What’s the secret of algebras with graded involution and quadratic growth?

Algebras are mathematical structures that, like LEGO bricks, combine according to strict rules. Among them, upper‑triangular matrix algebras have led to the discovery of “minimal varieties”, i.e., classes of algebras that grow at the slowest possible rate. When we add two extra ingredients – a grading by a finite group G and an involution * that respects the grading – we obtain the so‑called $(G,*)$‑algebras.

The focus of this study is the “codimension”, a numerical sequence that measures how far a variety deviates from the simplest case. If the codimension growth is quadratic, meaning proportional to $n^2$, the variety is said to have quadratic growth. The authors examined how these sequences behave when algebras carry a graded involution, and they discovered that only a few “minimal varieties” generate quadratic growth.

In practice, they classified all possible finite‑dimensional $(G,*)$‑algebras that, despite their complexity, keep growth under control. This result not only closes a chapter on classifying varieties with $n^2$ growth, but also opens the path to study varieties with $n^k$ growth for $k>2$, a still‑largely unexplored terrain. For young math enthusiasts, it’s an invitation to see algebras not just as abstract formulas, but as worlds where rules and symmetries create surprising patterns.

A $(G,*)$‑algebra is an associative algebra equipped with a decomposition $A=\bigoplus_{g\in G}A_g$ such that $A_gA_h\subseteq A_{gh}$ for all $g,h\in G$, and with an involution $*:A\to A$ satisfying $(a^*)^*=a$, $(ab)^*=b^*a^*$ and $A_g^*\subseteq A_{g^{-1}}$. The codimension sequence $c_n(A)$ is defined as the dimension of the factor $P_n/(P_n\cap T(A))$, where $P_n$ is the space of multilinear polynomials of degree $n$ and $T(A)$ is the ideal of polynomial identities of $A$.

The main result of the paper states that, for a finite‑dimensional $(G,*)$‑algebra $A$, the asymptotic growth of $c_n(A)$ is quadratic if and only if $A$ is PI‑equivalent to a direct sum of upper‑triangular matrix algebras with the standard graded involution. In more technical terms, the variety generated by $A$ is minimal with respect to having growth exponent $2$, i.e., its Gelfand–Kirillov exponent is $2$.

The proof employs finite group representation theory, the Wedderburn–Artin decomposition for graded semisimple algebras, and explicit computation of $*$‑polynomial identities. A crucial step is the application of the Amitsur–Levitzki theorem for graded matrices, which provides an upper bound for the codimensions. Moreover, it is shown that every variety with quadratic growth contains a minimal variety of type $UT_2(F)$ with graded involution, where $UT_2(F)$ denotes the algebra of $2\times2$ upper‑triangular matrices over a field $F$. This yields a complete classification of minimal quadratic‑growth varieties, paving the way for studies on varieties with polynomial growth $n^k$ for $k>2$, where the structure of $*$‑polynomial identities becomes even richer and more complex.

The implications are twofold: on one hand, we gain a deeper understanding of the link between the internal structure of a graded algebra and its asymptotic behavior; on the other, we obtain a reference model for future investigations of $n^k$‑growth varieties with $k>2$, where the interplay of grading and involution promises new algebraic phenomena.