Log-Ozone Group in Poisson Algebras

• Log-Ozone Group is a Poisson-algebraic invariant defined via derivations of the form f⁻¹{-,f} from regular Poisson normal elements in positive characteristic.
• It generalizes the associative ozone group by employing logarithmic derivations to capture center-fixing symmetries beyond mere Hamiltonian derivations.
• This invariant provides insights into the structure of skew-symmetric polynomial Poisson algebras, highlighting properties like unimodularity and Gorenstein centers.

The log-ozone group is a Poisson-algebraic invariant introduced for Poisson algebras over fields of positive characteristic. It is defined not as a group of automorphisms, but as the collection of Poisson derivations of the form $f^{-1}\{-,f\}$ arising from regular Poisson normal elements $f$. This construction was introduced as a Poisson-theoretic analogue of the associative ozone group, while correcting a mismatch that appears in positive characteristic if one works only with derivations annihilating the Poisson center (Chan et al., 9 Jul 2025). In the associative literature, by contrast, the ozone group means the subgroup of automorphisms fixing the center pointwise, $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$ (Chan et al., 2023, Chan et al., 2023, Liu et al., 24 Dec 2025).

1. Terminology, provenance, and conceptual placement

The term “ozone group” first appears in the associative setting. For an algebra $A$ with center $Z(A)$, the ozone group is

$$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$

so it records center-fixing automorphisms (Chan et al., 2023). In PI Artin–Schelter regular contexts, this invariant became structurally significant: it controls normal elements, characterizes skew polynomial rings in maximal-rank cases, and in the PI AS-regular setting is finite abelian (Chan et al., 2023, Liu et al., 24 Dec 2025).

The log-ozone group is a later, distinct notion. It belongs to Poisson algebra, not to associative automorphism theory, and it was introduced precisely because the earlier Poisson notion of an “ozone derivation” was too weak in positive characteristic: every Hamiltonian derivation annihilates the Poisson center, so vanishing on $Z(P)$ does not isolate the correct structure (Chan et al., 9 Jul 2025). The new construction instead mirrors the associative viewpoint in which ozone symmetries are induced by normal elements. In the Poisson setting, the analogue of a normal element is a Poisson normal element, and the associated derivation is logarithmic in form.

This terminological distinction matters. Several associative papers explicitly do not define or use a log-ozone group, only the ozone group itself (Chan et al., 2023, Chan et al., 2023, Liu et al., 24 Dec 2025). The same distinction is reiterated in the study of the Calabi–Yau family $B_q(f)$, where the relevant noncommutative object is the ozone group of automorphisms, whereas the logarithmic variant appears only in the Poisson analogue $A(f)$ discussed at the end of that paper (Gaddis et al., 10 Dec 2025).

2. Formal definition in Poisson algebra

Let $A$ be a Poisson algebra over a field $f$0 of characteristic $f$1. Its Poisson center is

$f$2

An element $f$3 is Poisson normal if

$f$4

The set of Poisson normal elements is denoted

$f$5

For a regular Poisson normal element $f$6, the associated log-ozone derivation is

$f$7

The log-ozone group is then defined by

$f$8

These are the paper’s exact basic definitions (Chan et al., 9 Jul 2025).

The adjective “log” refers to the formal resemblance

$f$9

so $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$0 behaves like a logarithmic Hamiltonian derivation. When $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$1 is a unit, the paper identifies $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$2 with the standard log-Hamiltonian form (Chan et al., 9 Jul 2025).

No separate “log-center” is introduced. Instead the paper defines

$\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$3

the common kernel of all log-ozone derivations. One always has $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$4, and for skew-symmetric polynomial Poisson algebras these two algebras coincide (Chan et al., 9 Jul 2025).

3. Elementary structure and algebraic properties

The first structural fact is multiplicativity of Poisson normal elements and additivity of the associated derivations. If $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$5, then

$\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$6

In characteristic $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$7, one also has

$\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$8

Consequently, $\mathrm{Oz}(A)=\mathrm{Aut}_{Z(A)}(A)$9 is not merely a set: it is an additive subgroup of $A$0, and in fact an elementary abelian $A$1-group, equivalently an $A$2-vector space (Chan et al., 9 Jul 2025).

In graded polynomial Poisson settings, Poisson normal elements interact particularly well with the grading. If $A$3 is graded and $A$4 decomposes into homogeneous summands of distinct degrees,

$A$5

then each $A$6 is again Poisson normal and satisfies

$A$7

This allows reduction to homogeneous Poisson normal elements in many arguments (Chan et al., 9 Jul 2025).

The same paper proves further compatibility statements. For connected graded Poisson domains,

$A$8

and in particular

$A$9

For Poisson Ore extensions $Z(A)$0, the log-ozone group can also be described explicitly in terms of $Z(A)$1, $Z(A)$2, and divisibility conditions on $Z(A)$3 or $Z(A)$4 for Poisson normal $Z(A)$5 (Chan et al., 9 Jul 2025).

A further graded phenomenon underlies the later classification theory. If $Z(A)$6 in a graded polynomial Poisson algebra, then

$Z(A)$7

and the corresponding log-ozone derivations commute: $Z(A)$8 This commuting, often diagonalizable, family of derivations is what makes $Z(A)$9 usable as a recognition invariant (Chan et al., 9 Jul 2025).

4. Maximal log-ozone groups and skew-symmetric Poisson structures

The central structural class is the skew-symmetric polynomial Poisson algebra

$$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$0

where $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$1 is skew-symmetric (Chan et al., 9 Jul 2025). In this setting every generator $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$2 is Poisson normal, hence determines a log-ozone derivation $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$3. These derivations act diagonally on $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$4: $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$5

The paper introduces three auxiliary notions for graded polynomial Poisson algebras. An algebra is “inferable” if every $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$6 acts diagonalizably on $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$7; “quasi-inferable” if every nonzero $$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$8 has some nonzero eigenvalue; and “loz-decomposable” if

$$\mathrm{Oz}(A):=\{\sigma\in \mathrm{Aut}(A)\mid \sigma(z)=z\text{ for all }z\in Z(A)\},$$9

These conditions organize the main recognition theorem (Chan et al., 9 Jul 2025).

The principal classification statement is Theorem 2.12. For an algebraically closed field $Z(P)$0 and a graded polynomial Poisson algebra $Z(P)$1, the following are equivalent:

$Z(P)$2

$Z(P)$3

$Z(P)$4

$Z(P)$5

Thus the log-ozone group is maximal exactly on the skew-symmetric class, once one adds the relevant eigenspace-decomposability hypothesis (Chan et al., 9 Jul 2025). This is the Poisson analogue of earlier associative characterizations of skew polynomial rings via ozone groups (Chan et al., 2023).

For $Z(P)$6, the Poisson center admits an explicit monomial description. If

$Z(P)$7

then

$Z(P)$8

Concretely, a monomial $Z(P)$9 lies in the center exactly when

$B_q(f)$0

In the same skew-symmetric setting, the common kernel of all log-ozone derivations recovers the center: $B_q(f)$1 This makes $B_q(f)$2 a direct tool for detecting central structure (Chan et al., 9 Jul 2025).

5. Examples and explicit computations

The two-variable skew-symmetric case is the model example. Let $B_q(f)$3, $B_q(f)$4, and

$B_q(f)$5

Then

$B_q(f)$6

and $B_q(f)$7 is generated by $B_q(f)$8 and $B_q(f)$9. Its size is

$A(f)$0

while the center can be recovered as

$A(f)$1

This is the clearest illustration of the maximal-rank phenomenon (Chan et al., 9 Jul 2025).

A contrasting example is the Poisson Jordan plane,

$A(f)$2

Here

$A(f)$3

but

$A(f)$4

Moreover,

$A(f)$5

This shows that a large center alone does not force maximality of the log-ozone group (Chan et al., 9 Jul 2025).

Three-dimensional Jacobian-type examples sharpen this contrast. For the unimodular skew-symmetric case

$A(f)$6

the paper computes

$A(f)$7

and $A(f)$8 is inferable. By contrast, if $A(f)$9 is irreducible, then

$A$0

Other examples show intermediate behavior: loz-decomposable but not quasi-inferable, or quasi-inferable without the skew-symmetric form (Chan et al., 9 Jul 2025).

6. Unimodularity, centers, and relation to the broader ozone framework

The log-ozone group is closely tied to unimodularity. For a polynomial Poisson algebra $A$1, the modular derivation is

$A$2

For a skew-symmetric algebra $A$3,

$A$4

so $A$5 is unimodular exactly when

$A$6

The paper proves that if $A$7 is unimodular, then $A$8 is Gorenstein. It also proves a dimension-three extension: if $A$9 and $f$00 is a graded polynomial Poisson algebra of dimension $f$01, then unimodularity implies that $f$02 is Gorenstein (Chan et al., 9 Jul 2025).

Vanishing of the log-ozone group has a sharp three-dimensional description. Assuming $f$03, for

$f$04

one has

$f$05

if and only if $f$06 is unimodular and isomorphic to $f$07 where $f$08 is irreducible or $f$09. A key mechanism is that if $f$10 is not unimodular, then its modular derivation lies in $f$11; hence triviality of $f$12 forces unimodularity (Chan et al., 9 Jul 2025).

This places the log-ozone group within a broader ozone-theoretic sequence. In associative algebra, ozone groups encode center-fixing automorphisms and, in the PI AS-regular case, are finite abelian (Chan et al., 2023, Liu et al., 24 Dec 2025). In the Poisson setting, the logarithmic variant replaces automorphisms by derivations induced from Poisson normal elements (Chan et al., 9 Jul 2025). The connection is explicit in the study of the Poisson analogues

$f$13

where the relevant analogue is again log-ozone derivations rather than an associative ozone group. That paper notes that $f$14 is the semiclassical limit of $f$15, recalls that the log-ozone group of $f$16 is trivial, and asks whether the log-ozone group of $f$17 is trivial for general $f$18 (Gaddis et al., 10 Dec 2025).

In this sense, the log-ozone group is the positive-characteristic Poisson counterpart of the associative ozone-group program, but with a distinctly differential rather than automorphic character. It packages Poisson normality, center structure, eigenspace decompositions, unimodularity, and Gorenstein properties into a single invariant that is both computable in examples and strong enough to characterize the skew-symmetric polynomial Poisson class (Chan et al., 9 Jul 2025).