On left-symmetric conformal bialgebras

Abstract

The notion of left-symmetric bialgebra was introduced in [C. M. Bai, Left-symmetric bialgebra and an analogue of the classical Yang-Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221-260] which is equivalent to a parakähler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure. In this paper, we study a conformal analog of left-symmetric bialgebras. The notions of left-symmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a C[∂]-module is equivalent to a parakähler Lie conformal algebra. We also obtain a conformal analog of the S-equation (see [C. M. Bai, Left-symmetric bialgebras and an analogue of the classical Yang-Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221-260]), and give a construction of the conformal symplectic double.

Left-symmetric conformal algebras and vertex algebras

“…We always use the notation λ (n) in the following context. If χ is Lie, associative or Novikov, by Dong's lemma, similar to that in [18] or [28], we can get the following proposition.…”

“…Proof. One can refer to the proof of Lemma 2.8 in [28]. 2 By Dong's lemma, if (A, F) is a formal distribution χ-algebra (here, χ is Lie, associative or Novikov), then F consists of mutually local formal distributions.…”

“…Let V be a vector space over C. In the theory of vertex algebras [6,28], a field on V is a series of the form a(z) = n∈Z a (n) z −n−1 where a (n) ∈ End V , and for each v ∈ V one has a (n) (v) = 0 for sufficiently large n. [28].) A vertex algebra is a complex vector space V endowed with the state-field correspondence Y which is a C-linear map of V to the space of fields a → Y (a, z) = n∈Z a (n) z −n−1 , a (n) ∈ End V, a vacuum element |0 and an operator T ∈ End V satisfying the following axioms (a, b ∈ V ): The locality condition for two fields is equivalent to the commutative formula (see [28])…”

“…A vertex algebra is a complex vector space V endowed with the state-field correspondence Y which is a C-linear map of V to the space of fields a → Y (a, z) = n∈Z a (n) z −n−1 , a (n) ∈ End V, a vacuum element |0 and an operator T ∈ End V satisfying the following axioms (a, b ∈ V ): The locality condition for two fields is equivalent to the commutative formula (see [28])…”

“…satisfies the axioms of a Lie conformal algebra introduced by Kac in [28,29]. It is also known as vertex Lie algebra in [19,39] or Lie pseudoalgebra [2] over C[T ] (the definition can be found in Section 2).…”