Bosonic Ghosts Lie Conformal Algebra¶
The Bosonic-ghosts or \(\beta-\gamma\)-system Lie conformal algebra with \(2n\) generators is the H-graded Lie conformal algebra generated by \(\beta_i, \gamma_i, i = 1,\ldots,n\) and a central element \(K\), with non-vanishing \(\lambda\)-brackets:
The generators \(\beta_i\) have degree \(1\) while the generators \(\gamma_i\) have degree \(0\).
AUTHORS:
- Reimundo Heluani (2020-06-15): Initial implementation. 
- class sage.algebras.lie_conformal_algebras.bosonic_ghosts_lie_conformal_algebra.BosonicGhostsLieConformalAlgebra(R, ngens=2, names=None, index_set=None)[source]¶
- Bases: - GradedLieConformalAlgebra- The Bosonic ghosts or \(\beta-\gamma\)-system Lie conformal algebra. - INPUT: - R– a commutative ring
- ngens– an even positive Integer (default:- 2); the number of non-central generators of this Lie conformal algebra.
- names– list of- str; alternative names for the generators
- index_set– an enumerated set; an indexing set for the generators
 - OUTPUT: - The Bosonic Ghosts Lie conformal algebra with generators \(\beta_i,\gamma_i, i=1,\ldots,n\) and \(K\), where \(2n\) is - ngens.- EXAMPLES: - sage: R = lie_conformal_algebras.BosonicGhosts(QQ); R The Bosonic ghosts Lie conformal algebra with generators (beta, gamma, K) over Rational Field sage: R.inject_variables(); beta.bracket(gamma) Defining beta, gamma, K {0: K} sage: beta.degree() 1 sage: gamma.degree() 0 sage: R = lie_conformal_algebras.BosonicGhosts(QQbar, ngens = 4, names = 'abcd'); R The Bosonic ghosts Lie conformal algebra with generators (a, b, c, d, K) over Algebraic Field sage: R.structure_coefficients() Finite family {('a', 'c'): ((0, K),), ('b', 'd'): ((0, K),), ('c', 'a'): ((0, -K),), ('d', 'b'): ((0, -K),)} - >>> from sage.all import * >>> R = lie_conformal_algebras.BosonicGhosts(QQ); R The Bosonic ghosts Lie conformal algebra with generators (beta, gamma, K) over Rational Field >>> R.inject_variables(); beta.bracket(gamma) Defining beta, gamma, K {0: K} >>> beta.degree() 1 >>> gamma.degree() 0 >>> R = lie_conformal_algebras.BosonicGhosts(QQbar, ngens = Integer(4), names = 'abcd'); R The Bosonic ghosts Lie conformal algebra with generators (a, b, c, d, K) over Algebraic Field >>> R.structure_coefficients() Finite family {('a', 'c'): ((0, K),), ('b', 'd'): ((0, K),), ('c', 'a'): ((0, -K),), ('d', 'b'): ((0, -K),)}