Examples of a finite dimensional Lie algebra with basis¶
- class sage.categories.examples.finite_dimensional_lie_algebras_with_basis.AbelianLieAlgebra(R, n=None, M=None, ambient=None)[source]¶
- Bases: - Parent,- UniqueRepresentation- An example of a finite dimensional Lie algebra with basis: the abelian Lie algebra. - Let \(R\) be a commutative ring, and \(M\) an \(R\)-module. The abelian Lie algebra on \(M\) is the \(R\)-Lie algebra obtained by endowing \(M\) with the trivial Lie bracket (\([a, b] = 0\) for all \(a, b \in M\)). - This class illustrates a minimal implementation of a finite dimensional Lie algebra with basis. - INPUT: - R– base ring
- n– (optional) a nonnegative integer (default:- None)
- M– an \(R\)-module (default: the free \(R\)-module of rank- n) to serve as the ground space for the Lie algebra
- ambient– (optional) a Lie algebra; if this is set, then the resulting Lie algebra is declared a Lie subalgebra of- ambient
 - OUTPUT: - The abelian Lie algebra on \(M\). - class Element(parent, value)[source]¶
- Bases: - Element- Initialize - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: TestSuite(a).run() - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> TestSuite(a).run() - lift()[source]¶
- Return the lift of - selfto the universal enveloping algebra.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: elt = 2*a + 2*b + 3*c sage: elt.lift() # needs sage.combinat sage.libs.singular 2*b0 + 2*b1 + 3*b2 - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> elt = Integer(2)*a + Integer(2)*b + Integer(3)*c >>> elt.lift() # needs sage.combinat sage.libs.singular 2*b0 + 2*b1 + 3*b2 
 - monomial_coefficients(copy=True)[source]¶
- Return the monomial coefficients of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: elt = 2*a + 2*b + 3*c sage: elt.monomial_coefficients() {0: 2, 1: 2, 2: 3} - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> elt = Integer(2)*a + Integer(2)*b + Integer(3)*c >>> elt.monomial_coefficients() {0: 2, 1: 2, 2: 3} 
 - to_vector(order=None, sparse=False)[source]¶
- Return - selfas a vector in- self.parent().module().- See the docstring of the latter method for the meaning of this. - EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: elt = 2*a + 2*b + 3*c sage: elt.to_vector() (2, 2, 3) - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> elt = Integer(2)*a + Integer(2)*b + Integer(3)*c >>> elt.to_vector() (2, 2, 3) 
 
 - ambient()[source]¶
- 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 - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> S = L.subalgebra([Integer(2)*a+b, b + c]) >>> S.ambient() == L True 
 - basis()[source]¶
- Return the basis of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.basis() Finite family {0: (1, 0, 0), 1: (0, 1, 0), 2: (0, 0, 1)} - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.basis() Finite family {0: (1, 0, 0), 1: (0, 1, 0), 2: (0, 0, 1)} 
 - basis_matrix()[source]¶
- Return the basis matrix of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.basis_matrix() [1 0 0] [0 1 0] [0 0 1] - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.basis_matrix() [1 0 0] [0 1 0] [0 0 1] 
 - from_vector(v, order=None)[source]¶
- Return the element of - selfcorresponding to the vector- vin- 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 - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> u = L.from_vector(vector(QQ, (Integer(1), Integer(0), Integer(0)))); u (1, 0, 0) >>> parent(u) is L True 
 - gens()[source]¶
- Return the generators of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.gens() ((1, 0, 0), (0, 1, 0), (0, 0, 1)) - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.gens() ((1, 0, 0), (0, 1, 0), (0, 0, 1)) 
 - ideal(gens)[source]¶
- Return the Lie subalgebra of - selfgenerated by the elements of the iterable- gens.- This currently requires the ground ring \(R\) to be a field. - EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: L.subalgebra([2*a+b, 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] - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> L.subalgebra([Integer(2)*a+b, 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] 
 - is_ideal(A)[source]¶
- Return if - selfis an ideal of the ambient space- A.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: L.is_ideal(L) True sage: S1 = L.subalgebra([2*a+b, b + c]) sage: S1.is_ideal(L) True sage: S2 = L.subalgebra([2*a+b]) sage: S2.is_ideal(S1) True sage: S1.is_ideal(S2) False - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> L.is_ideal(L) True >>> S1 = L.subalgebra([Integer(2)*a+b, b + c]) >>> S1.is_ideal(L) True >>> S2 = L.subalgebra([Integer(2)*a+b]) >>> S2.is_ideal(S1) True >>> S1.is_ideal(S2) False 
 - leading_monomials()[source]¶
- Return the set of leading monomials of the basis of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: I = L.ideal([2*a + b, b + c]) sage: I.leading_monomials() ((1, 0, 0), (0, 1, 0)) - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> I = L.ideal([Integer(2)*a + b, b + c]) >>> I.leading_monomials() ((1, 0, 0), (0, 1, 0)) 
 - lie_algebra_generators()[source]¶
- Return the basis of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.basis() Finite family {0: (1, 0, 0), 1: (0, 1, 0), 2: (0, 0, 1)} - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.basis() Finite family {0: (1, 0, 0), 1: (0, 1, 0), 2: (0, 0, 1)} 
 - lift(x)[source]¶
- Return the lift of - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.gens() sage: L.lift(a) b0 sage: L.lift(b).parent() is L.universal_enveloping_algebra() True sage: I = L.ideal([a + 2*b, b + 3*c]) sage: I.lift(I.basis()[0]) (1, 0, -6) - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.gens() >>> L.lift(a) b0 >>> L.lift(b).parent() is L.universal_enveloping_algebra() True >>> I = L.ideal([a + Integer(2)*b, b + Integer(3)*c]) >>> I.lift(I.basis()[Integer(0)]) (1, 0, -6) 
 - module()[source]¶
- Return an \(R\)-module which is isomorphic to the underlying \(R\)-module of - self.- See - sage.categories.lie_algebras.LieAlgebras.module()for an explanation.- In this particular example, this returns the module \(M\) that was used to construct - self.- EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: L.module() Vector space of dimension 3 over Rational Field sage: a, b, c = L.lie_algebra_generators() sage: S = L.subalgebra([2*a+b, b + c]) sage: S.module() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1/2] [ 0 1 1] - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.module() Vector space of dimension 3 over Rational Field >>> a, b, c = L.lie_algebra_generators() >>> S = L.subalgebra([Integer(2)*a+b, b + c]) >>> S.module() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1/2] [ 0 1 1] 
 - subalgebra(gens)[source]¶
- Return the Lie subalgebra of - selfgenerated by the elements of the iterable- gens.- This currently requires the ground ring \(R\) to be a field. - EXAMPLES: - sage: L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() sage: a, b, c = L.lie_algebra_generators() sage: L.subalgebra([2*a+b, 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] - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> a, b, c = L.lie_algebra_generators() >>> L.subalgebra([Integer(2)*a+b, 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] 
 - universal_enveloping_algebra()[source]¶
- 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: {} - >>> from sage.all import * >>> L = LieAlgebras(QQ).FiniteDimensional().WithBasis().example() >>> L.universal_enveloping_algebra() Noncommutative Multivariate Polynomial Ring in b0, b1, b2 over Rational Field, nc-relations: {} 
 
- sage.categories.examples.finite_dimensional_lie_algebras_with_basis.Example[source]¶
- alias of - AbelianLieAlgebra