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
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- stror a- CartanType; the Cartan Type for the corresponding finite dimensional Lie algebra. It must correspond to a simple finite dimensional Lie algebra.
- names– list of- stror- None(default:- None); alternative names for the generators. If- Nonethe 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[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 - >>> 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: 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] - >>> 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]