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In this paper we consider the representation theory of a non-standard quantization of sl ( 2 ) . This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl ( 2 ) . The results of this paper are used extensively in [4] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT). [ABSTRACT FROM AUTHOR]

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [3] , in studying vertex algebra, we have to deal with such a question: Do there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie algebras? In this paper, we study this question. We introduce the definitions of left-symmetric conformal algebra and Novikov conformal algebra. Many examples of these algebras are obtained. As an application, we present a construction of vertex algebra using left-symmetric conformal algebra. [ABSTRACT FROM AUTHOR]