Affine Lie Algebras¶
AUTHORS:
Travis Scrimshaw (2013-05-03): Initial version
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class
sage.algebras.lie_algebras.affine_lie_algebra.AffineLieAlgebra(g, kac_moody)¶ Bases:
sage.algebras.lie_algebras.lie_algebra.FinitelyGeneratedLieAlgebraAn (untwisted) affine Lie algebra.
Let \(R\) be a ring. Given a finite-dimensional simple Lie algebra \(\mathfrak{g}\) over \(R\), the affine Lie algebra \(\widehat{\mathfrak{g}}^{\prime}\) associated to \(\mathfrak{g}\) is defined as
\[\widehat{\mathfrak{g}}' = \bigl( \mathfrak{g} \otimes R[t, t^{-1}] \bigr) \oplus R c,\]where \(c\) is the canonical central element and \(R[t, t^{-1}]\) is the Laurent polynomial ring over \(R\). The Lie bracket is defined as
\[[x \otimes t^m + \lambda c, y \otimes t^n + \mu c] = [x, y] \otimes t^{m+n} + m \delta_{m,-n} ( x | y ) c,\]where \(( x | y )\) is the Killing form on \(\mathfrak{g}\).
There is a canonical derivation \(d\) on \(\widehat{\mathfrak{g}}'\) that is defined by
\[d(x \otimes t^m + \lambda c) = a \otimes m t^m,\]or equivalently by \(d = t \frac{d}{dt}\).
The affine Kac-Moody algebra \(\widehat{\mathfrak{g}}\) is formed by adjoining the derivation \(d\) such that
\[\widehat{\mathfrak{g}} = \bigl( \mathfrak{g} \otimes R[t,t^{-1}] \bigr) \oplus R c \oplus R d.\]Specifically, the bracket on \(\widehat{\mathfrak{g}}\) is defined as
\[[t^m \otimes x \oplus \lambda c \oplus \mu d, t^n \otimes y \oplus \lambda_1 c \oplus \mu_1 d] = \bigl( t^{m+n} [x,y] + \mu n t^n \otimes y - \mu_1 m t^m \otimes x\bigr) \oplus m \delta_{m,-n} (x|y) c .\]Note that the derived subalgebra of the Kac-Moody algebra is the affine Lie algebra.
INPUT:
Can be one of the following:
a base ring and an affine Cartan type: constructs the affine (Kac-Moody) Lie algebra of the classical Lie algebra in the bracket representation over the base ring
a classical Lie algebra: constructs the corresponding affine (Kac-Moody) Lie algebra
There is the optional argument
kac_moody, which can be set toFalseto obtain the affine Lie algebra instead of the affine Kac-Moody algebra.EXAMPLES:
We begin by constructing an affine Kac-Moody algebra of type \(G_2^{(1)}\) from the classical Lie algebra of type \(G_2\):
sage: g = LieAlgebra(QQ, cartan_type=['G',2]) sage: A = g.affine() sage: A Affine Kac-Moody algebra of ['G', 2] in the Chevalley basis
Next, we construct the generators and perform some computations:
sage: A.inject_variables() Defining e1, e2, f1, f2, h1, h2, e0, f0, c, d sage: e1.bracket(f1) (h1)#t^0 sage: e0.bracket(f0) (-h1 - 2*h2)#t^0 + 8*c sage: e0.bracket(f1) 0 sage: A[d, f0] (-E[3*alpha[1] + 2*alpha[2]])#t^-1 sage: A([[e0, e2], [[[e1, e2], [e0, [e1, e2]]], e1]]) (-6*E[-3*alpha[1] - alpha[2]])#t^2 sage: f0.bracket(f1) 0 sage: f0.bracket(f2) (E[3*alpha[1] + alpha[2]])#t^-1 sage: A[h1+3*h2, A[[[f0, f2], f1], [f1,f2]] + f1] - f1 (2*E[alpha[1]])#t^-1
We can construct its derived subalgebra, the affine Lie algebra of type \(G_2^{(1)}\). In this case, there is no canonical derivation, so the generator \(d\) is \(0\):
sage: D = A.derived_subalgebra() sage: D.d() 0
REFERENCES:
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basis()¶ Return the basis of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['D',4,1]) sage: B = g.basis() sage: al = RootSystem(['D',4]).root_lattice().simple_roots() sage: B[al[1]+al[2]+al[4],4] (E[alpha[1] + alpha[2] + alpha[4]])#t^4 sage: B[-al[1]-2*al[2]-al[3]-al[4],2] (E[-alpha[1] - 2*alpha[2] - alpha[3] - alpha[4]])#t^2 sage: B[al[4],-2] (E[alpha[4]])#t^-2 sage: B['c'] c sage: B['d'] d
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c()¶ Return the canonical central element \(c\) of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['A',3,1]) sage: g.c() c
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cartan_type()¶ Return the Cartan type of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['C',3,1]) sage: g.cartan_type() ['C', 3, 1]
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classical()¶ Return the classical Lie algebra of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['F',4,1]) sage: g.classical() Lie algebra of ['F', 4] in the Chevalley basis sage: so5 = lie_algebras.so(QQ, 5, 'matrix') sage: A = so5.affine() sage: A.classical() == so5 True
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d()¶ Return the canonical derivation \(d\) of
self.If
selfis the affine Lie algebra, then this returns \(0\).EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['A',3,1]) sage: g.d() d sage: D = g.derived_subalgebra() sage: D.d() 0
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derived_series()¶ Return the derived series of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',3,1]) sage: g.derived_series() [Affine Kac-Moody algebra of ['B', 3] in the Chevalley basis, Affine Lie algebra of ['B', 3] in the Chevalley basis] sage: g.lower_central_series() [Affine Kac-Moody algebra of ['B', 3] in the Chevalley basis, Affine Lie algebra of ['B', 3] in the Chevalley basis] sage: D = g.derived_subalgebra() sage: D.derived_series() [Affine Lie algebra of ['B', 3] in the Chevalley basis]
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derived_subalgebra()¶ Return the derived subalgebra of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',3,1]) sage: g Affine Kac-Moody algebra of ['B', 3] in the Chevalley basis sage: D = g.derived_subalgebra(); D Affine Lie algebra of ['B', 3] in the Chevalley basis sage: D.derived_subalgebra() == D True
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is_nilpotent()¶ Return
Falseasselfis semisimple.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',3,1]) sage: g.is_nilpotent() False sage: g.is_solvable() False
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is_solvable()¶ Return
Falseasselfis semisimple.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',3,1]) sage: g.is_nilpotent() False sage: g.is_solvable() False
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lie_algebra_generators()¶ Return the Lie algebra generators of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['A',1,1]) sage: list(g.lie_algebra_generators()) [(E[alpha[1]])#t^0, (E[-alpha[1]])#t^0, (h1)#t^0, (E[-alpha[1]])#t^1, (E[alpha[1]])#t^-1, c, d]
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lower_central_series()¶ Return the derived series of
self.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',3,1]) sage: g.derived_series() [Affine Kac-Moody algebra of ['B', 3] in the Chevalley basis, Affine Lie algebra of ['B', 3] in the Chevalley basis] sage: g.lower_central_series() [Affine Kac-Moody algebra of ['B', 3] in the Chevalley basis, Affine Lie algebra of ['B', 3] in the Chevalley basis] sage: D = g.derived_subalgebra() sage: D.derived_series() [Affine Lie algebra of ['B', 3] in the Chevalley basis]
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monomial(m)¶ Construct the monomial indexed by
m.EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['B',4,1]) sage: al = RootSystem(['B',4]).root_lattice().simple_roots() sage: g.monomial((al[1]+al[2]+al[3],4)) (E[alpha[1] + alpha[2] + alpha[3]])#t^4 sage: g.monomial((-al[1]-al[2]-2*al[3]-2*al[4],2)) (E[-alpha[1] - alpha[2] - 2*alpha[3] - 2*alpha[4]])#t^2 sage: g.monomial((al[4],-2)) (E[alpha[4]])#t^-2 sage: g.monomial('c') c sage: g.monomial('d') d
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zero()¶ Return the element \(0\).
EXAMPLES:
sage: g = LieAlgebra(QQ, cartan_type=['F',4,1]) sage: g.zero() 0