Affine Lie Conformal Algebra#

The affine Kac-Moody Lie conformal algebra associated to the finite dimensional simple Lie algebra \(\mathfrak{g}\). For a commutative ring \(R\), it is the \(R[T]\)-module freely generated by \(\mathfrak{g}\) plus a central element \(K\) satisfying \(TK = 0\). The non-vanishing \(\lambda\)-brackets are given by

\[[a_\lambda b] = [a,b] + \lambda (a,b)K,\]

where \(a,b \in \mathfrak{g}\) and \((a,b)\) is the normalized form of \(\mathfrak{g}\) so that its longest root has square-norm \(2\).

AUTHORS:

Reimundo Heluani (2019-08-09): Initial implementation.

class sage.algebras.lie_conformal_algebras.affine_lie_conformal_algebra.AffineLieConformalAlgebra(R, ct, names=None, prefix=None, bracket=None)[source]#

Bases: GradedLieConformalAlgebra

The current or affine Kac-Moody Lie conformal algebra.

INPUT:

R – a commutative Ring; the base ring for this Lie conformal algebra.
ct – a str or a CartanType; the Cartan Type for the corresponding finite dimensional Lie algebra. It must correspond to a simple finite dimensional Lie algebra.
names – a list of str or None (default: None) ; alternative names for the generators. If None the generators are labeled by the corresponding root and coroot vectors.
prefix – a str; parameter passed to IndexedGenerators
bracket – a str; parameter passed to IndexedGenerators.

EXAMPLES:

Sage

sage: R = lie_conformal_algebras.Affine(QQ, 'A1')
sage: R
The affine Lie conformal algebra of type ['A', 1] over Rational Field
sage: R.an_element()
B[alpha[1]] + B[alphacheck[1]] + B[-alpha[1]] + B['K']

sage: R = lie_conformal_algebras.Affine(QQ, 'A1', names = ('e', 'h','f'))
sage: R.inject_variables()
Defining e, h, f, K
sage: Family(e.bracket(f.T(3)))
Finite family {0: 6*T^(3)h, 1: 6*T^(2)h, 2: 6*Th, 3: 6*h, 4: 24*K}

sage: V = lie_conformal_algebras.Affine(QQ, CartanType(["A",2,1]))
Traceback (most recent call last):
...
ValueError: only affine algebras of simple finite dimensionalLie algebras are implemented

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.Affine(QQ, 'A1')
>>> R
The affine Lie conformal algebra of type ['A', 1] over Rational Field
>>> R.an_element()
B[alpha[1]] + B[alphacheck[1]] + B[-alpha[1]] + B['K']

>>> R = lie_conformal_algebras.Affine(QQ, 'A1', names = ('e', 'h','f'))
>>> R.inject_variables()
Defining e, h, f, K
>>> Family(e.bracket(f.T(Integer(3))))
Finite family {0: 6*T^(3)h, 1: 6*T^(2)h, 2: 6*Th, 3: 6*h, 4: 24*K}

>>> V = lie_conformal_algebras.Affine(QQ, CartanType(["A",Integer(2),Integer(1)]))
Traceback (most recent call last):
...
ValueError: only affine algebras of simple finite dimensionalLie algebras are implemented

OUTPUT:

The Affine Lie conformal algebra associated with the finite dimensional simple Lie algebra of Cartan type ct.

cartan_type()[source]#

The Cartan type of this Lie conformal algbera.

EXAMPLES:

Sage

sage: R = lie_conformal_algebras.Affine(QQ, 'B3')
sage: R
The affine Lie conformal algebra of type ['B', 3] over Rational Field
sage: R.cartan_type()
['B', 3]

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.Affine(QQ, 'B3')
>>> R
The affine Lie conformal algebra of type ['B', 3] over Rational Field
>>> R.cartan_type()
['B', 3]