Free Bosons Lie Conformal Algebra#

Given an \(R\)-module \(M\) with a symmetric, bilinear pairing \((\cdot, \cdot): M\otimes_R M \rightarrow R\). The Free Bosons Lie conformal algebra associated to this datum is the free \(R[T]\)-module generated by \(M\) plus a central vector \(K\) satisfying \(TK=0\). The remaining \(\lambda\)-brackets are given by:

\[[v_\lambda w] = \lambda (v,w)K,\]

where \(v,w \in M\).

This is an H-graded Lie conformal algebra where every generator \(v \in M\) has degree 1.

AUTHORS:

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

class sage.algebras.lie_conformal_algebras.free_bosons_lie_conformal_algebra.FreeBosonsLieConformalAlgebra(R, ngens=None, gram_matrix=None, names=None, index_set=None)#

Bases: GradedLieConformalAlgebra

The Free Bosons Lie conformal algebra.

INPUT:

  • R – a commutative ring.

  • ngens – a positive Integer (default 1); the number of non-central generators of this Lie conformal algebra.

  • gram_matrix: a symmetric square matrix with coefficients in R (default: identity_matrix(ngens)); the Gram matrix of the inner product

  • names – a tuple of str; alternative names for the generators

  • index_set – an enumerated set; alternative indexing set for the generators.

OUTPUT:

The Free Bosons Lie conformal algebra with generators

\(\alpha_i\), \(i=1,...,n\) and \(\lambda\)-brackets

\[[{\alpha_i}_{\lambda} \alpha_j] = \lambda M_{ij} K,\]

where \(n\) is the number of generators ngens and \(M\) is the gram_matrix. This Lie conformal algebra is \(H\)-graded where every generator has conformal weight \(1\).

EXAMPLES:

sage: R = lie_conformal_algebras.FreeBosons(AA); R
The free Bosons Lie conformal algebra with generators (alpha, K) over Algebraic Real Field
sage: R.inject_variables()
Defining alpha, K
sage: alpha.bracket(alpha)
{1: K}
sage: M = identity_matrix(QQ,2); R = lie_conformal_algebras.FreeBosons(QQ,gram_matrix=M, names='alpha,beta'); R
The free Bosons Lie conformal algebra with generators (alpha, beta, K) over Rational Field
sage: R.inject_variables(); alpha.bracket(beta)
Defining alpha, beta, K
{}
sage: alpha.bracket(alpha)
{1: K}
sage: R = lie_conformal_algebras.FreeBosons(QQbar, ngens=3); R
The free Bosons Lie conformal algebra with generators (alpha0, alpha1, alpha2, K) over Algebraic Field
gram_matrix()#

The Gram matrix that specifies the \(\lambda\)-brackets of the generators.

EXAMPLES:

sage: R = lie_conformal_algebras.FreeBosons(QQ,ngens=2);
sage: R.gram_matrix()
[1 0]
[0 1]