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