Lie Algebras

AUTHORS:

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

class sage.categories.lie_algebras.LieAlgebras(base, name=None)

Bases: Category_over_base_ring

The category of Lie algebras.

EXAMPLES:

sage: C = LieAlgebras(QQ); C
Category of Lie algebras over Rational Field
sage: sorted(C.super_categories(), key=str)
[Category of vector spaces over Rational Field]

We construct a typical parent in this category, and do some computations with it:

sage: A = C.example(); A                                                        # needs sage.groups sage.modules
An example of a Lie algebra: the Lie algebra from the associative
 algebra Symmetric group algebra of order 3 over Rational Field
 generated by ([2, 1, 3], [2, 3, 1])

sage: A.category()                                                              # needs sage.groups sage.modules
Category of Lie algebras over Rational Field

sage: A.base_ring()                                                             # needs sage.groups sage.modules
Rational Field

sage: a, b = A.lie_algebra_generators()                                         # needs sage.groups sage.modules
sage: a.bracket(b)                                                              # needs sage.groups sage.modules
-[1, 3, 2] + [3, 2, 1]
sage: b.bracket(2*a + b)                                                        # needs sage.groups sage.modules
2*[1, 3, 2] - 2*[3, 2, 1]

sage: A.bracket(a, b)                                                           # needs sage.groups sage.modules
-[1, 3, 2] + [3, 2, 1]

Please see the source code of \(A\) (with A??) for how to implement other Lie algebras.

Todo

Many of these tests should use Lie algebras that are not the minimal example and need to be added after github issue #16820 (and github issue #16823).

class ElementMethods

Bases: object

bracket(rhs)

Return the Lie bracket [self, rhs].

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L = LieAlgebras(QQ).example()
sage: x,y = L.lie_algebra_generators()
sage: x.bracket(y)
-[1, 3, 2] + [3, 2, 1]
sage: x.bracket(0)
0

exp(lie_group=None)

Return the exponential of self in lie_group.

INPUT:

• lie_group – (optional) the Lie group to map into; If lie_group is not given, the Lie group associated to the parent Lie algebra of self is used.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: g = (X + Y + Z).exp(); g                                          # needs sage.symbolic
exp(X + Y + Z)
sage: h = X.exp(); h                                                    # needs sage.symbolic
exp(X)
sage: g.parent()                                                        # needs sage.symbolic
Lie group G of Free Nilpotent Lie algebra on 3 generators (X, Y, Z)
 over Rational Field
sage: g.parent() is h.parent()                                          # needs sage.symbolic
True

The Lie group can be specified explicitly:

sage: # needs sage.combinat sage.modules sage.symbolic
sage: H = L.lie_group('H')
sage: k = Z.exp(lie_group=H); k
exp(Z)
sage: k.parent()
Lie group H of Free Nilpotent Lie algebra on 3 generators (X, Y, Z)
 over Rational Field
sage: g.parent() == k.parent()
False

killing_form(x)

Return the Killing form of self and x.

EXAMPLES:

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

lift()

Return the image of self under the canonical lift from the Lie algebra to its universal enveloping algebra.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: elt = 3*a + b - c
sage: elt.lift()
3*b0 + b1 - b2

sage: L.<x,y> = LieAlgebra(QQ, abelian=True)                            # needs sage.combinat sage.modules
sage: x.lift()                                                          # needs sage.combinat sage.modules
x

to_vector(order=None)

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 LieAlgebras.module() for how this is to be done.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()     # needs sage.modules
sage: u = L((1, 0, 0)).to_vector(); u                                   # needs sage.modules
(1, 0, 0)
sage: parent(u)                                                         # needs sage.modules
Vector space of dimension 3 over Rational Field

class FiniteDimensional(base_category)

Bases: CategoryWithAxiom_over_base_ring

WithBasis

alias of FiniteDimensionalLieAlgebrasWithBasis

extra_super_categories()

Implements the fact that a finite dimensional Lie algebra over a finite ring is finite.

EXAMPLES:

sage: LieAlgebras(IntegerModRing(4)).FiniteDimensional().extra_super_categories()
[Category of finite sets]
sage: LieAlgebras(ZZ).FiniteDimensional().extra_super_categories()
[]
sage: C = LieAlgebras(GF(5)).FiniteDimensional()
sage: C.is_subcategory(Sets().Finite())
True
sage: C = LieAlgebras(ZZ).FiniteDimensional()
sage: C.is_subcategory(Sets().Finite())
False
sage: C = LieAlgebras(GF(5)).WithBasis().FiniteDimensional()
sage: C.is_subcategory(Sets().Finite())
True

Graded

alias of GradedLieAlgebras

class Nilpotent(base_category)

Bases: CategoryWithAxiom_over_base_ring

Category of nilpotent Lie algebras.

class ParentMethods

Bases: object

is_nilpotent()

Return True since self is nilpotent.

EXAMPLES:

sage: h = lie_algebras.Heisenberg(ZZ, oo)                           # needs sage.combinat sage.modules
sage: h.is_nilpotent()                                              # needs sage.combinat sage.modules
True

step()

Return the nilpotency step of self.

EXAMPLES:

sage: h = lie_algebras.Heisenberg(ZZ, oo)                           # needs sage.combinat sage.modules
sage: h.step()                                                      # needs sage.combinat sage.modules
2

class ParentMethods

Bases: object

baker_campbell_hausdorff(X, Y, prec=None)

Return the element \(\log(\exp(X)\exp(Y))\).

The BCH formula is an expression for \(\log(\exp(X)\exp(Y))\) as a sum of Lie brackets of X ` and ``Y with rational coefficients. It is only defined if the base ring of self has a coercion from the rationals.

INPUT:

• X – an element of self

• Y – an element of self

• prec – an integer; the maximum length of Lie brackets to be considered in the formula

EXAMPLES:

The BCH formula for the generators of a free nilpotent Lie algebra of step 4:

sage: L = LieAlgebra(QQ, 2, step=4)                                     # needs sage.combinat sage.modules
sage: L.inject_variables()                                              # needs sage.combinat sage.modules
Defining X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222
sage: L.bch(X_1, X_2)                                                   # needs sage.combinat sage.modules
X_1 + X_2 + 1/2*X_12 + 1/12*X_112 + 1/12*X_122 + 1/24*X_1122

An example of the BCH formula in a quotient:

sage: Q = L.quotient(X_112 + X_122)                                     # needs sage.combinat sage.modules
sage: x, y = Q.basis().list()[:2]                                       # needs sage.combinat sage.modules
sage: Q.bch(x, y)                                                       # needs sage.combinat sage.modules
X_1 + X_2 + 1/2*X_12 - 1/24*X_1112

The BCH formula for a non-nilpotent Lie algebra requires the precision to be explicitly stated:

sage: L.<X,Y> = LieAlgebra(QQ)                                          # needs sage.combinat sage.modules
sage: L.bch(X, Y)                                                       # needs sage.combinat sage.modules
Traceback (most recent call last):
...
ValueError: the Lie algebra is not known to be nilpotent,
so you must specify the precision
sage: L.bch(X, Y, 4)                                                    # needs sage.combinat sage.modules
X + 1/12*[X, [X, Y]] + 1/24*[X, [[X, Y], Y]]
+ 1/2*[X, Y] + 1/12*[[X, Y], Y] + Y

The BCH formula requires a coercion from the rationals:

sage: L.<X,Y,Z> = LieAlgebra(ZZ, 2, step=2)                             # needs sage.combinat sage.modules
sage: L.bch(X, Y)                                                       # needs sage.combinat sage.modules
Traceback (most recent call last):
...
TypeError: the BCH formula is not well defined
since Integer Ring has no coercion from Rational Field

bch(X, Y, prec=None)

Return the element \(\log(\exp(X)\exp(Y))\).

The BCH formula is an expression for \(\log(\exp(X)\exp(Y))\) as a sum of Lie brackets of X ` and ``Y with rational coefficients. It is only defined if the base ring of self has a coercion from the rationals.

INPUT:

• X – an element of self

• Y – an element of self

• prec – an integer; the maximum length of Lie brackets to be considered in the formula

EXAMPLES:

The BCH formula for the generators of a free nilpotent Lie algebra of step 4:

sage: L = LieAlgebra(QQ, 2, step=4)                                     # needs sage.combinat sage.modules
sage: L.inject_variables()                                              # needs sage.combinat sage.modules
Defining X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222
sage: L.bch(X_1, X_2)                                                   # needs sage.combinat sage.modules
X_1 + X_2 + 1/2*X_12 + 1/12*X_112 + 1/12*X_122 + 1/24*X_1122

An example of the BCH formula in a quotient:

sage: Q = L.quotient(X_112 + X_122)                                     # needs sage.combinat sage.modules
sage: x, y = Q.basis().list()[:2]                                       # needs sage.combinat sage.modules
sage: Q.bch(x, y)                                                       # needs sage.combinat sage.modules
X_1 + X_2 + 1/2*X_12 - 1/24*X_1112

The BCH formula for a non-nilpotent Lie algebra requires the precision to be explicitly stated:

sage: L.<X,Y> = LieAlgebra(QQ)                                          # needs sage.combinat sage.modules
sage: L.bch(X, Y)                                                       # needs sage.combinat sage.modules
Traceback (most recent call last):
...
ValueError: the Lie algebra is not known to be nilpotent,
so you must specify the precision
sage: L.bch(X, Y, 4)                                                    # needs sage.combinat sage.modules
X + 1/12*[X, [X, Y]] + 1/24*[X, [[X, Y], Y]]
+ 1/2*[X, Y] + 1/12*[[X, Y], Y] + Y

The BCH formula requires a coercion from the rationals:

sage: L.<X,Y,Z> = LieAlgebra(ZZ, 2, step=2)                             # needs sage.combinat sage.modules
sage: L.bch(X, Y)                                                       # needs sage.combinat sage.modules
Traceback (most recent call last):
...
TypeError: the BCH formula is not well defined
since Integer Ring has no coercion from Rational Field

bracket(lhs, rhs)

Return the Lie bracket [lhs, rhs] after coercing lhs and rhs into elements of self.

If lhs and rhs are Lie algebras, then this constructs the product space, and if only one of them is a Lie algebra, then it constructs the corresponding ideal.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L = LieAlgebras(QQ).example()
sage: x, y = L.lie_algebra_generators()
sage: L.bracket(x, x + y)
-[1, 3, 2] + [3, 2, 1]
sage: L.bracket(x, 0)
0
sage: L.bracket(0, x)
0

Constructing the product space:

sage: L = lie_algebras.Heisenberg(QQ, 1)                                # needs sage.combinat sage.modules
sage: Z = L.bracket(L, L); Z                                            # needs sage.combinat sage.modules
Ideal (z) of Heisenberg algebra of rank 1 over Rational Field
sage: L.bracket(L, Z)                                                   # needs sage.combinat sage.modules
Ideal () of Heisenberg algebra of rank 1 over Rational Field

Constructing ideals:

sage: p, q, z = L.basis(); p, q, z                                      # needs sage.combinat sage.modules
(p1, q1, z)
sage: L.bracket(3*p, L)                                                 # needs sage.combinat sage.modules
Ideal (3*p1) of Heisenberg algebra of rank 1 over Rational Field
sage: L.bracket(L, q + p)                                               # needs sage.combinat sage.modules
Ideal (p1 + q1) of Heisenberg algebra of rank 1 over Rational Field

from_vector(v, order=None, coerce=False)

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

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

EXAMPLES:

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

ideal(*gens, **kwds)

Return the ideal of self generated by gens.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()     # needs sage.modules
sage: a, b, c = L.lie_algebra_generators()                              # needs sage.modules
sage: L.ideal([2*a - c, b + c])                                         # needs sage.modules
An example of a finite dimensional Lie algebra with basis:
 the 2-dimensional abelian Lie algebra over Rational Field
 with basis matrix:
[   1    0 -1/2]
[   0    1    1]

sage: L = LieAlgebras(QQ).example()                                     # needs sage.combinat sage.modules
sage: x, y = L.lie_algebra_generators()                                 # needs sage.combinat sage.modules
sage: L.ideal([x + y])                                                  # needs sage.combinat sage.modules
Traceback (most recent call last):
...
NotImplementedError: ideals not yet implemented: see #16824

is_abelian()

Return True if this Lie algebra is abelian.

A Lie algebra \(\mathfrak{g}\) is abelian if \([x, y] = 0\) for all \(x, y \in \mathfrak{g}\).

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L = LieAlgebras(QQ).example()
sage: L.is_abelian()
False
sage: R = QQ['x,y']
sage: L = LieAlgebras(QQ).example(R.gens())
sage: L.is_abelian()
True

sage: # not implemented, needs sage.combinat sage.modules
sage: L.<x> = LieAlgebra(QQ, 1)
sage: L.is_abelian()
True
sage: L.<x,y> = LieAlgebra(QQ, 2)
sage: L.is_abelian()
False

is_commutative()

Return if self is commutative. This is equivalent to self being abelian.

EXAMPLES:

sage: L = LieAlgebras(QQ).example()                                     # needs sage.combinat sage.modules
sage: L.is_commutative()                                                # needs sage.combinat sage.modules
False

sage: L.<x> = LieAlgebra(QQ, 1)         # not implemented               # needs sage.combinat sage.modules
sage: L.is_commutative()        # not implemented                       # needs sage.combinat sage.modules
True

is_ideal(A)

Return if self is an ideal of A.

EXAMPLES:

sage: L = LieAlgebras(QQ).example()                                     # needs sage.combinat sage.modules
sage: L.is_ideal(L)                                                     # needs sage.combinat sage.modules
True

is_nilpotent()

Return if self is a nilpotent Lie algebra.

EXAMPLES:

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

is_solvable()

Return if self is a solvable Lie algebra.

EXAMPLES:

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

killing_form(x, y)

Return the Killing form of x and y.

EXAMPLES:

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

lie_group(name='G', **kwds)

Return the simply connected Lie group related to self.

INPUT:

• name – string (default: 'G'); the name (symbol) given to the Lie group

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 1)                                # needs sage.combinat sage.modules
sage: G = L.lie_group('G'); G                                           # needs sage.combinat sage.modules sage.symbolic
Lie group G of Heisenberg algebra of rank 1 over Rational Field

lift()

Construct the lift morphism from self to the universal enveloping algebra of self (the latter is implemented as universal_enveloping_algebra()).

This is a Lie algebra homomorphism. It is injective if self is a free module over its base ring, or if the base ring is a \(\QQ\)-algebra.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: lifted = L.lift(2*a + b - c); lifted
2*b0 + b1 - b2
sage: lifted.parent() is L.universal_enveloping_algebra()
True

module()

Return an \(R\)-module which is isomorphic to the underlying \(R\)-module of self.

The rationale behind this method is to enable linear algebraic functionality on self (such as computing the span of a list of vectors in self) via an isomorphism from self to an \(R\)-module (typically, although not always, an \(R\)-module of the form \(R^n\) for an \(n \in \NN\)) on which such functionality already exists. For this method to be of any use, it should return an \(R\)-module which has linear algebraic functionality that self does not have.

For instance, if self has ordered basis \((e, f, h)\), then self.module() will be the \(R\)-module \(R^3\), and the elements \(e\), \(f\) and \(h\) of self will correspond to the basis vectors \((1, 0, 0)\), \((0, 1, 0)\) and \((0, 0, 1)\) of self.module().

This method module() needs to be set whenever a finite-dimensional Lie algebra with basis is intended to support linear algebra (which is, e.g., used in the computation of centralizers and lower central series). One then needs to also implement the \(R\)-module isomorphism from self to self.module() in both directions; that is, implement:

• a to_vector ElementMethod which sends every element of self to the corresponding element of self.module();

• a from_vector ParentMethod which sends every element of self.module() to an element of self.

The from_vector method will automatically serve as an element constructor of self (that is, self(v) for any v in self.module() will return self.from_vector(v)).

Todo

Ensure that this is actually so.

EXAMPLES:

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

representation(f=None, index_set=None, on_basis=False, **kwargs)

Return a representation of self.

If no arguments are given, then this returns the trivial representation.

Currently the only implemented method of constructing a representation is by explicitly specifying the action of

• the elements of self by matrices;

• the basis elements of self using a dict or a Family();

• a function on basis elements (either passed as on_basis or setting on_basis=True).

INPUT:

• f – the function that defines the action

• index_set – the index set of the representation

• on_basis – (optional) see above

See also

RepresentationByMorphism

EXAMPLES:

sage: L.<x,y> = LieAlgebra(QQ, {('x','y'): {'y':1}})
sage: f = {x: Matrix([[1,0],[0,0]]), y: Matrix([[0,1],[0,0]])}
sage: L.representation(f)
Representation of Lie algebra on 2 generators (x, y) over Rational Field defined by:
       [1 0]
x |--> [0 0]
       [0 1]
y |--> [0 0]
sage: L.representation()
Trivial representation of Lie algebra on 2 generators (x, y) over Rational Field

subalgebra(gens, names=None, index_set=None, category=None)

Return the subalgebra of self generated by gens.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()     # needs sage.modules
sage: a, b, c = L.lie_algebra_generators()                              # needs sage.modules
sage: L.subalgebra([2*a - c, b + c])                                    # needs sage.modules
An example of a finite dimensional Lie algebra with basis:
 the 2-dimensional abelian Lie algebra over Rational Field
 with basis matrix:
[   1    0 -1/2]
[   0    1    1]

sage: L = LieAlgebras(QQ).example()                                     # needs sage.combinat sage.modules
sage: x,y = L.lie_algebra_generators()                                  # needs sage.combinat sage.modules
sage: L.subalgebra([x + y])                                             # needs sage.combinat sage.modules
Traceback (most recent call last):
...
NotImplementedError: subalgebras not yet implemented: see #17416

trivial_representation()

Return the trivial representation of self.

EXAMPLES:

sage: L = lie_algebras.strictly_upper_triangular_matrices(QQ, 4)
sage: L.trivial_representation()
Trivial representation of Lie algebra of 4-dimensional
 strictly upper triangular matrices over Rational Field

universal_enveloping_algebra()

Return the universal enveloping algebra of self.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()     # needs sage.modules
sage: L.universal_enveloping_algebra()                                  # needs sage.combinat sage.modules
Noncommutative Multivariate Polynomial Ring in b0, b1, b2
 over Rational Field, nc-relations: {}

sage: L = LieAlgebra(QQ, 3, 'x', abelian=True)                          # needs sage.combinat sage.modules
sage: L.universal_enveloping_algebra()                                  # needs sage.combinat sage.modules
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field

See also

lift()

class SubcategoryMethods

Bases: object

Nilpotent()

Return the full subcategory of nilpotent objects of self.

A Lie algebra \(L\) is nilpotent if there exist an integer \(s\) such that all iterated brackets of \(L\) of length more than \(s\) vanish. The integer \(s\) is called the nilpotency step. For instance any abelian Lie algebra is nilpotent of step 1.

EXAMPLES:

sage: LieAlgebras(QQ).Nilpotent()
Category of nilpotent Lie algebras over Rational Field
sage: LieAlgebras(QQ).WithBasis().Nilpotent()
Category of nilpotent Lie algebras with basis over Rational Field

WithBasis

alias of LieAlgebrasWithBasis

example(gens=None)

Return an example of a Lie algebra as per Category.example.

EXAMPLES:

sage: LieAlgebras(QQ).example()                                             # needs sage.groups sage.modules
An example of a Lie algebra: the Lie algebra from the associative algebra
 Symmetric group algebra of order 3 over Rational Field
 generated by ([2, 1, 3], [2, 3, 1])

Another set of generators can be specified as an optional argument:

sage: F.<x,y,z> = FreeAlgebra(QQ)                                           # needs sage.combinat sage.modules
sage: LieAlgebras(QQ).example(F.gens())                                     # needs sage.combinat sage.modules
An example of a Lie algebra: the Lie algebra from the associative algebra
 Free Algebra on 3 generators (x, y, z) over Rational Field
 generated by (x, y, z)

super_categories()

EXAMPLES:

sage: LieAlgebras(QQ).super_categories()
[Category of vector spaces over Rational Field]

class sage.categories.lie_algebras.LiftMorphism(domain, codomain)

Bases: Morphism

The natural lifting morphism from a Lie algebra to its enveloping algebra.