Finite Dimensional Nilpotent Lie Algebras With Basis¶
AUTHORS:
Eero Hakavuori (2018-08-16): initial version
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class
sage.categories.finite_dimensional_nilpotent_lie_algebras_with_basis.
FiniteDimensionalNilpotentLieAlgebrasWithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
Category of finite dimensional nilpotent Lie algebras with basis.
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class
ParentMethods
¶ Bases:
object
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is_nilpotent
()¶ Return
True
sinceself
is nilpotent.EXAMPLES:
sage: L = LieAlgebra(QQ, {('x','y'): {'z': 1}}, nilpotent=True) sage: L.is_nilpotent() True
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lie_group
(name='G', **kwds)¶ Return the Lie group associated to
self
.INPUT:
name
– string (default:'G'
); the name (symbol) given to the Lie group
EXAMPLES:
We define the Heisenberg group:
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
We test multiplying elements of the group:
sage: p,q,z = L.basis() sage: g = G.exp(p); g exp(p1) sage: h = G.exp(q); h exp(q1) sage: g*h exp(p1 + q1 + 1/2*z)
We extend an element of the Lie algebra to a left-invariant vector field:
sage: X = G.left_invariant_extension(2*p + 3*q, name='X'); X Vector field X on the Lie group G of Heisenberg algebra of rank 1 over Rational Field sage: X.at(G.one()).display() X = 2 d/dx_0 + 3 d/dx_1 sage: X.display() X = 2 d/dx_0 + 3 d/dx_1 + (3/2*x_0 - x_1) d/dx_2
See also
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step
()¶ Return the nilpotency step of
self
.EXAMPLES:
sage: L = LieAlgebra(QQ, {('X','Y'): {'Z': 1}}, nilpotent=True) sage: L.step() 2 sage: sc = {('X','Y'): {'Z': 1}, ('X','Z'): {'W': 1}} sage: LieAlgebra(QQ, sc, nilpotent=True).step() 3
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class