We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.

Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di-or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di-or tri-Var-dendriform algebras is Koszul dual to the operad governing di-or trialgebras corresponding to Var ! . We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a RotaBaxter algebra of nonzero weight in Var. MSC:18D50, 17B69

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A double algebra is a linear space V equipped with linear map V ⊗ V → V ⊗ V . Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space V is naturally described by a linear operator R on the algebra End V of linear transformations of V . Double Lie algebras correspond in this sense to skew-symmetric Rota-Baxter operators, double associative algebra structures-to (left) averaging operators.

We describe irreducible conformal subalgebras of Cend N and build the structure theory of associative conformal algebras with finite faithful representation.1991 Mathematics Subject Classification. 16S32, 16S99, 16W20.