Matrix-Element Bialgebras Determined by Quadratic Coordinate Algebras

We develop the theory of Manin′s construction of quantum groups from finitely generated quadratic algebras. In general, this construction yields a bialgebra with matrix comultiplication. We give formulae for the relations in the algebra and sufficient conditions for the existence of an antipode and for polynomiality of the algebra; these are more systematic than the calculational approach of previous treatments.