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)[source]¶

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 – tuple of strings; 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

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

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.FreeBosons(AA); R
The free Bosons Lie conformal algebra with generators (alpha, K) over Algebraic Real Field
>>> R.inject_variables()
Defining alpha, K
>>> alpha.bracket(alpha)
{1: K}
>>> M = identity_matrix(QQ,Integer(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
>>> R.inject_variables(); alpha.bracket(beta)
Defining alpha, beta, K
{}
>>> alpha.bracket(alpha)
{1: K}
>>> R = lie_conformal_algebras.FreeBosons(QQbar, ngens=Integer(3)); R
The free Bosons Lie conformal algebra with generators (alpha0, alpha1, alpha2, K) over Algebraic Field

gram_matrix()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.FreeBosons(QQ,ngens=Integer(2));
>>> R.gram_matrix()
[1 0]
[0 1]