# Lie Algebras¶

AUTHORS:

• Travis Scrimshaw (07-15-2013): Initial implementation
class sage.categories.lie_algebras.LieAlgebras(base, name=None)

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
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()
Category of Lie algebras over Rational Field

sage: A.base_ring()
Rational Field

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

sage: A.bracket(a, b)
-[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 trac ticket #16820 (and trac ticket #16823).

class ElementMethods

Bases: object

bracket(rhs)

Return the Lie bracket [self, rhs].

EXAMPLES:

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


The Lie group can be specified explicitly:

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()
sage: a, b, c = L.lie_algebra_generators()
sage: a.killing_form(b)
0

lift()

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

EXAMPLES:

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)
sage: x.lift()
x

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

EXAMPLES:

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

class FiniteDimensional(base_category)
WithBasis
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: LieAlgebras(GF(5)).FiniteDimensional().is_subcategory(Sets().Finite())
True
sage: LieAlgebras(ZZ).FiniteDimensional().is_subcategory(Sets().Finite())
False
sage: LieAlgebras(GF(5)).WithBasis().FiniteDimensional().is_subcategory(Sets().Finite())
True

Graded
class Nilpotent(base_category)

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)
sage: h.is_nilpotent()
True

step()

Return the nilpotency step of self.

EXAMPLES:

sage: h = lie_algebras.Heisenberg(ZZ, oo)
sage: h.step()
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)
sage: L.inject_variables()
Defining X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222
sage: L.bch(X_1, X_2)
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)
sage: x, y = Q.basis().list()[:2]
sage: Q.bch(x, y)
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)
sage: L.bch(X, Y)
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)
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)
sage: L.bch(X, Y)
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)
sage: L.inject_variables()
Defining X_1, X_2, X_12, X_112, X_122, X_1112, X_1122, X_1222
sage: L.bch(X_1, X_2)
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)
sage: x, y = Q.basis().list()[:2]
sage: Q.bch(x, y)
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)
sage: L.bch(X, Y)
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)
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)
sage: L.bch(X, Y)
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: 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)
sage: Z = L.bracket(L, L); Z
Ideal (z) of Heisenberg algebra of rank 1 over Rational Field
sage: L.bracket(L, Z)
Ideal () of Heisenberg algebra of rank 1 over Rational Field


Constructing ideals:

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

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 the latter 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

ideal(*gens, **kwds)

Return the ideal of self generated by gens.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: L.ideal([2*a - c, b + c])
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()
sage: x,y = L.lie_algebra_generators()
sage: L.ideal([x + y])
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: 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: L.<x> = LieAlgebra(QQ,1)  # todo: not implemented - #16823
sage: L.is_abelian()  # todo: not implemented - #16823
True
sage: L.<x,y> = LieAlgebra(QQ,2)  # todo: not implemented - #16823
sage: L.is_abelian()  # todo: not implemented - #16823
False

is_commutative()

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

EXAMPLES:

sage: L = LieAlgebras(QQ).example()
sage: L.is_commutative()
False

sage: L.<x> = LieAlgebra(QQ, 1) # todo: not implemented - #16823
sage: L.is_commutative() # todo: not implemented - #16823
True

is_ideal(A)

Return if self is an ideal of A.

EXAMPLES:

sage: L = LieAlgebras(QQ).example()
sage: L.is_ideal(L)
True

is_nilpotent()

Return if self is a nilpotent Lie algebra.

EXAMPLES:

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

is_solvable()

Return if self is a solvable Lie algebra.

EXAMPLES:

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

killing_form(x, y)

Return the Killing form of x and y.

EXAMPLES:

sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example()
sage: a, b, c = L.lie_algebra_generators()
sage: L.killing_form(a, b+c)
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)
sage: G = L.lie_group('G'); G
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: 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()
sage: L.module()
Vector space of dimension 3 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()
sage: a, b, c = L.lie_algebra_generators()
sage: L.subalgebra([2*a - c, b + c])
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()
sage: x,y = L.lie_algebra_generators()
sage: L.subalgebra([x + y])
Traceback (most recent call last):
...
NotImplementedError: subalgebras not yet implemented: see #17416

universal_enveloping_algebra()

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: {}

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

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
example(gens=None)

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

EXAMPLES:

sage: LieAlgebras(QQ).example()
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)
sage: LieAlgebras(QQ).example(F.gens())
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)

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