What’s the secret behind graded algebras with quadratic growth?

Imagine a mathematical structure, an algebra, split into “colored” pieces by a finite group G. Each piece carries a label, and the operations respect those labels. This is a G‑graded algebra. To gauge how “complicated” an algebra is, mathematicians study its polynomial identities: equations that hold for every element. They measure this complexity with the G‑graded codimensions cⁿᴳ(A). Looking at the sequence of these numbers as n grows, two behaviors emerge: either the growth is slow, like a polynomial, or it’s fast, like an exponential.

The new research focuses on the middle ground, where growth is quadratic, i.e., proportional to n². The authors classified all “varieties” – sets of algebras sharing the same identities – generated by unitary algebras with this growth. The most striking finding is that every such variety can be broken down into a direct sum of smaller algebras, each generating a minimal variety. In practice, it’s like dismantling a complex puzzle into simpler pieces, each with its own fundamental rules. This result deepens our understanding of graded algebras and paves the way for broader classifications in areas like group theory and theoretical physics, where symmetries are key.

The G‑graded codimensions cⁿᴳ(A) are defined as the dimension of the factor space Vₙ⁽ᴳ⁾/Idⁿᴳ(A), where Vₙ⁽ᴳ⁾ is the space of multilinear G‑polynomials of degree n and Idⁿᴳ(A) is its intersection with the identities of the algebra. Quadratic growth means there exists a constant K>0 such that K⁻¹ n² ≤ cⁿᴳ(A) ≤ K n² for sufficiently large n. This behavior typically appears in algebras that, while infinite‑dimensional, have a limited representation structure.

The proposed classification relies on analyzing the simple components in the Wedderburn‑Artin decomposition for G‑graded algebras. Each simple component has the form M_{d}(F)⊗F[G/H] with H≤G, where M_{d}(F) is a d×d matrix algebra and F[G/H] is the group algebra of the quotient. Quadratic growth emerges when d=1 and the index of the coset G/H is bounded, or when d=2 but the grading annihilates most cross‑products. In both cases, the polynomial identities are generated by second‑order Capelli polynomials, imposing partial commutativity relations.

Minimal varieties correspond to algebras with a single non‑trivial simple component. Their direct sum preserves quadratic growth because codimensions add almost independently, thanks to the additivity property of codimensions with respect to direct products of algebras. This result gives a complete picture of quadratic‑growth varieties, linking them to well‑known representation structures and opening the way to extensions toward cubic or quartic growth, where identity complexity rises dramatically.