Kac-Moody Algebras With Triangular Decomposition Basis¶
AUTHORS:
Travis Scrimshaw (07-15-2017): Initial implementation
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class
sage.categories.triangular_kac_moody_algebras.
TriangularKacMoodyAlgebras
(base, name=None)¶ Bases:
sage.categories.category_types.Category_over_base_ring
Category of Kac-Moody algebras with a distinguished basis that respects the triangular decomposition.
We require that the grading group is the root lattice of the appropriate Cartan type.
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class
ElementMethods
¶ Bases:
object
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part
()¶ Return whether the element
v
is in the lower, zero, or upper part ofself
.OUTPUT:
\(-1\) if
v
is in the lower part, \(0\) if in the zero part, or \(1\) if in the upper partEXAMPLES:
sage: L = LieAlgebra(QQ, cartan_type="F4") sage: L.inject_variables() Defining e1, e2, e3, e4, f1, f2, f3, f4, h1, h2, h3, h4 sage: e1.part() 1 sage: f4.part() -1 sage: (h2 + h3).part() 0 sage: (f1.bracket(f2) + 4*f4).part() -1 sage: (e1 + f1).part() Traceback (most recent call last): ... ValueError: element is not in one part
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class
ParentMethods
¶ Bases:
object
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e
(i=None)¶ Return the generators \(e\) of
self
.INPUT:
i
– (optional) if specified, return just the generator \(e_i\)
EXAMPLES:
sage: L = lie_algebras.so(QQ, 5) sage: L.e() Finite family {1: E[alpha[1]], 2: E[alpha[2]]} sage: L.e(1) E[alpha[1]]
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f
(i=None)¶ Return the generators \(f\) of
self
.INPUT:
i
– (optional) if specified, return just the generator \(f_i\)
EXAMPLES:
sage: L = lie_algebras.so(QQ, 5) sage: L.f() Finite family {1: E[-alpha[1]], 2: E[-alpha[2]]} sage: L.f(1) E[-alpha[1]]
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verma_module
(la, basis_key=None, **kwds)¶ Return the Verma module with highest weight
la
overself
.INPUT:
basis_key
– (optional) a key function for the indexing set of the basis elements ofself
EXAMPLES:
sage: L = lie_algebras.sl(QQ, 3) sage: P = L.cartan_type().root_system().weight_lattice() sage: La = P.fundamental_weights() sage: M = L.verma_module(La[1]+La[2]) sage: M Verma module with highest weight Lambda[1] + Lambda[2] of Lie algebra of ['A', 2] in the Chevalley basis
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super_categories
()¶ EXAMPLES:
sage: from sage.categories.triangular_kac_moody_algebras import TriangularKacMoodyAlgebras sage: TriangularKacMoodyAlgebras(QQ).super_categories() [Join of Category of graded lie algebras with basis over Rational Field and Category of kac moody algebras over Rational Field]
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class