Post-Hopf Group Algebras: Theory & Applications

• Post-Hopf group algebras are group algebras kG endowed with an extra coalgebra homomorphism that lifts post-Lie identities to the Hopf level.
• They generalize classical Hopf algebras by inducing a secondary Hopf structure via the Grossman–Larson product, connecting to skew braces and relative Rota–Baxter operators.
• Extensions to twisted, Yetter–Drinfeld, and group-graded settings further bridge post-Hopf constructions with Hopf trusses and matched pairs in Yang–Baxter solutions.

Searching arXiv for papers on post-Hopf algebras, post-Hopf group algebras, and related structures. I’m going to look up recent arXiv records on post-Hopf algebras and related constructions. Post-Hopf group algebras are group-algebra realizations of post-Hopf structures: a group algebra $kG$ equipped with an additional coalgebra-compatible binary operation that lifts post-Lie identities from infinitesimal algebra to Hopf level. In the cocommutative theory introduced by Li–Sheng–Tang, these objects connect post-Lie algebras, Hopf braces, relative Rota–Baxter operators, matched pairs, and Yang–Baxter constructions; later work enlarged the framework to twisted, Yetter–Drinfeld, and group-graded settings, where post-Hopf group algebras interact with Hopf trusses, twisted relative Rota–Baxter operators, and Hopf $\pi$-braces (Li et al., 2022, Vilaboa et al., 2024, Ning et al., 27 Jul 2025).

1. Axiomatic framework

A post-Hopf algebra is a pair $(H,\rhd)$ in which $H=(H,\cdot,1,\Delta,\varepsilon,S)$ is a Hopf algebra and $\rhd:H\otimes H\to H$ is a coalgebra homomorphism satisfying three structural requirements: a module-algebra type identity,

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

a post-associativity type identity,

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

and convolution invertibility of the left multiplication map

$$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$

From the coalgebra condition and the Hopf axioms one obtains the derived identities

$$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$

A morphism of post-Hopf algebras is a Hopf algebra morphism preserving $\rhd$ (Li et al., 2022).

The primitive part inherits a post-Lie structure. If

$\pi$0

then $\pi$1 restricts to $\pi$2, and $\pi$3 is a post-Lie algebra, where the bracket is the commutator induced by $\pi$4. This places post-Hopf algebras in the same formal position relative to post-Lie algebras as universal enveloping Hopf algebras occupy relative to Lie algebras (Li et al., 2022).

A trivial post-Hopf structure is given by $\pi$5. Consequently, post-Hopf algebras strictly extend ordinary Hopf algebras. In later variants, two relaxations became important: weak post-Hopf structures, where convolution invertibility is dropped, and twisted post-Hopf structures, where the axioms are modified by a coalgebra endomorphism $\pi$6 (Vilaboa et al., 2024).

2. Classical cocommutative theory and the sub-adjacent Hopf algebra

In the cocommutative case, a post-Hopf structure determines a second Hopf multiplication on the same coalgebra. The fundamental construction is the generalized Grossman–Larson product

$\pi$7

If $\pi$8 is cocommutative, then

$\pi$9

is a Hopf algebra, called the subadjacent Hopf algebra. The original Hopf algebra $(H,\rhd)$0 becomes a left $(H,\rhd)$1-module bialgebra via $(H,\rhd)$2 (Li et al., 2022).

The universal enveloping algebra of a post-Lie algebra is the basic source of examples. If $(H,\rhd)$3 is post-Lie, then $(H,\rhd)$4 carries a natural post-Hopf structure extending $(H,\rhd)$5, and its subadjacent Hopf algebra is canonically isomorphic to $(H,\rhd)$6, where

$(H,\rhd)$7

This places the post-Hopf Grossman–Larson product in direct correspondence with the subadjacent Lie bracket of post-Lie theory (Li et al., 2022).

A Cartier–Quillen–Milnor–Moore theorem also exists in the post-Hopf setting. For a cocommutative connected post-Hopf algebra $(H,\rhd)$8 in characteristic $(H,\rhd)$9,

$H=(H,\cdot,1,\Delta,\varepsilon,S)$0

as post-Hopf algebras. This shows that connected cocommutative post-Hopf algebras are entirely controlled by their primitive post-Lie algebra. The enveloping-algebra model is therefore the canonical connected model of post-Hopf theory (Catoire, 2024).

This also clarifies a recurrent misconception. Ordinary group algebras $H=(H,\cdot,1,\Delta,\varepsilon,S)$1 are cocommutative, but in general they are not connected as coalgebras: the coradical is spanned by all group-like elements. Hence the post-Hopf Cartier–Quillen–Milnor–Moore theorem governs enveloping-algebra realizations more directly than raw discrete group algebras $H=(H,\cdot,1,\Delta,\varepsilon,S)$2 (Catoire, 2024). In parallel, explicit combinatorial formulas for the antipode of $H=(H,\cdot,1,\Delta,\varepsilon,S)$3 and for the inverse Oudom–Guin isomorphism were obtained via a twisted product $H=(H,\cdot,1,\Delta,\varepsilon,S)$4, with the Grossman–Larson Hopf algebra of ordered trees as the model example (Li, 2024).

3. Specialization to ordinary group algebras $H=(H,\cdot,1,\Delta,\varepsilon,S)$5

Let $H=(H,\cdot,1,\Delta,\varepsilon,S)$6 be a group and $H=(H,\cdot,1,\Delta,\varepsilon,S)$7 its group algebra, with Hopf structure

$H=(H,\cdot,1,\Delta,\varepsilon,S)$8

Because $H=(H,\cdot,1,\Delta,\varepsilon,S)$9 is group-like, any coalgebra map $\rhd:H\otimes H\to H$0 sends basis elements to group-like elements. Thus a post-Hopf structure on $\rhd:H\otimes H\to H$1 is determined by a set-theoretic operation

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

extended linearly (Li et al., 2022).

The post-Hopf axioms then become purely group-theoretic. For fixed $\rhd:H\otimes H\to H$3, the map

$\rhd:H\otimes H\to H$4

is a group endomorphism because

$\rhd:H\otimes H\to H$5

The post-associativity axiom becomes

$\rhd:H\otimes H\to H$6

Together with the convolution-invertibility condition, this is precisely the group-level specialization of the abstract post-Hopf axioms (Li et al., 2022).

The associated subadjacent product is

$\rhd:H\otimes H\to H$7

extended linearly to $\rhd:H\otimes H\to H$8. When $\rhd:H\otimes H\to H$9 is post-Hopf, this defines a second Hopf algebra structure on the same coalgebra, usually denoted $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$0. In this sense, a post-Hopf group algebra is a group algebra carrying two compatible Hopf multiplications: the original group-algebra product and the Grossman–Larson-type product coming from $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$1 (Li et al., 2022).

In the cocommutative case, this Hopf-level structure admits a direct group-level interpretation. A post-Hopf structure on $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$2 encodes a post-group structure on $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$3; in the terminology used later in the literature, it is also described as equivalent to a skew brace structure on $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$4, or equivalently to a relative Rota–Baxter operator on $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$5. The Hopf algebra $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$6 is then the linearization of that underlying group-theoretic data (Sciandra, 2024).

4. Categorical correspondences and Yang–Baxter consequences

A central feature of post-Hopf group algebras is that they are one presentation of a larger equivalence pattern. For cocommutative Hopf algebras, a post-Hopf algebra $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$7 gives a relative Rota–Baxter operator by taking the identity map

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

where $x\rhd (y\cdot z)=(x_1\rhd y)\cdot(x_2\rhd z),$9 is the subadjacent Hopf algebra. Conversely, a relative Rota–Baxter operator

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

on a left $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$1-module bialgebra $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$2 induces a post-Hopf structure on $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$3 by

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

In the cocommutative setting these constructions form an adjunction between post-Hopf algebras and relative Rota–Baxter operators (Li et al., 2022).

The same data produce matched pairs of Hopf algebras. From a cocommutative relative Rota–Baxter operator one constructs a descendent Hopf algebra $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$5, a right action on $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$6, and hence a matched pair. Specializing to a cocommutative post-Hopf algebra $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$7, one obtains a matched pair $$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$8, with

$$x\rhd (y\rhd z)=\big(x_1\cdot(x_2\rhd y)\big)\rhd z,$$9

This is the Hopf-algebraic analogue of brace-type factorization (Li et al., 2022).

From the matched pair one obtains a Yang–Baxter operator

$$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$0

which is a coalgebra isomorphism satisfying the Yang–Baxter equation. For $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$1, the construction frequently restricts to set-theoretic data on group-like elements, so post-Hopf group algebras supply algebraic linearizations of brace-type Yang–Baxter solutions on groups (Li et al., 2022).

This correspondence explains why, in the cocommutative case, post-Hopf algebras are often described as equivalent to Hopf braces. The post-Hopf product $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$2 is not merely an auxiliary operation; it is the mechanism that reconstructs the second Hopf multiplication, the matched pair, and the resulting braiding operator (Li et al., 2022).

5. Twisted and Yetter–Drinfeld extensions

The twisted theory replaces the single operation $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$3 by a triple $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$4, where $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$5 is a coalgebra morphism and $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$6 is a coalgebra endomorphism. A weak twisted post-Hopf algebra satisfies twisted analogues of post-associativity, associativity of $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$7, and compatibility with the original multiplication. If $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$8 preserves the unit and a convolution-invertibility condition holds, one obtains a twisted post-Hopf algebra in the non-weak sense. The untwisted specialization $$\alpha_{\rhd}:H\to \operatorname{End}(H),\qquad \alpha_{\rhd,x}(y)=x\rhd y.$$9 recovers the post-Hopf algebras of Li–Sheng–Tang (Vilaboa et al., 2024).

In the cocommutative case, and therefore for $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$0, the extra “class conditions” appearing in the braided formalism are automatic. A coalgebra map

$$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$1

is determined by a set map $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$2, and a coalgebra map

$$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$3

amounts, on basis elements, to a secondary group-like operation. The paper defines the associated second multiplication

$$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$4

which on basis elements reads

$$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$5

This makes $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$6 into a Hopf truss, and conversely any Hopf truss on $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$7 yields a twisted post-Hopf structure (Vilaboa et al., 2024).

The main categorical statement is that, under the relevant conditions, weak twisted post-Hopf algebras, Hopf trusses, and weak twisted relative Rota–Baxter operators are equivalent categories. Under cocommutativity these conditions are automatic, so cocommutative weak twisted post-Hopf algebras are equivalent to cocommutative Hopf trusses, and via invertible operators also to cocommutative twisted relative Rota–Baxter operators. For group algebras this gives a precise twisted enlargement of the classical post-Hopf/Hopf-brace correspondence (Vilaboa et al., 2024).

A different extension is the Yetter–Drinfeld theory. A Yetter–Drinfeld post-Hopf algebra is a post-Hopf object in the braided category $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$8, with a subadjacent Hopf algebra $$x\rhd 1=\varepsilon(x)1,\qquad 1\rhd x=x,\qquad S(x\rhd y)=x\rhd S(y).$$9 and a braided compatibility replacing ordinary cocommutativity. The category of Yetter–Drinfeld post-Hopf algebras is isomorphic to the category of Yetter–Drinfeld braces and equivalent to a subcategory of Yetter–Drinfeld relative Rota–Baxter operators. When the underlying Hopf algebra is cocommutative, the braided conditions collapse to the classical ones, so on $\rhd$0 a Yetter–Drinfeld post-Hopf structure is exactly an ordinary post-Hopf structure on the group algebra (Sciandra, 2024).

6. Group-graded generalization and terminological scope

A recent shift in terminology occurs in the theory of Hopf $\rhd$1-algebras, where $\rhd$2 is an abelian group and

$\rhd$3

is a family of coalgebras with multiplications $\rhd$4. In that context, a post-Hopf $\rhd$5-algebra is a Hopf $\rhd$6-algebra equipped with a family of coalgebra homomorphisms

$\rhd$7

satisfying the group-graded analogues

$\rhd$8

and

$\rhd$9

together with convolution invertibility of the induced endomorphisms (Ning et al., 27 Jul 2025).

In that paper, “post-Hopf group algebra” is exactly this group-graded notion. Under cocommutativity, one defines a second multiplication

$\pi$00

and obtains a subadjacent Hopf $\pi$01-algebra $\pi$02. The main theorem states that cocommutative post-Hopf $\pi$03-algebras are in bijection with cocommutative Hopf $\pi$04-braces. In the same framework, a Rota–Baxter Hopf $\pi$05-algebra canonically produces a Hopf $\pi$06-brace, hence also a post-Hopf $\pi$07-algebra (Ning et al., 27 Jul 2025).

Accordingly, current usage supports two closely related meanings of the expression “post-Hopf group algebra.” In the earlier cocommutative literature it refers to an ordinary group algebra $\pi$08 endowed with a post-Hopf structure. In the later group-graded literature it refers to a post-Hopf $\pi$09-algebra, that is, a Hopf group algebra in Turaev’s sense equipped with a graded post-operation. The two meanings are compatible rather than competing: the latter is a genuine group-graded enlargement of the former (Ning et al., 27 Jul 2025).

At the structural level, both usages preserve the same organizing principle. A post-Hopf group algebra is a coalgebra whose original multiplication can be deformed, split, or reassembled by a post-type operation into a second multiplication; this second multiplication is the Hopf-brace or Hopf-truss side of the theory, while the original operation also encodes relative Rota–Baxter data, matched pairs, and braiding constructions. In that sense, post-Hopf group algebras are the Hopf-algebraic linear avatars of post-group, truss, and brace structures on groups and group-graded systems.