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 to False 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

REFERENCES: