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
inlie_group
.INPUT:
lie_group
– (optional) the Lie group to map into; Iflie_group
is not given, the Lie group associated to the parent Lie algebra ofself
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
andx
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: a, b, c = L.lie_algebra_generators() # needs sage.combinat sage.modules sage: a.killing_form(b) # needs sage.combinat 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 elementself
ofg
(whereg
is the parent ofself
).Implement this if you implement
g.module()
. SeeLieAlgebras.module()
for how this is to be done.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: u = L((1, 0, 0)).to_vector(); u # needs sage.combinat sage.modules (1, 0, 0) sage: parent(u) # needs sage.combinat sage.modules Vector space of dimension 3 over Rational Field
- class FiniteDimensional(base_category)#
Bases:
CategoryWithAxiom_over_base_ring
- 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: C = LieAlgebras(GF(5)).FiniteDimensional() # needs sage.rings.finite_rings sage: C.is_subcategory(Sets().Finite()) # needs sage.rings.finite_rings True sage: C = LieAlgebras(ZZ).FiniteDimensional() sage: C.is_subcategory(Sets().Finite()) False sage: C = LieAlgebras(GF(5)).WithBasis().FiniteDimensional() # needs sage.rings.finite_rings sage: C.is_subcategory(Sets().Finite()) # needs sage.rings.finite_rings 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
sinceself
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 ofself
has a coercion from the rationals.INPUT:
X
– an element ofself
Y
– an element ofself
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 ofself
has a coercion from the rationals.INPUT:
X
– an element ofself
Y
– an element ofself
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 coercinglhs
andrhs
into elements ofself
.If
lhs
andrhs
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 vectorv
inself.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.combinat sage.modules sage: u = L.from_vector(vector(QQ, (1, 0, 0))); u # needs sage.combinat sage.modules (1, 0, 0) sage: parent(u) is L # needs sage.combinat sage.modules True
- ideal(*gens, **kwds)#
Return the ideal of
self
generated bygens
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: a, b, c = L.lie_algebra_generators() # needs sage.combinat sage.modules sage: L.ideal([2*a - c, b + c]) # needs sage.combinat 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 toself
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 ofA
.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.combinat sage.modules sage: L.is_nilpotent() # needs sage.combinat sage.modules True
- is_solvable()#
Return if
self
is a solvable Lie algebra.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: L.is_solvable() # needs sage.combinat sage.modules True
- killing_form(x, y)#
Return the Killing form of
x
andy
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: a, b, c = L.lie_algebra_generators() # needs sage.combinat sage.modules sage: L.killing_form(a, b + c) # needs sage.combinat 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 ofself
(the latter is implemented asuniversal_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 inself
) via an isomorphism fromself
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 thatself
does not have.For instance, if
self
has ordered basis \((e, f, h)\), thenself.module()
will be the \(R\)-module \(R^3\), and the elements \(e\), \(f\) and \(h\) ofself
will correspond to the basis vectors \((1, 0, 0)\), \((0, 1, 0)\) and \((0, 0, 1)\) ofself.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 fromself
toself.module()
in both directions; that is, implement:a
to_vector
ElementMethod which sends every element ofself
to the corresponding element ofself.module()
;a
from_vector
ParentMethod which sends every element ofself.module()
to an element ofself
.
The
from_vector
method will automatically serve as an element constructor ofself
(that is,self(v)
for anyv
inself.module()
will returnself.from_vector(v)
).Todo
Ensure that this is actually so.
EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: L.module() # needs sage.combinat sage.modules Vector space of dimension 3 over Rational Field
- subalgebra(gens, names=None, index_set=None, category=None)#
Return the subalgebra of
self
generated bygens
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat sage.modules sage: a, b, c = L.lie_algebra_generators() # needs sage.combinat sage.modules sage: L.subalgebra([2*a - c, b + c]) # needs sage.combinat 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
- universal_enveloping_algebra()#
Return the universal enveloping algebra of
self
.EXAMPLES:
sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() # needs sage.combinat 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
- 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]