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Fundamental Algebras and Varieties with Quadratic Codimension Growth

This paper classifies associative algebras over a field of characteristic zero with quadratic codimension growth up to PI-equivalence by establishing a structural decomposition that shows every such algebra is PI-equivalent to a finite direct sum of standard generators for minimal linear and quadratic varieties, a nilpotent summand, and at least one quadratic generator.

Original authors: Wesley Quaresma Cota

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wesley Quaresma Cota

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a branch dedicated to understanding the hidden rules that govern how things combine. Imagine a collection of objects where you can mix them together in specific ways, following a set of laws that dictate the outcome of every possible combination. Mathematicians call these structures algebras. While the objects themselves might be numbers, matrices, or abstract symbols, the laws they obey—known as polynomial identities—are often so complex that listing them all is impossible. To make sense of this chaos, researchers look for a way to measure how fast these rules grow as the combinations become more intricate. They do this by counting the number of independent rules that appear at each level of complexity. This count, known as the codimension sequence, acts like a ruler for the size of the algebra's universe. If the count grows slowly, the algebra is simple and predictable; if it explodes, the algebra is wild and chaotic. For decades, mathematicians knew that these growth rates fell into two distinct camps: either they grew slowly, following a smooth curve, or they exploded exponentially. The question that remained unanswered was exactly what those slow, smooth curves looked like when they were not just straight lines, but gentle curves that bent slightly upward.

A researcher has now mapped this specific territory, providing a complete catalog of every algebra that grows at this specific, quadratic rate. Their work, led by Wesley Quaresma Cota at the University of São Paulo, solves a puzzle that had been open since the mid-2000s. Before this, mathematicians knew the five simplest algebras that produced this quadratic growth, but they did not know if there were other, more complicated algebras that could also grow at this rate without being built from those five. The new study proves that there are no hidden surprises. Every algebra that grows at this specific quadratic speed is essentially a combination of those five known building blocks, perhaps mixed with a few simpler linear ones and a bit of "noise" that eventually disappears. The researcher achieved this by developing a new method to dissect the internal structure of these algebras, stripping away the unnecessary parts to reveal a core "detector" that perfectly records the algebra's behavior.

The journey to this conclusion began with a focus on the most fundamental pieces of the puzzle. In the world of algebra, complex structures can often be broken down into smaller, irreducible components called fundamental algebras. These are the atoms of the field; any larger algebra is essentially a collection of these fundamental pieces glued together. The researcher realized that to understand the quadratic growth, they only needed to understand the fundamental pieces that grow at this rate. They identified a specific property of these pieces, related to how their internal "radical" parts—essentially the messy, non-repeating components—interact with each other. When these interactions are just right, the algebra grows quadratically.

To solve the problem, the researcher invented a clever tool they call a detector algebra. Imagine taking a complex algebra and looking at its internal layers. The detector is a simplified model built from the five known quadratic building blocks. It is constructed by checking which specific interactions occur within the original algebra. If the original algebra has a certain type of interaction between its parts, the detector includes a specific building block. If it lacks that interaction, the detector leaves that block out. The researcher proved that this detector is not just a rough sketch; it is a perfect mirror. The original algebra and its detector follow exactly the same rules. This means that to understand the complex, original object, one only needs to look at the simple detector made of the known pieces.

The power of this discovery lies in what it rules out. For years, there was a lingering doubt that there might be some exotic, unknown algebra that grew quadratically but could not be built from the five known minimal varieties. The new proof eliminates this possibility entirely. It shows that no matter how complex an algebra appears, if its growth rate is quadratic, it is guaranteed to be equivalent to a combination of the five known quadratic generators and the two known linear generators, plus a nilpotent part that vanishes in the long run. The researcher demonstrated that if an algebra lacks the specific internal interactions that trigger quadratic growth, it will only grow linearly. Conversely, if it does have those interactions, it must contain at least one of the five quadratic building blocks.

This result unifies the entire field of algebras with slow growth. Previously, the linear growth algebras were understood, and the five minimal quadratic algebras were known, but the general case remained a mystery. Now, the picture is complete. Any algebra that grows at a quadratic rate is simply a finite sum of these specific, well-understood components. The work does not just list these algebras; it provides a structural description that explains why they are the only ones possible. By reducing the infinite variety of complex algebras to a finite set of building blocks, the researcher has provided a definitive map for this corner of mathematics. The classification is absolute: there are no other quadratic growth patterns waiting to be discovered, and the five algebras identified in 2006 are the only genuine sources of this specific type of complexity. The study stands as a complete proof, leaving no room for ambiguity or further exceptions in this specific domain.

Technical Summary: Fundamental Algebras and Varieties with Quadratic Codimension Growth

Problem Statement
The paper addresses the classification of associative algebras over a field of characteristic zero whose codimension sequences exhibit quadratic growth. While the asymptotic behavior of codimension sequences $c_n(A)$ is a fundamental invariant in the theory of polynomial identities (PI-theory), distinguishing between different rates of polynomial growth requires more than just the existence of a bound. Previous work by Giambruno and La Mattina [3] classified algebras with linear growth, and Alves Jorge and Vieira [11] identified the five minimal varieties of quadratic growth ( $var(A_2), var(A_4), var(A_4^*), var(A_5), var(A_6)$ ). However, a gap remained in the general classification: it was unknown whether an arbitrary variety of quadratic growth could contain components not arising from these five minimal varieties or from varieties of linear growth. Specifically, the structural description of arbitrary algebras with quadratic growth, particularly those that are not necessarily unital, remained open.

Methodology
The author employs the framework of fundamental algebras, which serve as the essential finite-dimensional building blocks for PI-equivalence classes. The methodology proceeds through the following steps:

Reduction to Fundamental Algebras: Utilizing Kemer's theory and recent results by Giambruno, Quintino, and Vieira [7], the study of an arbitrary algebra of polynomial growth is reduced (up to a nilpotent summand) to a single fundamental algebra. For algebras with quadratic growth, this fundamental algebra must have a radical parameter $s_A = 2$ (where $s_A$ is the nilpotency index of the Jacobson radical minus one) and a one-dimensional semisimple part.
Structural Decomposition: The Jacobson radical $J(A)$ of a fundamental algebra $A = F + J(A)$ is decomposed into subspaces $J_{ik}$ based on the action of the identity element of the semisimple part. This yields components $J_{10}, J_{01}, J_{00}, J_{11}$ .
Detector Algebra Construction: The core technical innovation is the construction of a detector algebra $B(A)$ . This algebra is a finite direct sum of specific "model" algebras ( $A_1, A_1^*, A_2, A_4, A_4^*, A_5, A_6$ ). The presence of each summand in $B(A)$ is determined by indicator parameters ( $\varepsilon_i$ ) that record the non-vanishing of specific products and commutators within the radical components of $A$ (e.g., $J_{10}J_{00}$ , $[J_{11}, J_{11}]$ , $J_{01}J_{10}$ ).
PI-Equivalence Proof: The author proves that $A$ is PI-equivalent to $B(A)$ ( $A \sim_T B(A)$ ). This is achieved by showing that the T-ideals of $A$ and $B(A)$ coincide. This involves demonstrating that any multilinear polynomial identity satisfied by $B(A)$ is also satisfied by $A$ , leveraging the specific multiplication rules of the radical components and the fact that $J^3 = \{0\}$ .

Key Contributions and Results

Structural Characterization: The paper establishes that for every finite-dimensional nonnilpotent fundamental algebra $A$ with $s_A = 2$ , the algebra is PI-equivalent to a direct sum of the minimal generators of linear and quadratic varieties.
The Detector Algebra: The introduction of $B(A)$ provides an explicit mechanism to "detect" the quadratic growth components within the radical structure of $A$ . The summands of $B(A)$ correspond exactly to the non-zero radical interactions that generate quadratic growth.
Completeness of Minimal Varieties: The main theorem (Theorem 1.1) proves that every variety of quadratic growth is generated, up to a nilpotent component, by a finite direct sum of algebras from the set $\{A_1, A_1^*, A_2, A_4, A_4^*, A_5, A_6\}$ . Crucially, the set must contain at least one algebra from the quadratic minimal set $\{A_2, A_4, A_4^*, A_5, A_6\}$ .
Classification Theorem: The paper provides a complete classification (Corollary 5.3) stating that an algebra $A$ has quadratic codimension growth if and only if:
$A \sim_T N \oplus \bigoplus_{M \in S} M$
where $N$ is a nilpotent algebra, $S$ is a subset of the known minimal generators, and $S$ intersects the set of quadratic minimal generators.

Significance
The paper claims to complete the classification of varieties with quadratic codimension growth, a problem that remained open since the 2006 classification of minimal varieties [11]. The significance lies in demonstrating that the five minimal varieties of quadratic growth identified in previous literature are sufficient to generate all varieties of quadratic growth; no new "genuinely quadratic" building blocks are required beyond those already known.

By reducing the general case to fundamental algebras and constructing the detector algebra, the author bridges the gap between the study of minimal varieties and arbitrary varieties. This result unifies the description of varieties with at most quadratic growth, showing they are generated by finite direct sums of minimal varieties (linear or quadratic) and nilpotent algebras. The work reinforces the role of fundamental algebras as the primary structural units in PI-theory and provides a concrete, algorithmic method (via the detector algebra) to determine the PI-equivalence class of any algebra with quadratic growth.

Technical Summary: Fundamental Algebras and Varieties with Quadratic Codimension Growth

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