# 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)

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]