Lie Conformal Algebras - Category Framework

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# Lie Conformal Algebras

Let \(R\) be a commutative ring, a super Lie conformal algebra [Kac1997] over \(R\) (also known as a vertex Lie algebra) is an \(R[T]\) super module \(L\) together with a \(\mathbb{Z}/2\mathbb{Z}\)-graded \(R\)-bilinear operation (called the \(\lambda\)-bracket) \(L\otimes L \rightarrow L[\lambda]\) (polynomials in \(\lambda\) with coefficients in \(L\)), \(a \otimes b \mapsto [a_\lambda b]\) satisfying

1. Sesquilinearity:

   \[[Ta_\lambda b] = - \lambda [a_\lambda b], \qquad [a_\lambda Tb] = (\lambda+ T) [a_\lambda b].\]

2. Skew-Symmetry:

   \[[a_\lambda b] = - (-1)^{p(a)p(b)} [b_{-\lambda - T} a],\]

   where \(p(a)\) is \(0\) if \(a\) is even and \(1\) if \(a\) is odd. The bracket in the RHS is computed as follows. First we evaluate \([b_\mu a]\) with the formal parameter \(\mu\) to the left, then replace each appearance of the formal variable \(\mu\) by \(-\lambda - T\). Finally apply \(T\) to the coefficients in \(L\).

3. Jacobi identity:

   \[[a_\lambda [b_\mu c]] = [ [a_{\lambda + \mu} b]_\mu c] + (-1)^{p(a)p(b)} [b_\mu [a_\lambda c]],\]

   which is understood as an equality in \(L[\lambda, \mu]\).

   \(T\) is usually called the translation operation or the derivative. For an element \(a \in L\) we will say that \(Ta\) is the derivative of \(a\). We define the \(n\)-th products \(a_{(n)} b\) for \(a,b \in L\) by

   \[[a_\lambda b] = \sum_{n \geq 0} \frac{\lambda^n}{n!} a_{(n)} b.\]

   A Lie conformal algebra is called H-Graded [DSK2006] if there exists a decomposition \(L = \oplus L_n\) such that the \(\lambda\)-bracket becomes graded of degree \(-1\), that is:

   \[a_{(n)} b \in L_{p + q -n -1} \qquad a \in L_p, \: b \in L_q, \: n \geq 0.\]

   In particular this implies that the action of \(T\) increases degree by \(1\).

Note

In the literature arbitrary gradings are allowed. In this implementation we only support nonnegative rational gradings.

EXAMPLES:

1. The Virasoro Lie conformal algebra \(Vir\) over a ring \(R\) where \(12\) is invertible has two generators \(L, C\) as an \(R[T]\)-module. It is the direct sum of a free module of rank \(1\) generated by \(L\), and a free rank one \(R\) module generated by \(C\) satisfying \(TC = 0\). \(C\) is central (the \(\lambda\)-bracket of \(C\) with any other vector vanishes). The remaining \(\lambda\)-bracket is given by

   \[[L_\lambda L] = T L + 2 \lambda L + \frac{\lambda^3}{12} C.\]

2. The affine or current Lie conformal algebra \(L(\mathfrak{g})\) associated to a finite dimensional Lie algebra \(\mathfrak{g}\) with non-degenerate, invariant \(R\)-bilinear form \((,)\) is given as a central extension of the free \(R[T]\) module generated by \(\mathfrak{g}\) by a central element \(K\). The \(\lambda\)-bracket of generators is given by

   \[[a_\lambda b] = [a,b] + \lambda (a,b) K, \qquad a,b \in \mathfrak{g}\]

3. The Weyl Lie conformal algebra, or \(\beta-\gamma\) system is given as the central extension of a free \(R[T]\) module with two generators \(\beta\) and \(\gamma\), by a central element \(K\). The only non-trivial brackets among generators are

   \[[\beta_\lambda \gamma] = - [\gamma_\lambda \beta] = K\]

4. The Neveu-Schwarz super Lie conformal algebra is a super Lie conformal algebra which is an extension of the Virasoro Lie conformal algebra. It consists of a Virasoro generator \(L\) as in example 1 above and an odd generator \(G\). The remaining brackets are given by:

   \[[L_\lambda G] = \left( T + \frac{3}{2} \lambda \right) G \qquad [G_\lambda G] = 2 L + \frac{\lambda^2}{3} C\]

See also

- `sage.algebras.lie_conformal_algebras.lie_conformal_algebra`

- `sage.algebras.lie_conformal_algebras.examples`

AUTHORS:

- Reimundo Heluani (2019-10-05): Initial implementation.

class sage.categories.lie_conformal_algebras.LieConformalAlgebras(base, name=None)

  Bases: `Category_over_base_ring`

  The category of Lie conformal algebras.

  This is the base category for all Lie conformal algebras. Subcategories with axioms are `FinitelyGenerated` and `WithBasis`. A finitely generated Lie conformal algebra is a Lie conformal algebra over \(R\) which is finitely generated as an \(R[T]\)-module. A Lie conformal algebra with basis is one with a preferred basis as an \(R\)-module.

  EXAMPLES:

  The base category:

  Sage

      sage: C = LieConformalAlgebras(QQ); C
      Category of Lie conformal algebras over Rational Field
      sage: C.is_subcategory(VectorSpaces(QQ))
      True

  Python

      >>> from sage.all import *
      >>> C = LieConformalAlgebras(QQ); C
      Category of Lie conformal algebras over Rational Field
      >>> C.is_subcategory(VectorSpaces(QQ))
      True

  Some subcategories:

  Sage

      sage: LieConformalAlgebras(QQbar).FinitelyGenerated().WithBasis()               # needs sage.rings.number_field
      Category of finitely generated Lie conformal algebras with basis
       over Algebraic Field

  [Python]

      >>> from sage.all import *
      >>> LieConformalAlgebras(QQbar).FinitelyGenerated().WithBasis()               # needs sage.rings.number_field
      Category of finitely generated Lie conformal algebras with basis
       over Algebraic Field

  In addition we support functorial constructions `Graded` and `Super`. These functors commute:

  Sage

      sage: CGS = LieConformalAlgebras(AA).Graded().Super(); CGS                      # needs sage.rings.number_field
      Category of H-graded super Lie conformal algebras over Algebraic Real Field
      sage: CGS is LieConformalAlgebras(AA).Super().Graded()                          # needs sage.rings.number_field
      True

  Python

      >>> from sage.all import *
      >>> CGS = LieConformalAlgebras(AA).Graded().Super(); CGS                      # needs sage.rings.number_field
      Category of H-graded super Lie conformal algebras over Algebraic Real Field
      >>> CGS is LieConformalAlgebras(AA).Super().Graded()                          # needs sage.rings.number_field
      True

  That is, we only consider gradings on super Lie conformal algebras that are compatible with the \(\ZZ/2\ZZ\) grading.

  The base ring needs to be a commutative ring:

  Sage

      sage: LieConformalAlgebras(QuaternionAlgebra(2))                                # needs sage.combinat sage.modules
      Traceback (most recent call last):
      ValueError: base must be a commutative ring
      got Quaternion Algebra (-1, -1) with base ring Rational Field

  [Python]

      >>> from sage.all import *
      >>> LieConformalAlgebras(QuaternionAlgebra(Integer(2)))                                # needs sage.combinat sage.modules
      Traceback (most recent call last):
      ValueError: base must be a commutative ring
      got Quaternion Algebra (-1, -1) with base ring Rational Field

  class ElementMethods

    Bases: `object`

    is_even_odd()

      Return `0` if this element is even and `1` if it is odd.

      Note

      This method returns `0` by default since every Lie conformal algebra can be thought as a purely even Lie conformal algebra. In order to implement a super Lie conformal algebra, the user needs to implement this method.

      EXAMPLES:

      Sage

          sage: R = lie_conformal_algebras.NeveuSchwarz(QQ)                       # needs sage.combinat sage.modules
          sage: R.inject_variables()                                              # needs sage.combinat sage.modules
          Defining L, G, C
          sage: G.is_even_odd()                                                   # needs sage.combinat sage.modules
          1

      Python

          >>> from sage.all import *
          >>> R = lie_conformal_algebras.NeveuSchwarz(QQ)                       # needs sage.combinat sage.modules
          >>> R.inject_variables()                                              # needs sage.combinat sage.modules
          Defining L, G, C
          >>> G.is_even_odd()                                                   # needs sage.combinat sage.modules
          1

  FinitelyGeneratedAsLambdaBracketAlgebra

    alias of `FinitelyGeneratedLieConformalAlgebras`

  Graded

    alias of `GradedLieConformalAlgebras`

  class ParentMethods

    Bases: `object`

  Super

    alias of `SuperLieConformalAlgebras`

  WithBasis

    alias of `LieConformalAlgebrasWithBasis`

  example()

    An example of parent in this category.

    EXAMPLES:

    Sage

        sage: LieConformalAlgebras(QQ).example()                                    # needs sage.combinat sage.modules
        The Virasoro Lie conformal algebra over Rational Field

    Python

        >>> from sage.all import *
        >>> LieConformalAlgebras(QQ).example()                                    # needs sage.combinat sage.modules
        The Virasoro Lie conformal algebra over Rational Field

  super_categories()

    The list of super categories of this category.

    EXAMPLES:

    Sage

        sage: C = LieConformalAlgebras(QQ)
        sage: C.super_categories()
        [Category of Lambda bracket algebras over Rational Field]
        sage: C = LieConformalAlgebras(QQ).FinitelyGenerated(); C
        Category of finitely generated Lie conformal algebras over Rational Field
        sage: C.super_categories()
        [Category of finitely generated lambda bracket algebras over Rational Field,
         Category of Lie conformal algebras over Rational Field]
        sage: C.all_super_categories()
        [Category of finitely generated Lie conformal algebras over Rational Field,
         Category of finitely generated lambda bracket algebras over Rational Field,
         Category of Lie conformal algebras over Rational Field,
         Category of Lambda bracket algebras over Rational Field,
         Category of vector spaces over Rational Field,
         Category of modules over Rational Field,
         Category of bimodules over Rational Field on the left and Rational Field on the right,
         Category of right modules over Rational Field,
         Category of left modules over Rational Field,
         Category of commutative additive groups,
         Category of additive groups,
         Category of additive inverse additive unital additive magmas,
         Category of commutative additive monoids,
         Category of additive monoids,
         Category of additive unital additive magmas,
         Category of commutative additive semigroups,
         Category of additive commutative additive magmas,
         Category of additive semigroups,
         Category of additive magmas,
         Category of sets,
         Category of sets with partial maps,
         Category of objects]

    Python

        >>> from sage.all import *
        >>> C = LieConformalAlgebras(QQ)
        >>> C.super_categories()
        [Category of Lambda bracket algebras over Rational Field]
        >>> C = LieConformalAlgebras(QQ).FinitelyGenerated(); C
        Category of finitely generated Lie conformal algebras over Rational Field
        >>> C.super_categories()
        [Category of finitely generated lambda bracket algebras over Rational Field,
         Category of Lie conformal algebras over Rational Field]
        >>> C.all_super_categories()
        [Category of finitely generated Lie conformal algebras over Rational Field,
         Category of finitely generated lambda bracket algebras over Rational Field,
         Category of Lie conformal algebras over Rational Field,
         Category of Lambda bracket algebras over Rational Field,
         Category of vector spaces over Rational Field,
         Category of modules over Rational Field,
         Category of bimodules over Rational Field on the left and Rational Field on the right,
         Category of right modules over Rational Field,
         Category of left modules over Rational Field,
         Category of commutative additive groups,
         Category of additive groups,
         Category of additive inverse additive unital additive magmas,
         Category of commutative additive monoids,
         Category of additive monoids,
         Category of additive unital additive magmas,
         Category of commutative additive semigroups,
         Category of additive commutative additive magmas,
         Category of additive semigroups,
         Category of additive magmas,
         Category of sets,
         Category of sets with partial maps,
         Category of objects]