# 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)

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: 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] + B[alphacheck] + B[-alpha] + 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


OUTPUT:

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

cartan_type()

The Cartan type of this Lie conformal algbera.

EXAMPLES:

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]