Affine Lie Conformal Algebra - Algebras

Skip to content
Back to top
View this page
Edit this page

# 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)

  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` – 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()

    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]