Affine Lie Algebras¶
AUTHORS:
- Travis Scrimshaw (2013-05-03): Initial version
-
sage.algebras.lie_algebras.affine_lie_algebra.
AffineLieAlgebra
¶ An (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 toFalse
to 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] (E[-alpha[1]])#t^0 + (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
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