Fundamental Algebras and Varieties with Quadratic Codimension Growth

We classify, up to PI-equivalence, associative algebras over a field of characteristic zero whose codimension sequences have quadratic growth. The decisive step is a structural description of fundamental algebras, which provide the essential finite-dimensional building blocks in the study of varieties of polynomial growth. For every finite-dimensional nonnilpotent fundamental algebra $A$ with radical parameter $s_A=2$, we use the decomposition of $J(A)$ to construct an explicit detector algebra $B(A)$ whose summands record the nonzero radical components and the relevant products among them. We prove that $A{\sim }_{T}B(A)$. Consequently, every algebra with quadratic codimension growth is PI-equivalent to a finite direct sum of the standard generators of the minimal linear and quadratic varieties, together with a possible nilpotent summand, and at least one quadratic generator must occur.