  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.
