An Embedding of a Dialgebra into an Associative Conformal Algebra

We prove that a dialgebra (coming from K-theory) can be embedded into an associative conformal algebra (coming from conformal field theory). This shows that these notions are related in the same way as ordinary associative algebras related to Lie algebras. The notion of a dialgebra originally appear in K-theory as an analogue of the universal associative envelope for Leibnitz algebras [L1]. On the other hand, a conformal algebra was introduced in [K1] as a tool for studying operator product expansion (OPE) in conformal field theory. In this note we show that these two structures (coming from different origins) are closely related: an associative conformal algebra can be considered as a dialgebra, and an arbitrary dialgebra can be embedded into an appropriate associative conformal algebra. 1. Definitions Throughout the paper, k be a field of characteristic zero, Z+ stands for the set of non-negative integers. Definition 1 ([L1, L2]). An (associative) dialgebra is a k-linear space A equipped with two linear operations ⊣: A⊗A→ A, ⊢: A⊗A→ A such that a ⊣ (b ⊣ c) = (a ⊣ b) ⊣ c, a ⊢ (b ⊢ c) = (a ⊢ b) ⊢ c, (1) (a ⊣ b) ⊣ c = a ⊣ (b ⊢ c), (2) (a ⊢ b) ⊣ c = a ⊢ (b ⊣ c), (3) (a ⊣ b) ⊢ c = a ⊢ (b ⊢ c) (4) for all a, b, c ∈ A. Definition 2 ([K1]). A conformal algebra is a linear space C endowed with a linear map D : C → C and with a family of linear operations 1