Abelian Hessian algebra and commutative Frobenius algebra

In this paper we study the structure of an abelian Hessian algebra. First we show that it can be decomposed into unital abelian Hessian algebras and a complete abelian Hessian algebra (abbreviated by CAHA). Then we show that a unital one is in fact a hyperbolic extension of a CAHA. Next we investigate the structure of CAHA by studying double filtration obtained canonically from lower and upper annihilator series. This double filtration together with j-invariant of ternary cubic form give a complete classification of CAHA up to dimension 6.