Examples of a Lie algebra¶
- sage.categories.examples.lie_algebras.Example[source]¶
- alias of - LieAlgebraFromAssociative
- class sage.categories.examples.lie_algebras.LieAlgebraFromAssociative(gens)[source]¶
- Bases: - Parent,- UniqueRepresentation- An example of a Lie algebra: a Lie algebra generated by a set of elements of an associative algebra. - This class illustrates a minimal implementation of a Lie algebra. - Let \(R\) be a commutative ring, and \(A\) an associative \(R\)-algebra. The Lie algebra \(A\) (sometimes denoted \(A^-\)) is defined to be the \(R\)-module \(A\) with Lie bracket given by the commutator in \(A\): that is, \([a, b] := ab - ba\) for all \(a, b \in A\). - What this class implements is not precisely \(A^-\), however; it is the Lie subalgebra of \(A^-\) generated by the elements of the iterable - gens. This specific implementation does not provide a reasonable containment test (i.e., it does not allow you to check if a given element \(a\) of \(A^-\) belongs to this Lie subalgebra); it, however, allows computing inside it.- INPUT: - gens– a nonempty iterable consisting of elements of an associative algebra \(A\)
 - OUTPUT: - The Lie subalgebra of \(A^-\) generated by the elements of - gens- EXAMPLES: - We create a model of \(\mathfrak{sl}_2\) using matrices: - sage: gens = [matrix([[0,1],[0,0]]), matrix([[0,0],[1,0]]), matrix([[1,0],[0,-1]])] sage: for g in gens: ....: g.set_immutable() sage: L = LieAlgebras(QQ).example(gens) sage: e,f,h = L.lie_algebra_generators() sage: e.bracket(f) == h True sage: h.bracket(e) == 2*e True sage: h.bracket(f) == -2*f True - >>> from sage.all import * >>> gens = [matrix([[Integer(0),Integer(1)],[Integer(0),Integer(0)]]), matrix([[Integer(0),Integer(0)],[Integer(1),Integer(0)]]), matrix([[Integer(1),Integer(0)],[Integer(0),-Integer(1)]])] >>> for g in gens: ... g.set_immutable() >>> L = LieAlgebras(QQ).example(gens) >>> e,f,h = L.lie_algebra_generators() >>> e.bracket(f) == h True >>> h.bracket(e) == Integer(2)*e True >>> h.bracket(f) == -Integer(2)*f True - class Element[source]¶
- Bases: - ElementWrapper- Wrap an element as a Lie algebra element. 
 - lie_algebra_generators()[source]¶
- Return the generators of - selfas a Lie algebra.- EXAMPLES: - sage: L = LieAlgebras(QQ).example() # needs sage.combinat sage.groups sage: L.lie_algebra_generators() # needs sage.combinat sage.groups Family ([2, 1, 3], [2, 3, 1]) - >>> from sage.all import * >>> L = LieAlgebras(QQ).example() # needs sage.combinat sage.groups >>> L.lie_algebra_generators() # needs sage.combinat sage.groups Family ([2, 1, 3], [2, 3, 1]) 
 - zero()[source]¶
- Return the element 0. - EXAMPLES: - sage: L = LieAlgebras(QQ).example() # needs sage.combinat sage.groups sage: L.zero() # needs sage.combinat sage.groups 0 - >>> from sage.all import * >>> L = LieAlgebras(QQ).example() # needs sage.combinat sage.groups >>> L.zero() # needs sage.combinat sage.groups 0