Graded Generalized Exponent in Algebras

Updated 1 December 2025

Graded Generalized Exponent is an invariant measuring the exponential growth rate of graded codimensions in algebras with structured gradings.
It generalizes the classical PI-exponent by accounting for additional algebraic structures such as group actions, semigroup gradings, and module parameters.
Recent advances apply combinatorial, representation-theoretic, and functional approaches to compute these exponents in both associative and non-associative modular contexts.

A graded generalized exponent is an invariant describing the asymptotic growth of graded polynomial identities in a graded algebraic structure, typically defined as the exponential rate of growth of a sequence of graded codimensions associated with a chosen grading and, in some cases, with additional structure such as group actions, centrality, or polynomial module parameters. It generalizes the classical PI-exponent of polynomial identity theory to settings where the algebra is equipped with a grading by a semigroup, group, or more general structure, and the identities considered are required to respect this grading or additional compatible actions.

1. Definitions and Framework

Let $A$ be a finite-dimensional algebra over a field $F$ , equipped with a grading by a set, semigroup, or group $T$ : $A = \bigoplus_{t\in T} A_t$ . The sequence of graded codimensions $\{c_n^T(A)\}_{n\geq1}$ is defined via the relatively free $T$ -graded algebra generated by graded variables and the ideal of graded identities satisfied by $A$ . The $n$ th graded codimension $c_n^T(A)$ measures the dimension of multilinear graded polynomials of total degree $n$ that are not graded identities for $A$ .

The graded generalized exponent (or graded PI-exponent) is

$\exp^T(A) = \limsup_{n\to\infty} \sqrt[n]{c_n^T(A)}$

when the limit exists, and describes the exponential growth rate of graded codimensions associated with the algebra and grading. In the presence of extra structure—such as a group action compatible with the grading, a $T$ -graded group $G$ , a graded $W$ -action, or a proper-central identity context—the exponent is modified analogously to capture the associated graded codimension growth (Repovš et al., 2017, Gordienko, 2023, Busalacchi et al., 27 Nov 2025, Benanti et al., 12 May 2025).

In the context of non-associative or modular algebras, the term also refers to algebraic constructions involving generalized exponentials, such as Laguerre polynomials or the Artin–Hasse exponential, which are used to switch gradings via derivations and functional equations (sometimes described as "graded generalized exponentials") (Avitabile et al., 2013, Avitabile et al., 2018).

2. Existence, Integrality, and General Properties

For finite-dimensional algebras over a field of characteristic zero, existence of the graded generalized exponent is well established under broad conditions:

For associative or Lie algebras graded by a finite group or abelian semigroup, or for simple Lie superalgebras with $\mathbb{Z}_2$ -grading, the limit defining the exponent exists and is an integer (Repovš et al., 2016, Repovš et al., 2017, Gordienko, 2023, Gordienko et al., 2015). In the group-graded case, for any such algebra one finds

$\exp^G(A) = \max\{\dim B_i\}$

where the $B_i$ are the simple components in the semisimple decomposition of the algebra as a $G$ -graded module (Gordienko, 2023).

For $T$ a commutative semigroup, and $A$ graded-simple, the exponent exists (Repovš et al., 2017). For non-graded-simple finite-dimensional algebras, sufficient structure (such as unitality and graded Wedderburn–Artin decompositions) ensures existence and integrality (Gordienko, 2014, Gordienko et al., 2015).
In the presence of compatible group actions or more intricate bimodule structures (for instance, $G$ -graded $W$ -algebras), the generalized exponent coincides with the ordinary graded exponent and is an integer (Busalacchi et al., 27 Nov 2025).

Non-integral exponents appear in the setting of semigroup gradings where $T$ is not a group or when the algebra is not semisimple. In this case, the graded PI-exponent can be strictly less than the dimension of the algebra and take non-integer values, controlled by the combinatorial structure of the grading and radical interactions (Gordienko, 2014, Gordienko et al., 2015).

3. Variants: Proper-Central, Generalized, and Group-Action Exponents

Several extensions of the graded generalized exponent exist to capture more specialized phenomena:

Proper central $G$ -exponent: For a variety of $G$ -graded algebras, the growth of the sequence of proper central $G$ -polynomials (those vanishing in the center) is measured by $c_n^{G,\delta}(A)$ , with corresponding exponent

$\exp^{G,\delta}(A) = \lim_{n\to\infty}\sqrt[n]{c_n^{G,\delta}(A)}$

which is again an integer and is characterized in terms of the dimensions of centrally admissible semisimple graded subalgebras. There is a finite list of minimal graded algebras whose inclusion in a variety $\mathcal{V}$ forces the proper-central exponent to exceed $2$ (Benanti et al., 12 May 2025).

Generalized $W$ -exponent: For a $G$ -graded algebra $A$ equipped with a graded action by an algebra $W$ (via the multiplier algebra), the codimension sequences $c_n^{G,W}(A)$ define an exponent $\exp_W^G(A)$ , which coincides with the usual graded exponent and remains integer-valued (Busalacchi et al., 27 Nov 2025).
Graded group action exponent: When the grading interacts with a group action by graded pseudoautomorphisms (i.e., compatibly permuting the graded components up to scalars), the associated codimension growth is bounded by an integer exponent determined by the maximal dimension of simple components in the semisimple quotient of the algebra (Gordienko, 2023).
Lie superalgebras: For finite-dimensional simple Lie superalgebras over characteristic zero, the $\mathbb{Z}_2$ -graded PI-exponent exists and is at most the dimension of the algebra. The underlying proof uses combinatorial "gluing" and Young diagram methods to analyze cocharacters (Repovš et al., 2016).

4. Calculation Techniques and Examples

Computation or estimation of graded generalized exponents relies on analyzing the asymptotics of the codimension sequences, typically via combinatorial constructions, representation theory of symmetric groups (Young diagrams), and structure theory of graded-simple algebras:

For $A = M_n(F)$ with group grading, $\exp^G(A) = n^2$ .
For $UT_2(F)$ with canonical $\mathbb{Z}_2$ -grading, and various $W$ -actions, $\exp_W^{\mathbb{Z}_2}(UT_2) = 2$ in all explored cases (Busalacchi et al., 27 Nov 2025).
In the semigroup-graded case, explicit formulas are given in terms of continuous optimization over polytopes determined by matrix-unit weights, with non-integral possible values (e.g., exponents of the form $1 + m + \sqrt{m}$ for right-zero bands of order $m+1$ ) (Gordienko et al., 2015, Gordienko, 2014).

In summary, for associative algebras graded by finite groups, the graded PI-exponent equals the maximal dimension among the simple graded constituents; for suitably "non-group" or semigroup-graded settings, non-integral values arise, determined by radical extensions and the combinatorics of the grading parameters (Gordienko et al., 2015, Gordienko, 2014).

5. Graded Generalized Exponentials in Modular and Non-Associative Contexts

In the theory of modular non-associative algebras (e.g. Lie algebras over fields of positive prime characteristic) and in grading switching methods, "graded generalized exponent" refers to operator-valued functional calculus derived from generalized exponentials such as:

Artin–Hasse exponential ( $E_p(X)$ ) and its use for grading switching via nilpotent derivations, with functional equations modulo $p$ -torsion that behave similarly to $\exp(X+Y) = \exp(X)\exp(Y)$ (Avitabile et al., 2013).
Generalized Laguerre polynomials ( $L_{p-1}^{(\alpha)}(X)$ ), which play the role of a "graded exponential" for general derivations, and their compositional inverse ( $G^{(\alpha)}(X)$ ) is viewed as a "graded generalized logarithm" (Avitabile et al., 2013, Avitabile et al., 2018).

When applied to an algebra $A$ with a derivation $D$ of prime characteristic, these operator-theoretic generalized exponentials allow one to "switch" gradings by producing new decompositions of $A$ , ensuring that the underlying algebraic structure (e.g., commutativity or associativity) is preserved up to $p$ -torsion corrections. These constructions generalize toral switching and provide a toolkit for constructing new, often inaccessible, group gradings in modular settings (Avitabile et al., 2013, Avitabile et al., 2018).

6. Real-Exponent and Polyhedral Generalizations

Recent extensions consider grading structures over real polyhedral cones, as in the paper of modules over $k[\mathbb{R}_{\geq 0}^n]$ . In this setting, the notion of "generalized exponent" is refined via socle and top functors, which classify the degrees or faces along which modules "die" or are "born." Here, the graded generalized exponent reflects subtle features of support, birth, and death in the module, crucial for applications such as multidimensional persistence modules in topological data analysis (Miller, 2020).

Socle and top constructions in this context provide categorical tools analogous to those in classic PI-theory, allowing for primary, secondary, and irreducible decompositions, and Matlis duality, but now with refined gradings and continuous support sets (Miller, 2020).

7. Structural and Classification Theorems

For group-graded, finite-dimensional algebras, the graded PI-exponent is always integer and determined by block size (Gordienko, 2023, Aljadeff, 2010).
For semigroup gradings, only when $T$ is a zero-band or cancellative semigroup (and $A$ is unital), do graded and ungraded PI-exponents coincide and are integral (Gordienko, 2014, Gordienko et al., 2015).
Classification results exist for varieties of group-graded algebras with proper central exponents, exhibiting a finite minimal set of "obstruction" algebras such that containing one increases the proper central exponent above $2$ (Benanti et al., 12 May 2025).
For graded simple algebras over commutative semigroups, the existence and explicit computation of the exponent is controlled by the structure of the relatively free algebra and multiplicity bounds via polynomial growth of colength (Repovš et al., 2017).
For Lie superalgebras, the $\mathbb{Z}_2$ -graded PI-exponent exists, is integer-valued, and is bounded by the algebra's dimension (Repovš et al., 2016).

Key references:

Repovš–Zaicev ("Graded PI-exponents of simple Lie superalgebras") (Repovš et al., 2016); Gordienko et al. ("Semigroup graded algebras and graded PI-exponent") (Gordienko et al., 2015); Busalacchi–Martino–Rizzo ("Varieties of graded $W$ -algebras and asymptotic behavior of codimension growth") (Busalacchi et al., 27 Nov 2025); E. Miller ("Essential graded algebra over polynomial rings with real exponents") (Miller, 2020); Avitabile–Mattarei ("A generalized truncated logarithm") (Avitabile et al., 2018).