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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.

Let A be a left-symmetric (resp. Novikov) algebra, E be a vector space containing A as a subspace and V be a complement of A in E. The extending structures problem which asks for the classification of all leftsymmetric (resp. Novikov) algebra structures on E such that A is a subalgebra of E is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on E such that A is a subalgebra of E if and only if E is isomorphic to a unified product of A and V . Two cohomological type objects H 2 A (V, A) and H 2 (V, A) are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension A ⊂ E of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several special examples are provided in details. 1 2 YANYONG HONGThis problem is called extending structures problem. The extending structures problems for groups, associative algebras, Hopf algebras, Lie algebras, and Leibniz algebras have been studied in [4,3,6,5,7] respectively. Since left-symmetric algebras and Novikov algebras are important algebraic structures in mathematics and mathematical physics, it is interesting and meaningful to study the extending structures problems for left-symmetric algebras and Novikov algebras. Certainly, this problem is very difficult. When A = {0}, this problem is equivalent to classify all left-symmetric algebras and Novikov algebras on an arbitrary vector space E. Of course, when the dimension of E is large, it is impossible to classify all algebraic structures. In fact, the classifications of left-symmetric algebras in dimension 2 and dimension 3 over C were obtained in [11,10], and the classifications of Novikov algebras in dimension 2 and dimension 3 over R and C were obtained in [12,16]. But, as far as we know, the complete classifications of left-symmetric algebras (resp. Novikov algebras) whose dimension are larger than 3 have not been obtained so far. There are only some papers on the classification of some special left-symmetric algebras and Novikov algebras of dimensions 4 and 5, such as [18,23,24,16]. Therefore, we will assume that A = {0} below. In fact, the extending structures problem generalizes some classic algebraic problems. For example, the extension of the left-symmetric algebra A by an abelian left-symmetric algebra C is characterized by the second cohomology group H 2 (C, A) (see [19]); the problem which asks for describing and classifying the extensions of the left-symmetric algebra A by another left-symmetric algebra B can be answered by bicrossed products of A and B (see [9]) and this problem is called the factorization problem. In this paper, our aim is to give a theoretical interpretation of the extending structures problem for left-symmetric (resp. Novikov) algebras using some cohomological type objects and provide some detailed answers to it in certain ...

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In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view of the corresponding Gel'fand-Dorfman bialgebras, some sufficient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several classes of new infinite simple Lie conformal algebras. These results will be useful for classification purposes.

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Let (g, [•, •], δ g ) be a fixed Lie bialgebra, E be a vector space containing g as a subspace and V be a complement of g in E. A natural problem is that how to classify all Lie bialgebraic structures on E such that (g, [•, •], δ g ) is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on g is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on E, called the unified co-product of (g, δ g ) by V . With this unified co-product and the unified product of (g, [•, •]) by V developed in [1], the unified bi-product of (g, [•, •], δ g ) by V is introduced. Moreover, we show that any E in the extending structures problem is isomorphic to a unified bi-product of (g, [•, •], δ g ) by V . Then an object HBI 2 g (V, g) is constructed to classify all E in the extending structures problem. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when dimV = 1 are investigated in detail.

In this article, a large class of simple modules over the Schrödinger-Virasoro algebra G are constructed, which include highest weight modules and Whittaker modules. These modules are determined by the simple modules over the finite-dimensional quotient algebras of some subalgebras. Moreover, we show that all simple modules of G with locally finite actions of elements in a certain positive part belong to this class of simple modules. Similarly, a large class of simple modules over the W -algebra W (2, 2) are constructed.