Towards the classification of finite-dimensional diagonally graded commutative algebras

Any finite-dimensional commutative (associative) graded algebra with all nonzero homogeneous subspaces one-dimensional is defined by a 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 $\leq 7$ over an arbitrary field are in bijection with the equivalence classes consisting of coefficient matrices with the same distribution of nonzero entries, while dramatically there may be infinitely many isomorphism classes of dimension $n$ corresponding to one equivalence class of coefficient matrices when $n\geq 8$. Furthermore, we adopt the Skjelbred-Sund method of central extensions to study the isomorphism classes of DGCAs, and associate any DGCA with a undirected simple graph to explicitly describe its corresponding second (graded) commutative cohomology group as an affine variety.