x_1 - 1/2*x_2 ∂/∂x_12
>>> livf[Integer(1)].display(coord_frame)
X_2 = ∂/∂x_2 + 1/2*x_1 ∂/∂x_12
>>> livf[Integer(2)].display(coord_frame)
X_12 = ∂/∂x_12

Examples of custom labeling for the frame:

Sage

sage: G.left_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
sage: G.left_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
sage: G.left_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

Python

>>> from sage.all import *
>>> G.left_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
>>> G.left_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
>>> G.left_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

left_translation(g)[source]¶

Return the left translation by g as an automorphism of self.

The left translation by \(g\) on a Lie group \(G\) is the map

\[G \to G, \qquad h \mapsto gh.\]

INPUT:

g – an element of self

EXAMPLES:

A left translation in the Heisenberg group:

Sage

sage: H = lie_algebras.Heisenberg(QQ, 1)
sage: p,q,z = H.basis()
sage: G = H.lie_group()
sage: g = G.exp(p)
sage: L_g = G.left_translation(g); L_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
sage: L_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
    (x_0, x_1, x_2) ↦ (x_0 + 1, x_1, 1/2*x_1 + x_2)

Python

>>> from sage.all import *
>>> H = lie_algebras.Heisenberg(QQ, Integer(1))
>>> p,q,z = H.basis()
>>> G = H.lie_group()
>>> g = G.exp(p)
>>> L_g = G.left_translation(g); L_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
>>> L_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
    (x_0, x_1, x_2) ↦ (x_0 + 1, x_1, 1/2*x_1 + x_2)

Left translation by a generic element:

Sage

sage: h = G.point([var('a'), var('b'), var('c')])
sage: L_h = G.left_translation(h)
sage: L_h.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, -1/2*b*x_0 + 1/2*a*x_1 + c + x_2)

Python

>>> from sage.all import *
>>> h = G.point([var('a'), var('b'), var('c')])
>>> L_h = G.left_translation(h)
>>> L_h.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, -1/2*b*x_0 + 1/2*a*x_1 + c + x_2)

lie_algebra()[source]¶

Return the Lie algebra of self.

EXAMPLES:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: G.lie_algebra() == L
True

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> G.lie_algebra() == L
True

livf(**kwds)[source]¶

Return the frame of left-invariant vector fields of self.

The labeling of the frame and the dual frame can be customized using keyword parameters as described in sage.manifolds.differentiable.manifold.DifferentiableManifold.vector_frame().

EXAMPLES:

The default left-invariant frame:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: livf = G.left_invariant_frame(); livf
Vector frame (G, (X_1,X_2,X_12))
sage: coord_frame = G.chart_exp1().frame()
sage: livf[0].display(coord_frame)
X_1 = ∂/∂x_1 - 1/2*x_2 ∂/∂x_12
sage: livf[1].display(coord_frame)
X_2 = ∂/∂x_2 + 1/2*x_1 ∂/∂x_12
sage: livf[2].display(coord_frame)
X_12 = ∂/∂x_12

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> livf = G.left_invariant_frame(); livf
Vector frame (G, (X_1,X_2,X_12))
>>> coord_frame = G.chart_exp1().frame()
>>> livf[Integer(0)].display(coord_frame)
X_1 = ∂/∂x_1 - 1/2*x_2 ∂/∂x_12
>>> livf[Integer(1)].display(coord_frame)
X_2 = ∂/∂x_2 + 1/2*x_1 ∂/∂x_12
>>> livf[Integer(2)].display(coord_frame)
X_12 = ∂/∂x_12

Examples of custom labeling for the frame:

Sage

sage: G.left_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
sage: G.left_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
sage: G.left_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

Python

>>> from sage.all import *
>>> G.left_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
>>> G.left_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
>>> G.left_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

log(x)[source]¶

Return the logarithm of the element x of self.

INPUT:

x – an element of self

The logarithm is by definition the inverse of exp().

If the Lie algebra of self does not admit symbolic coefficients, the logarithm is not defined for abstract, i.e. symbolic, points.

EXAMPLES:

The logarithm is the inverse of the exponential:

Sage

sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: G.log(G.exp(X)) == X
True
sage: G.log(G.exp(X)*G.exp(Y))
X + Y + 1/2*Z

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2), names=('X', 'Y', 'Z',)); (X, Y, Z,) = L._first_ngens(3)
>>> G = L.lie_group()
>>> G.log(G.exp(X)) == X
True
>>> G.log(G.exp(X)*G.exp(Y))
X + Y + 1/2*Z

The logarithm is not defined for abstract (symbolic) points:

Sage

sage: g = G.point([var('a'), 1, 2]); g
exp(a*X + Y + 2*Z)
sage: G.log(g)
Traceback (most recent call last):
...
TypeError: unable to convert a to a rational

Python

>>> from sage.all import *
>>> g = G.point([var('a'), Integer(1), Integer(2)]); g
exp(a*X + Y + 2*Z)
>>> G.log(g)
Traceback (most recent call last):
...
TypeError: unable to convert a to a rational

one()[source]¶

Return the identity element of self.

EXAMPLES:

Sage

sage: L = LieAlgebra(QQ, 2, step=4)
sage: G = L.lie_group()
sage: G.one()
exp(0)

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(4))
>>> G = L.lie_group()
>>> G.one()
exp(0)

right_invariant_extension(X, name=None)[source]¶

Return the right-invariant vector field that has the value X at the identity.

INPUT:

X – an element of the Lie algebra of self
name – (optional) a string to use as a name for the vector field; if nothing is given, the name of the vector X is used

EXAMPLES:

A right-invariant extension in the Heisenberg group:

Sage

sage: L = lie_algebras.Heisenberg(QQ, 1)
sage: p, q, z = L.basis()
sage: H = L.lie_group('H')
sage: X = H.right_invariant_extension(p); X
Vector field p1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
sage: X.display(H.chart_exp1().frame())
p1 = ∂/∂x_0 + 1/2*x_1 ∂/∂x_2

Python

>>> from sage.all import *
>>> L = lie_algebras.Heisenberg(QQ, Integer(1))
>>> p, q, z = L.basis()
>>> H = L.lie_group('H')
>>> X = H.right_invariant_extension(p); X
Vector field p1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
>>> X.display(H.chart_exp1().frame())
p1 = ∂/∂x_0 + 1/2*x_1 ∂/∂x_2

Default vs. custom naming for the invariant vector field:

Sage

sage: Y = H.right_invariant_extension(p + q); Y
Vector field p1 + q1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
sage: Z = H.right_invariant_extension(p + q, 'Z'); Z
Vector field Z on the Lie group H of Heisenberg algebra of rank 1 over Rational Field

Python

>>> from sage.all import *
>>> Y = H.right_invariant_extension(p + q); Y
Vector field p1 + q1 on the Lie group H of Heisenberg algebra of rank 1 over Rational Field
>>> Z = H.right_invariant_extension(p + q, 'Z'); Z
Vector field Z on the Lie group H of Heisenberg algebra of rank 1 over Rational Field

right_invariant_frame(**kwds)[source]¶

Return the frame of right-invariant vector fields of self.

The labeling of the frame and the dual frame can be customized using keyword parameters as described in sage.manifolds.differentiable.manifold.DifferentiableManifold.vector_frame().

EXAMPLES:

The default right-invariant frame:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: rivf = G.right_invariant_frame(); rivf
Vector frame (G, (XR_1,XR_2,XR_12))
sage: coord_frame = G.chart_exp1().frame()
sage: rivf[0].display(coord_frame)
XR_1 = ∂/∂x_1 + 1/2*x_2 ∂/∂x_12
sage: rivf[1].display(coord_frame)
XR_2 = ∂/∂x_2 - 1/2*x_1 ∂/∂x_12
sage: rivf[2].display(coord_frame)
XR_12 = ∂/∂x_12

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> rivf = G.right_invariant_frame(); rivf
Vector frame (G, (XR_1,XR_2,XR_12))
>>> coord_frame = G.chart_exp1().frame()
>>> rivf[Integer(0)].display(coord_frame)
XR_1 = ∂/∂x_1 + 1/2*x_2 ∂/∂x_12
>>> rivf[Integer(1)].display(coord_frame)
XR_2 = ∂/∂x_2 - 1/2*x_1 ∂/∂x_12
>>> rivf[Integer(2)].display(coord_frame)
XR_12 = ∂/∂x_12

Examples of custom labeling for the frame:

Sage

sage: G.right_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
sage: G.right_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
sage: G.right_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

Python

>>> from sage.all import *
>>> G.right_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
>>> G.right_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
>>> G.right_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

right_translation(g)[source]¶

Return the right translation by g as an automorphism of self.

The right translation by \(g\) on a Lie group \(G\) is the map

\[G \to G, \qquad h\mapsto hg.\]

INPUT:

g – an element of self

EXAMPLES:

A right translation in the Heisenberg group:

Sage

sage: H = lie_algebras.Heisenberg(QQ, 1)
sage: p,q,z = H.basis()
sage: G = H.lie_group()
sage: g = G.exp(p)
sage: R_g = G.right_translation(g); R_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
sage: R_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (x_0 + 1, x_1, -1/2*x_1 + x_2)

Python

>>> from sage.all import *
>>> H = lie_algebras.Heisenberg(QQ, Integer(1))
>>> p,q,z = H.basis()
>>> G = H.lie_group()
>>> g = G.exp(p)
>>> R_g = G.right_translation(g); R_g
Diffeomorphism of the Lie group G of Heisenberg algebra of rank 1 over Rational Field
>>> R_g.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (x_0 + 1, x_1, -1/2*x_1 + x_2)

Right translation by a generic element:

Sage

sage: h = G.point([var('a'), var('b'), var('c')])
sage: R_h = G.right_translation(h)
sage: R_h.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, 1/2*b*x_0 - 1/2*a*x_1 + c + x_2)

Python

>>> from sage.all import *
>>> h = G.point([var('a'), var('b'), var('c')])
>>> R_h = G.right_translation(h)
>>> R_h.display(chart1=G.chart_exp1(), chart2=G.chart_exp1())
G → G
   (x_0, x_1, x_2) ↦ (a + x_0, b + x_1, 1/2*b*x_0 - 1/2*a*x_1 + c + x_2)

rivf(**kwds)[source]¶

Return the frame of right-invariant vector fields of self.

The labeling of the frame and the dual frame can be customized using keyword parameters as described in sage.manifolds.differentiable.manifold.DifferentiableManifold.vector_frame().

EXAMPLES:

The default right-invariant frame:

Sage

sage: L = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: rivf = G.right_invariant_frame(); rivf
Vector frame (G, (XR_1,XR_2,XR_12))
sage: coord_frame = G.chart_exp1().frame()
sage: rivf[0].display(coord_frame)
XR_1 = ∂/∂x_1 + 1/2*x_2 ∂/∂x_12
sage: rivf[1].display(coord_frame)
XR_2 = ∂/∂x_2 - 1/2*x_1 ∂/∂x_12
sage: rivf[2].display(coord_frame)
XR_12 = ∂/∂x_12

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(2))
>>> G = L.lie_group()
>>> rivf = G.right_invariant_frame(); rivf
Vector frame (G, (XR_1,XR_2,XR_12))
>>> coord_frame = G.chart_exp1().frame()
>>> rivf[Integer(0)].display(coord_frame)
XR_1 = ∂/∂x_1 + 1/2*x_2 ∂/∂x_12
>>> rivf[Integer(1)].display(coord_frame)
XR_2 = ∂/∂x_2 - 1/2*x_1 ∂/∂x_12
>>> rivf[Integer(2)].display(coord_frame)
XR_12 = ∂/∂x_12

Examples of custom labeling for the frame:

Sage

sage: G.right_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
sage: G.right_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
sage: G.right_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

Python

>>> from sage.all import *
>>> G.right_invariant_frame(symbol='Y')
Vector frame (G, (Y_1,Y_2,Y_12))
>>> G.right_invariant_frame(symbol='Z', indices=None)
Vector frame (G, (Z_0,Z_1,Z_2))
>>> G.right_invariant_frame(symbol='W', indices=('a','b','c'))
Vector frame (G, (W_a,W_b,W_c))

step()[source]¶

Return the nilpotency step of self.

EXAMPLES:

Sage

sage: L = LieAlgebra(QQ, 2, step=4)
sage: G = L.lie_group()
sage: G.step()
4

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(4))
>>> G = L.lie_group()
>>> G.step()
4