Relaxed Weak Post-Hopf Algebras

• Relaxed weak post-Hopf algebras are defined by omitting the unitary constraint, while retaining the multiplicativity and post-associative coalgebra conditions.
• The classification on Sweedler’s H4 shows six distinct families, distinguishing genuine relaxed structures from traditional weak post-Hopf cases.
• This framework opens pathways to explore non-unitary behaviors, braided generalizations, and connections with quasi-Hopf algebroids and related algebraic variants.

Relaxed weak post-Hopf algebras are Hopf-algebraic structures obtained by weakening the unit condition in the weak post-Hopf formalism. In the current literature, the notion is introduced by omitting the unitary constraint from the definition of weak post-Hopf algebras: one retains the coalgebra-homomorphism condition and the two post-type compatibility identities, but no longer requires $1 \rhd x = x$. This places relaxed weak post-Hopf algebras downstream from the earlier theory of strict post-Hopf algebras, where the action map is convolution-invertible and cocommutativity drives the construction of sub-adjacent Hopf algebras, matched pairs, and Yang–Baxter operators; earlier papers explicitly did not define weak or relaxed variants (Quanguo, 26 Jul 2025, Li et al., 2022).

1. Definition and basic formalism

Over an algebraically closed field $k$ of characteristic zero, a weak post-Hopf algebra is recalled as a pair $(H,\rhd)$, where $H$ is a Hopf algebra and $\rhd:H\otimes H\to H$ is a coalgebra homomorphism such that, for all $x,y,z\in H$,

$x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$

$1 \rhd x = x.$

The first identity encodes multiplicativity of the action, the second is the post-associative law, and the third is the unitary constraint. The relaxed weak post-Hopf algebra is then defined by deleting the third axiom: a pair $(H,\rhd)$ is relaxed weak post-Hopf if $k$0 is a Hopf algebra, $k$1 is a coalgebra homomorphism, and only

$k$2

$k$3

are imposed (Quanguo, 26 Jul 2025).

A basic consequence survives in the relaxed setting:

$k$4

The derivation uses the multiplicativity identity specialized at the unit together with the assumption that $k$5 is a coalgebra morphism. Thus the action on $k$6 remains rigidly controlled by the counit even when the unitary constraint on the left is removed (Quanguo, 26 Jul 2025).

This asymmetry is the defining algebraic feature of the relaxed notion. In the weak post-Hopf case, both $k$7 and $k$8 hold; in the relaxed case only the first remains forced by the axioms. The resulting structures therefore admit genuinely non-unitary behaviors, rather than merely degenerate presentations of already known weak post-Hopf algebras (Quanguo, 26 Jul 2025).

2. Position within post-Hopf and post-Lie theory

The strict post-Hopf algebra introduced in 2022 is a pair $k$9 with $(H,\rhd)$0 a Hopf algebra and $(H,\rhd)$1 a coalgebra homomorphism satisfying

$(H,\rhd)$2

$(H,\rhd)$3

together with convolution invertibility of the left multiplication map $(H,\rhd)$4. From these axioms one derives

$(H,\rhd)$5

Restricting $(H,\rhd)$6 to primitive elements yields a post-Lie algebra structure on $(H,\rhd)$7, and for a post-Lie algebra $(H,\rhd)$8 the universal enveloping algebra $(H,\rhd)$9 carries a natural post-Hopf structure (Li et al., 2022).

In the cocommutative case, strict post-Hopf algebras admit a sub-adjacent Hopf algebra

$H$0

with antipode

$H$1

and this generalized Grossman–Larsson product supports matched-pair constructions and solutions of the Yang–Baxter equation. Relative Rota–Baxter operators, descendent Hopf algebras, and graph and module-bialgebra characterizations are all developed in that strong setting (Li et al., 2022).

The 2024 Cartier–Quillen–Milnor–Moore analysis broadens the formalism to left and right Post-Hopf algebras and proves that, in characteristic zero, a cocommutative connected right Post-Hopf algebra is isomorphic as a Post-Hopf algebra to the universal enveloping algebra of its primitive Post-Lie algebra. It also shows that, in the connected case, convolution invertibility can be deduced from the post-compatibility axioms in the right Post-Hopf setting (Catoire, 2024).

Against this background, relaxed weak post-Hopf algebras are a genuine weakening rather than a restatement. Earlier papers on post-Hopf algebras explicitly did not define or discuss “weak post-Hopf algebras” or “relaxed” variants; their main constructions rely on the strict post-Hopf axioms and, for Hopf-level consequences, on cocommutativity (Li et al., 2022, Li, 2024). The relaxed weak notion retains the two post-type compatibilities but removes the left unit requirement, and it does not incorporate the convolution-invertibility axiom that is central in the strict theory (Quanguo, 26 Jul 2025).

3. The Sweedler Hopf algebra and the complete classification

The first complete classification of relaxed weak post-Hopf algebra structures is carried out on Sweedler’s $H$2-dimensional Hopf algebra $H$3. This algebra is generated by $H$4 and $H$5 with relations

$H$6

with basis $H$7, comultiplication

$H$8

counit

$H$9

and antipode

$\rhd:H\otimes H\to H$0

Its group-like and primitive-type subspaces are

$\rhd:H\otimes H\to H$1

$\rhd:H\otimes H\to H$2

$\rhd:H\otimes H\to H$3

These spaces control the possible values of $\rhd:H\otimes H\to H$4 under the coalgebra-homomorphism constraint (Quanguo, 26 Jul 2025).

Every relaxed weak post-Hopf algebra structure on $\rhd:H\otimes H\to H$5 belongs to one of six families.

Family	Status	Defining description
(i)	Unitary	sign action of $\rhd:H\otimes H\to H$6, scalar $\rhd:H\otimes H\to H$7 on $\rhd:H\otimes H\to H$8-rows
(ii)	Unitary	$\rhd:H\otimes H\to H$9 collapses $x,y,z\in H$0 to $x,y,z\in H$1, $x,y,z\in H$2-rows determined by $x,y,z\in H$3
(iii)	Unitary	trivial $x,y,z\in H$4- and $x,y,z\in H$5-actions
(iv)	Relaxed	vanishing on $x,y,z\in H$6 and $x,y,z\in H$7 under $x,y,z\in H$8 and $x,y,z\in H$9
(v)	Relaxed	$x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$0 collapses $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$1 to $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$2
(vi)	Relaxed	$x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$3 collapses $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$4 to $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$5 and $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$6 does the same

In family (i), the unitary action is standard on the basis element $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$7 and on the first row,

$x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$8

while $x \rhd (yz) = (x_1 \rhd y)(x_2 \rhd z),$9 acts by sign on the skew-primitive part,

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$0

and the $x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$1- and $x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$2-rows are controlled by a scalar parameter $x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$3:

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$4

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$5

In family (ii), the distinctive feature is

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$6

and the $x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$7-rows take the form

$x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$8

with the same pattern for $x \rhd (y \rhd z) = (x_1(x_2 \rhd y)) \rhd z,$9 (Quanguo, 26 Jul 2025).

Family (iii) is the remaining unitary case:

$1 \rhd x = x.$0

while all actions by $1 \rhd x = x.$1 and $1 \rhd x = x.$2 vanish. The three unitary families are precisely the weak post-Hopf algebra structures previously obtained on $1 \rhd x = x.$3 (Quanguo, 26 Jul 2025).

The new phenomenon appears in families (iv)–(vi). In family (iv),

$1 \rhd x = x.$4

and all actions by $1 \rhd x = x.$5 and $1 \rhd x = x.$6 are zero. In family (v),

$1 \rhd x = x.$7

while again $1 \rhd x = x.$8 and the $1 \rhd x = x.$9- and $(H,\rhd)$0-rows vanish. In family (vi), the collapse is stronger:

$(H,\rhd)$1

with $(H,\rhd)$2 and all $(H,\rhd)$3- and $(H,\rhd)$4-actions trivial. These three cases fail the unitary condition and are the genuinely relaxed structures that diverge from the established weak post-Hopf frameworks (Quanguo, 26 Jul 2025).

4. Proof method, isomorphism classes, and structural interpretation

The classification on $(H,\rhd)$5 proceeds by combining the relaxed axioms with the coalgebra-homomorphism property and the decomposition of $(H,\rhd)$6 into group-like and $(H,\rhd)$7-primitive subspaces. The basic identity

$(H,\rhd)$8

first fixes

$(H,\rhd)$9

One then uses the coproduct constraints to force images into the appropriate primitive subspaces; for example,

$k$00

implies $k$01, while

$k$02

shows that $k$03. Iterating this analysis and applying the axioms

$k$04

yields the six normal forms (Quanguo, 26 Jul 2025).

Up to Hopf-algebra isomorphism, the parameter $k$05 in families (i) and (ii) is partially redundant. For each $k$06, the Hopf automorphism

$k$07

defines conjugate relaxed weak post-Hopf structures by

$k$08

Under this equivalence, nonzero values of $k$09 can often be normalized to a canonical representative such as $k$10. Consequently, families (i) and (ii) each typically contribute two isomorphism classes, namely $k$11 and $k$12, while families (iii)–(vi) contribute one class each (Quanguo, 26 Jul 2025).

The classification is specific to characteristic zero. The note records that in characteristic $k$13 the relation

$k$14

collapses to commutativity, changing both the algebra structure and the primitive subspaces, so the classification does not transfer unchanged (Quanguo, 26 Jul 2025).

Several structural observations are attached to the six families. Because $k$15 is a coalgebra homomorphism, the image of group-like and skew-primitive elements is sharply constrained. The relaxed axioms determine the action on the unit, but impose no further compatibility with the antipode. In families (iv)–(vi), the paper states that the relaxed nature reflects a partial action behavior, and the parameter $k$16 in families (i)–(ii) is viewed as controlling a simple deformation along the $k$17 and $k$18 directions (Quanguo, 26 Jul 2025).

5. Relation to broader weak and relaxed Hopf-type frameworks

Relaxed weak post-Hopf algebras sit at the intersection of two bodies of work that were originally separate. On one side is the post-Hopf/post-Lie program, where strict post-Hopf algebras, sub-adjacent Hopf algebras, relative Rota–Baxter operators, and Post-Hopf CQMM theorems are developed under strong axioms and often under cocommutativity. On the other side is the literature on variant Hopf notions—weak Hopf algebras, quasi-Hopf algebras, Hopf algebroids, multiplier Hopf algebras, and Hopf monads—where relaxation is organized by weakening monoidal or counital constraints (Li et al., 2022, Vercruysse, 2012).

The 2012 survey on variant notions of Hopf algebra does not define post-Hopf structures, but it makes explicit that weak and quasi relaxations correspond to categorical relaxations of the fiber functor: separable Frobenius monoidal functors lead to weak Hopf algebras, quasi-monoidal functors lead to quasi-Hopf algebras, and bimodule-valued forgetful functors lead to bialgebroids and Hopf algebroids. The same survey states that a meaningful framework for “relaxed weak post-Hopf algebras” would therefore lie in the weak/quasi/Hopf-algebroid/multiplier spectrum obtained by relaxing the categorical constraints underlying ordinary Hopf reconstruction (Vercruysse, 2012).

A categorical relaxation beyond classical Hopf monads is provided by slack Hopf monads. This framework does not use the term “post-Hopf,” but it introduces a modified fusion operator and a slack preservation of internal Homs that encompass quasi-Hopf algebras while retaining Hopf-monadic structure. In that precise sense, “slack” supplies a categorical model for relaxed Hopf-type behavior beyond strict Hopf monads (Bruguières et al., 2023).

A more direct extension appears at the algebroid level. Post-Hopf algebroids, introduced in 2025, generalize cocommutative post-Hopf algebras and the pre-Hopf algebroids used in exotic aromatic $k$19-series. The same paper defines weak post-Hopf algebroids by keeping cocommutativity and the post-compatibilities

$k$20

$k$21

$k$22

$k$23

but omitting the $k$24-data required for a full post-Hopf algebroid. It then proposes candidate relaxations toward “relaxed weak post-Hopf algebra/algebroid” by allowing non-cocommutativity, braided Grossman–Larson products, and weak source/target counits $k$25 in place of strict unit/counit identities (Laurent et al., 26 Dec 2025).

The algebraic classification on $k$26 and the algebroid proposals address different levels of generality. The former gives a complete small-dimensional list after removing unitarity; the latter identifies which parts of the post-Hopf algebroid formalism might still survive under braided or weak counital hypotheses. Together they show that relaxation can be pursued both concretely, by direct classification, and structurally, by weakening the Hopf backbone itself (Quanguo, 26 Jul 2025, Laurent et al., 26 Dec 2025).

6. Conceptual issues and open directions

Several misconceptions are explicitly ruled out by the literature. First, relaxed weak post-Hopf algebras are not part of the original post-Hopf definition: the 2022 paper and the 2024 work on sub-adjacent Hopf algebras both state that no weak or relaxed post-Hopf variant is defined there (Li et al., 2022, Li, 2024). Second, the new relaxed families on $k$27 are not merely reparametrizations of the known weak post-Hopf structures: families (iv)–(vi) fail $k$28, and in family (vi) even

$k$29

They therefore form a distinct class of relaxed weak post-Hopf algebraic configurations (Quanguo, 26 Jul 2025).

The existing strict theory also identifies the main obstructions that any broader relaxed theory must face. In the cocommutative strict setting, the generalized Grossman–Larson product

$k$30

is associative and yields a Hopf algebra, but the 2024 study of the sub-adjacent Hopf algebra emphasizes that if cocommutativity is dropped, this product may fail to be associative, and the clean antipode formulas depending on convolution invertibility may fail as well (Li, 2024). This suggests that extending relaxed weak post-Hopf algebras beyond small pointed examples will likely require extra braided compatibility, weak source/target counits, or graded/filtered substitutes for global invertibility.

The concrete open problems stated for the relaxed weak algebraic setting are the extension to other small Hopf algebras such as Taft algebras, the search for a cohomology governing deformations of weak post-Hopf or relaxed weak post-Hopf structures, a sharper analysis of $k$31-orbits of the parameter $k$32, the study of intermediate axioms between full unitarity and the fully relaxed regime, and a systematic relation with partial Hopf actions and measurements (Quanguo, 26 Jul 2025).

At the algebroid level, open questions concern the uniqueness and existence of the anti-automorphism $k$33, the minimal hypotheses on a braiding $k$34 and on $k$35 that ensure associativity of the Grossman–Larson product without cocommutativity, and the formulation of full weak bialgebroid axioms compatible with the post-structure (Laurent et al., 26 Dec 2025). A plausible implication is that the next stage of the subject will involve reconciling the concrete classification methods of the Sweedler case with the braided and weak-counital mechanisms proposed for post-Hopf algebroids.

In this form, relaxed weak post-Hopf algebras occupy a sharply delimited but conceptually significant position. They show that the post-Hopf identities admit non-unitary realizations already on $k$36, while the surrounding literature indicates how such realizations may connect to broader weak, braided, and algebroid generalizations of Hopf-type structure (Quanguo, 26 Jul 2025, Laurent et al., 26 Dec 2025).