Towards the classification of finite-dimensional diagonally graded commutative associative algebras

Any n-dimensional commutative (associative) graded algebra with all nonzero homogeneous subspaces one-dimensional and of degrees exactly from 1 to n is defined by an $$n\times n$$ symmetric coefficient matrix. This algebraic structure gives a basic kind of A-graded algebras originally studied by Arnold. In this paper, we call them diagonally graded commutative algebras (DGCAs) and verify that the isomorphism classes of DGCAs of dimension $$\le 5$$ over an arbitrary field are in bijection with the equivalence classes consisting of coefficient matrices with the same distribution of nonzero entries, while unexpectedly there may be infinitely many isomorphism classes of dimension n corresponding to one equivalence class of coefficient matrices when $$n\ge 8$$ . Furthermore, we adopt the Skjelbred-Sund method of central extensions to study the isomorphism classes of DGCAs, and associate any DGCA with an undirected simple graph to explicitly describe its corresponding second (graded) commutative cohomology group as an affine variety.