One-dimensional central extensions and simplicities of a class of left-symmetric conformal algebras

In this paper, we introduce the definition of pre-Gel’fand-Dorfman algebra and present several constructions. Moreover, we show that a class of left-symmetric conformal algebras named quadratic left-symmetric conformal algebras are one to one correspondence with pre-Gel’fand-Dorfman algebras. Then we investigate the simplicities and central extensions of quadratic left-symmetric conformal algebras by a one-dimensional centre from the point of view of pre-Gel’fand-Dorfman algebras. We show that under some conditions, central extensions of quadratic left-symmetric conformal algebras by a one-dimensional centre can be characterized by four bilinear forms on pre-Gel’fand-Dorfman algebras. Several methods to construct simple quadratic left-symmetric conformal algebras from pre-Gel’fand-Dorfman algebras are also given.