Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.


Introduction
The Yang-Baxter equation is an important subject in mathematical physics [40].Drinfeld highlighted the importance of the study of set-theoretical solutions of the Yang-Baxter equation in [12].The pioneer works on set-theoretical solutions are those of Etingof-Schedler-Soloviev [15], Lu-Yan-Zhu [29] and Gateva-Ivanova-Van den Bergh [19].To understand the structure of set-theoretical solutions, Rump introduced braces in [36] for abelian groups, which provide involutive nondegenerate solutions.See also [5,6] for more details about the finite simple solutions of the Yang-Baxter equation.Later Guarnieri and Vendramin generalized braces to the nonabelian case and introduced skew braces in [24], which provide nondegenerate set-theoretical solutions of the Yang-Baxter equation.Recently, Gateva-Ivanova [18] used braided groups and braces to study set-theoretical solutions of the Yang-Baxter equation.In [1], Angiono, Galindo and Vendramin introduced the notion of Hopf braces, generalizing Rump's braces and Guarnieri-Vendramin's skew-braces.Any Hopf brace produces a solution of the Yang-Baxter equation.
In this paper, we provide another approach to understand the structure of set-theoretical solutions of the Yang-Baxter equation in certain Hopf algebras.In particular, we introduce the notion of post-Hopf algebras, which naturally provide solutions of the Yang-Baxter equation in the underlying vector spaces.We also introduce the notion of relative Rota-Baxter operators on Hopf algebras, which naturally give rise to post-Hopf algebras, and thus to solutions of the Yang-Baxter equation.The whole theory is based on the fact that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra.Note that the classical Grossman-Larsson product was defined in the context of polynomials of ordered rooted trees [34], and have important applications in the studies of Magnus expansions [8,13] and Lie-Butcher series [33,34].
A post-Hopf algebra is a Hopf algebra H equipped with a coalgebra homomorphism from H ⊗ H to H satisfying some compatibility conditions (see Definition 2.1).Magma algebras, in particular ordered rooted trees, provide a class of examples of post-Hopf algebras.A cocommutative post-Hopf algebra gives rise to a new subadjacent Hopf algebra and a module bialgebra structure on itself.The terminology of post-Hopf algebras is justified by the fact that a post-Hopf algebra gives rise to a post-Lie algebra on the space of primitive elements.The notion of post-Lie algebras was introduced in [39], and have important applications in geometric numerical integration [10,11].In [14], Ebrahimi-Fard, Lundervold and Munthe-Kaas studied the Lie enveloping algebra of a post-Lie algebra, which turns out to be a post-Hopf algebra.They also find that there is a new Hopf algebra structure (the subadjacent Hopf algebra) on the Lie enveloping algebra of a post-Lie algebra, by which the Magnus expansions and Lie-Butcher series can be constructed.The subadjacent Hopf algebra is also the main ingredient in our construction of solutions of the Yang-Baxter equation.Moreover, we show that cocommutative post-Hopf algebras and cocommutative Hopf braces are equivalent.As a byproduct, we obtain the notion of pre-Hopf algebras as commutative post-Hopf algebras.
Rota-Baxter operators on Lie algebras and associative algebras have important applications in various fields, such as Connes-Kreimer's algebraic approach to renormalization of quantum field theory [9], the classical Yang-Baxter equation and integrable systems [2,28,38], splitting of operads [3], double Lie algebras [20] and etc. See the book [25] for more details.Recently, the notion of Rota-Baxter operators on groups was introduced in [26], and further studied in [4].One can obtain Rota-Baxter operators of weight 1 on Lie algebras from that on Lie groups by differentiation.Then in the remarkable work [21], Goncharov succeeded in defining Rota-Baxter operators on cocommutative Hopf algebras such that many classical results still hold in the Hopf algebra level.In this paper, we introduce a more general notion of relative Rota-Baxter operators on Hopf algebras containing Goncharov's Rota-Baxter operators as special cases.A cocommutative post-Hopf algebra naturally gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra, and conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra.
Remarkably, a relative Rota-Baxter operator on a cocommutative Hopf algebra naturally gives rise to a matched pair of Hopf algebras.In particular, for a cocommutative post-Hopf algebra, the original Hopf algebra and the subadjacent Hopf algebra form a matched pair of Hopf algebras satisfying certain good properties.Based on this fact, we construct solutions of the Yang-Baxter equation in a Hopf algebra using post-Hopf algebras as well as relative Rota-Baxter operators, and give explicit formulas of solutions for the post-Hopf algebras coming from ordered rooted trees.We further characterize relative Rota-Baxter operators using graphs in the smash product Hopf algebra and module structures.
The paper is organized as follows.In Section 2, first we introduce the notion of post-Hopf algebras and show that a cocommutative post-Hopf algebra gives rise to a subadjacent Hopf algebra together with a module bialgebra structure on itself.Then we show that there is a oneto-one correspondence between cocommutative post-Hopf algebras and cocommutative Hopf braces.In Section 3, we introduce the notion of relative Rota-Baxter operators and show that post-Hopf algebras are the underlying structures, and give rise to relative Rota-Baxter operators on the subadjacent Hopf algebras.In Section 4, we show that a relative Rota-Baxter operator gives rise to a matched pair of Hopf algebras.In particular, a cocommutative post-Hopf algebra gives rise to a matched pair of Hopf algebras.Consequently, one can construct solutions of the Yang-Baxter equation using post-Hopf algebras and relative Rota-Baxter operators.In Section 5, we give some alternative characterizations of relative Rota-Baxter operators using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.
Convention.In this paper, we fix an algebraically closed ground field k of characteristic 0. For any coalgebra (C, ∆, ε), we compress the Sweedler notation of the comultiplication ∆ as for simplicity.Furthermore, for n ≥ 1 we write Let (H, •, 1, ∆, ε, S ) be a Hopf algebra.Denote by G(H) the set of group-like elements in H, which is a group.Denote by P g,h (H) the subspace of (g, h)-primitive elements in H for g, h ∈ G(H).Denote by P(H) the subspace of primitive elements in H, which is a Lie algebra.For other basic notions of Hopf algebras, we follow the textbooks [32].

Post-Hopf algebras
In this section, first we introduce the notion of a post-Hopf algebra, and show that a cocommutative post-Hopf algebra gives rise to a subadjacent Hopf algebra together with a module bialgebra structure on itself.A post-Hopf algebra induces a post-Lie algebra structure on the space of primitive elements and conversely, there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra.Then we show that cocommutative post-Hopf algebras and cocommutative Hopf braces are equivalent.Finally, we introduce the notion of a pre-Hopf algebra which is a commutative post-Hopf algebra.
Recall from [17,39] that a post-Lie algebra and Eqs. ( 1)-( 2) equivalently mean that the linear map 2.1.Post-Hopf algebras and their basic properties.Definition 2.1.A post-Hopf algebra is a pair (H, ⊲), where H is a Hopf algebra and ⊲ : H ⊗ H → H is a coalgebra homomorphism satisfying the following equalities: for any x, y, z ∈ H, and the left multiplication α ⊲ : H → End(H) defined by α ⊲,x y = x ⊲ y, ∀x, y ∈ H, is convolution invertible in Hom(H, End(H)).Namely, there exists unique It is obvious that post-Hopf algebras and homomorphisms between post-Hopf algebras form a category, which is denoted by PH.We denote by cocPH the subcategory of PH consisting of cocommutative post-Hopf algebras and homomorphisms between them.
Remark 2.2.Similar axioms in the definition of a post-Hopf algebra also appeared in the definition of D-algebras [33,34] and D-bialgebras [31] with motivations from the studies of numerical Lie group integrators and the algebraic structure on the universal enveloping algebra of a post-Lie algebra.
Moreover, we have the following properties.Lemma 2.3.Let (H, ⊲) be a post-Hopf algebra.Then for all x, y ∈ H, we have Proof.Since ⊲ is a coalgebra homomorphism, we have By Eq. ( 5), we have α ⊲,1 β ⊲,1 = β ⊲,1 α ⊲,1 = id H , which means that α ⊲,1 is a linear automorphism of H. On the other hand, we have Finally we have Now we give the main result in this section.
Proof.Since ⊲ is a coalgebra homomorphism and H is cocommutative, we have for all x, y ∈ H, which implies that the comultiplication ∆ is an algebra homomorphism with respect to the multiplication * ⊲ .Moreover, we have which implies that the counit ε is also an algebra homomorphism with respect to the multiplication * ⊲ .Since the comultiplication ∆ is an algebra homomorphism with respect to the multiplication •, for all x, y, z ∈ H, we have which implies that the multiplication * ⊲ is associative.For any x ∈ H, by ( 6) and ( 7), we have Thus, (H, * ⊲ , 1, ∆, ε) is a cocommutative bialgebra.Since ⊲ is a coalgebra homomorphism and H is cocommutative, we know that and S ⊲ is a coalgebra homomorphism.Also, note that and it means that Moreover, we have Then by (3) and ( 6), (H, •, 1) is a left H ⊲ -module algebra.Since ⊲ is also a coalgebra homomorphism, (H, •, 1, ∆, ε, S ) is a left H ⊲ -module bialgebra via the action ⊲.
Example 2.6.Any Hopf algebra H has at least the following trivial post-Hopf algebra structure, x ⊲ y = ε(x)y, ∀x, y ∈ H.
In the sequel, we study the relation between post-Hopf algebras and post-Lie algebras.
Theorem 2.7.Let (H, ⊲) be a post-Hopf algebra.Then its subspace P(H) of primitive elements is a post-Lie algebra.
Proof.Since ⊲ is a coalgebra homomorphism, for all x, y ∈ P(H), we have Thus, we obtain a linear map ⊲ : P(H) ⊗ P(H) → P(H).By (3), for all x, y ∈ P(H), we have Thus, we have By (4), we have Thus, we have In [14,35] the authors studied the universal enveloping algebra of a pre-Lie algebra and also of a post-Lie algebra.By [14, Proposition 3.1, Theorem 3.4], the binary product ⊲ in a post-Lie algebra (h, [•, •] h , ⊲) can be extended to its universal enveloping algebra and induces a subadjacent Hopf algebra structure isomorphic to the universal enveloping algebra U(h ⊲ ) of the subadjacent Lie algebra h ⊲ .
We summarize their result in the setting of post-Hopf algebras as follows.We do not claim any originality (see [14,35] for details).
Theorem 2.8.Let (h, [•, •] h , ⊲) be a post-Lie algebra with its subadjacent Lie algebra h ⊲ .Then (U(h), ⊲) is a post-Hopf algebra, where ⊲ is the extension of ⊲ determined by Moreover, the subadjacent Hopf algebra U(h) ⊲ is isomorphic to the universal enveloping algebra U(h ⊲ ) of the subadjacent Lie algebra h ⊲ .
In a recent work [16], Foissy extended any magma operation on a vector space V, i.e. an arbitrary bilinear map ⊛ : V ⊗ V → V, to the coshuffle Hopf algebra (T V, •, ∆ cosh ) as follows: and According to the discussion in [16], it is straightforward to obtain the following result.
Let k{OT } be the free k-vector space generated by OT .The left grafting operator where τ • s ω is the ordered rooted tree resulting from attaching the root of τ to the node s of the tree ω from the left.For example, we have It is obvious that (k{OT }, ) is a magma algebra.By Theorem 2.9, (T k{OT }, •, ∆ cosh , ⊲) is a post-Hopf algebra, where the underlying coshuffle Hopf algebra (T k{OT }, •, ∆ cosh ) has the linear basis consisting of all ordered rooted forests and its antipode S is given by Moreover, it is the universal enveloping algebra of the free post-Lie algebra on one generator { }.See [16,33] for more details about free post-Lie algebras and their universal enveloping algebras.
Let B + : T k{OT } → k{OT } be the linear map producing an ordered tree τ from any ordered rooted forest τ 1 • • • τ m by grafting the m trees τ 1 , . . ., τ m on a new root in order.For example, we have Let B − : k{OT } → T k{OT } be the linear map producing an ordered forest from any ordered rooted tree τ by removing its root.For example, we have Moreover, the operation B − extends to T k{OT } by Note that the subadjacent Hopf algebra (T k{OT }, * ⊲ , ∆ cosh , S ⊲ ) is isomorphic to the Grossman-Larson Hopf algebra of ordered rooted trees defined in [22].Using the left grafting operation, the multiplication * ⊲ is given by for all ordered rooted forests X, Y, and the antipode S ⊲ can be recursively defined by where µ is the unit map and ε is the counit map.
Let (H 4 , ⊲) be a post-Hopf algebra structure on H 4 .Then Namely, g ⊲ g ∈ G(H 4 ) and g ⊲ x ∈ P 1,g⊲g (H 4 ).Since g ∈ G(H 4 ) implies that α ⊲,g is invertible by Eq. ( 5), we know that g ⊲ g = g and g ⊲ x ∈ P 1,g (H 4 ) \ {0}.Also, Therefore, g ⊲ x = x or −x.On the other hand, Then x ⊲ g ∈ P g,g (H 4 ), and thus x ⊲ g = 0.So That is, x ⊲ x ∈ P 1,g (H 4 ), and we can set x ⊲ x = ax for some a ∈ k.Then It implies that g ⊲ x = −x unless a = 0.In summary, one can easily check that there is the post-Hopf algebra structure (H 4 , ⊲ a ) for any a ∈ k illustrated as below, such that α ⊲ a has the convolution inverse α ⊲ −a .

2.2.
Post-Hopf algebras and Hopf braces.In this subsection, we establish the relation between Hopf braces and post-Hopf algebras.

Pre-Hopf algebras. A post-Lie algebra (h, [•,
•] h , ⊲) reduces to a pre-Lie algebra if the Lie bracket [•, •] h is abelian.More precisely, a pre-Lie algebra (h, ⊲) is a vector space h equipped with a binary product ⊲ : From this perspective, we introduce the notion of pre-Hopf algebras as special post-Hopf algebras.
Definition 2.14.A post-Hopf algebra (H, ⊲) is called a pre-Hopf algebra if H is a commutative Hopf algebra.
The above properties for post-Hopf algebras are still valid for pre-Hopf algebras.
Corollary 2.15.Let (H, ⊲) be a cocommutative pre-Hopf algebra.Then is a Hopf algebra, which is called the subadjacent Hopf algebra, where the multiplication * ⊲ and the antipode S ⊲ are given by ( 9) and (10) respectively.
Moreover, H is a left H ⊲ -module bialgebra via the action ⊲.
Corollary 2.16.Let (H, ⊲) be a pre-Hopf algebra.Then its subspace P(H) of primitive elements is a pre-Lie algebra.
Recall that a pre-Lie algebra (h, ⊲) also gives rise to a subadjacent Lie algebra h ⊲ in which the Lie bracket is defined by Corollary 2.17.Let (h, ⊲) be a pre-Lie algebra with its subadjacent Lie algebra h ⊲ .Then the product ⊲ can be extended to the one ⊲ on the symmetric algebra Sym(h), making it a pre-Hopf algebra.Moreover, the subadjacent Hopf algebra Sym(h) ⊲ is isomorphic to the universal enveloping algebra U(h ⊲ ) of the subadjacent Lie algebra h ⊲ .
Let k{T } be the free k-vector space generated by T .The grafting operator : k{T } ⊗ k{T } → k{T } is defined by where τ • s ω is the rooted tree resulting from attaching the root of τ to the node s of the tree ω.
For example, we have Moreover, Chapoton and Livernet [7] have shown that (k{T }, ) is the free pre-Lie algebra generated by { }.By Theorem 2.9, we deduce that (T k{T }, •, ∆ cosh , ⊲) is a post-Hopf algebra.Since (k{T }, ) is a pre-Lie algebra, the post-Hopf algebra structure reduces to the symmetric algebra S k{T }.Thus, we deduce that (S k{T }, •, ∆ cosh , ⊲) is a pre-Hopf algebra.Furthermore, it is the universal enveloping algebra of the free pre-Lie algebra (k{T }, ), and its subadjacent Hopf algebra (S k{T }, * ⊲ , ∆ cosh , S ⊲ ) is dual to the Connes-Kreimer Hopf algebra of rooted trees.

Relative Rota-Baxter operators on Hopf algebras
In this section, first we recall relative Rota-Baxter operators on Lie algebras and groups, and Rota-Baxter operators on cocommutative Hopf algebras.Then we introduce a more general notion of relative Rota-Baxter operators of weight 1 on cocommutative Hopf algebras with respect to module bialgebras.We establish the relation between the category of relative Rota-Baxter operators of weight 1 on cocommutative Hopf algebras and the category of post-Hopf algebras.
Let φ : h → Der(k) be an action of a Lie algebra (h, [•, •] h ) on a Lie algebra (k, [•, •] k ).A linear map T : k → h is called a relative Rota-Baxter operator (of weight 1) on h with respect to (k; φ) if (14) [T (u), Let Φ : H → Aut(K) be an action of a group H on a group K.A map T : K → H is called a relative Rota-Baxter operator (of weight 1) if ( 15) Given any Hopf algebra (H, ∆, ε, S ), define the adjoint action of H on itself by ad x y = x 1 yS (x 2 ).A Rota-Baxter operator (of weight 1) on a cocommutative Hopf algebra H was defined by Goncharov in [21], which is a coalgebra homomorphism B satisfying ( 16) In the sequel, all the (relative) Rota-Baxter operators under consideration are of weight 1, so we will not emphasize it anymore.Now we generalize the above adjoint action to arbitrary actions and introduce the notion of relative Rota-Baxter operators on Hopf algebras.Definition 3.1.Let H and K be two Hopf algebras such that K is a left H-module bialgebra via an action ⇀.A coalgebra homomorphism T : K → H is called a relative Rota-Baxter operator with respect to the left H-module bialgebra (K, ⇀) if the following equality holds: A homomorphism between two relative Rota-Baxter operators T : K → H and T ′ : It is obvious that relative Rota-Baxter operators on Hopf algebras and homomorphisms between them form a category, which is denoted by rRB.We denote by cocrRB the subcategory of rRB consisting of relative Rota-Baxter operators with respect to cocommutative left module bialgebras and homomorphisms between them.
A cocommutative post-Hopf algebra naturally gives rise to a relative Rota-Baxter operator.
so id H : H → H ⊲ is a relative Rota-Baxter operator with respect to the left H ⊲ -module bialgebra (H, ⊲).
Let g : H → H ′ be a post-Hopf algebra homomorphism from (H, ⊲) to (H ′ , ⊲ ′ ).Then (g, g) obviously satisfy Eq. ( 18).Since g is a coalgebra homomorphism and we deduce that g is a homomorphism from the Hopf algebra H ⊲ to H ′ ⊲ ′ .Therefore, (g, g) is a homomorphism from the relative Rota-Baxter operator id H : It is straightforward to check that this is indeed a functor.
It is well-known that a relative Rota-Baxter operator T : k → h on a Lie algebra h with respect to an action (k; φ) endows k with the following post-Lie algebra structure ⊲ T , (19) u Theorem 3.3.Let T : K → H be a relative Rota-Baxter operator with respect to a left Hmodule bialgebra (K, ⇀).Then there exists a post-Hopf algebra structure Let T : K → H and T ′ : K ′ → H ′ be two relative Rota-Baxter operators and ( f, g) a homomorphism between them.Then g is a homomorphism from the post-Hopf algebra (K, ⊲ T ) to (K ′ , ⊲ T ′ ).Consequently, we obtain a functor Ξ : rRB → PH from the category of relative Rota-Baxter operators on Hopf algebras to the category of post-Hopf algebras.
Moreover, the functor Ξ| cocrRB is right adjoint to the functor Υ given in Proposition 3.2.
Proof.Since T is a coalgebra homomorphism and ⇀ is the left module bialgebra action, we have which implies that ⊲ T is a coalgebra homomorphism.Similarly, we have Then by (17), we obtain Define linear map S T : K → K by S T (a) = S H (T (a 1 )) ⇀ S K (a 2 ).( 21) Then for all a ∈ K, we have = S H (T (a 1 ))T (a 2 )T (S T (a 3 )) = S H (T (a 1 ))T (a 2 (T (a 3 ) ⇀ S T (a 4 ))) = S H (T (a 1 ))T (a 2 (T (a 3 ) ⇀ (S H (T (a 4 )) ⇀ S K (a 5 )))) = S H (T (a 1 ))T (a 2 (T (a 3 )S H (T (a 4 )) ⇀ S K (a 5 ))) = S H (T (a 1 ))T (a 2 S K (a 3 )) = S H (T (a 1 ))T (ε K (a 2 )1) = S H (T (a)).( 22) For all a ∈ K, define β ⊲ T ,a ∈ End(K) by β ⊲ T ,a ≔ α ⊲ T ,S T (a) .That is, Then we have Therefore, α ⊲ T is convolution invertible.Hence, (K, ⊲ T ) is a post-Hopf algebra.Let ( f, g) be a homomorphism from the relative Rota-Baxter operator T to T ′ .Then we have which implies that g is a homomorphism from the post-Hopf algebra (K, ⊲ T ) to (K ′ , ⊲ T ′ ).It is straightforward to see that this is indeed a functor.Next we prove that Ξ| cocrRB : cocrRB → cocPH is right adjoint to Υ : cocPH → cocrRB.Namely, Hom cocrRB (id : , where T : K → H is a relative Rota-Baxter operator on a Hopf algebra H with respect to a cocommutative module bialgebra (K, ⇀) and (H ′ , ⊲ ′ ) is a cocommutative post-Hopf algebra.
Let g : (H ′ , ⊲ ′ ) → (K, ⊲ T ) be a post-Hopf algebra homomorphism.Let f = T g, which is obviously a coalgebra homomorphism.For all x, y ∈ H ′ , we have Hence, ( f, g) is a homomorphism between the relative Rota-Baxter operators id : Conversely, if ( f, g) is a homomorphism between the relative Rota-Baxter operators id : By Theorem 3.3 and Theorem 2.4, we immediately get the following result.
Corollary 3.4.Let T : K → H be a relative Rota-Baxter operator with respect to a cocommutative H-module bialgebra (K, ⇀).Then (K, * T , 1, ∆, ε, S T ) is a Hopf algebra, which is called the descendent Hopf algebra and denoted by K T , where the antipode S T is given by (21) and the multiplication * T is given by Moreover, T : K T → H is a Hopf algebra homomorphism.

Matched pairs of Hopf algebras and solutions of the Yang-Baxter equation
In this section, we show that a relative Rota-Baxter operator on cocommutative Hopf algebras naturally gives rise to a matched pair of Hopf algebras.As applications, we construct solutions of the Yang-Baxter equation using post-Hopf algebras and relative Rota-Baxter operators on cocommutative Hopf algebras.
First we recall the smash product and matched pairs of Hopf algebras.Let H and K be two Hopf algebras such that K is a cocommutative H-module bialgebra via an action ⇀.There is the following smash product on K ⊗ H, for any x, x ′ ∈ H, a, a ′ ∈ K, where a ⊗ x ∈ K ⊗ H is rewritten as a#x to emphasize this smash product.We denote such a smash product algebra by K ⋊ H.In particular, if H is also cocommutative, then K ⋊ H becomes a cocommutative Hopf algebra with the usual tensor product comultiplication and the antipode defined by S (a#x) = (S H (x 1 ) ⇀ S K (a))#S H (x 2 ).Definition 4.1.A matched pair of Hopf algebras is a 4-tuple (H, K, ⇀, ↼), where H and K are Hopf algebras, ⇀: H ⊗ K → K and ↼: H ⊗ K → H are linear maps such that K is a left H-module coalgebra and H is a right K-module coalgebra and the following compatibility conditions hold: for all x, y ∈ H and a, b ∈ K.
Let (H, K, ⇀, ↼) be a matched pair of Hopf algebras.The double crossproduct K ⊲⊳ H of K and H is the k-vector space K ⊗ H with the unit 1 K ⊗ 1 H , such that its product, coproduct, counit and antipode are given by ), (32) for all a, b ∈ K and x, y ∈ H. See [30] for further details of the double crossproducts.
By [30, is a linear isomorphism.
Let T : K → H be a relative Rota-Baxter operator with respect to a cocommutative H-module bialgebra (K, ⇀).Define a linear map ↼: Theorem 4.3.With the above notations, if H is also cocommutative, then it is a right K T -module coalgebra via the action ↼ given in Eq. (33).Moreover, the 4-tuple (H, K T , ⇀, ↼) is a matched pair of cocommutative Hopf algebras.
Proof.We define a linear map Φ T : K ⊗ H → K ⊗ H as following: Since T is a coalgebra homomorphism, the linear map Φ T is invertible.Moreover, we have Transfer the smash product Hopf algebra structure K ⋊ H to K ⊗ H via the linear isomorphism Then it is obvious that i K T and i H are injective Hopf algebra homomorphisms, and Therefore, we obtain that (K ⊗ H, • T , 1 T , ∆ T , ε T , S T ) is a Hopf algebra that can be factorized into Hopf algebras K T and H. Thus, we deduce that H is a right K T -module coalgebra via the action ↼ and K T is a left H-module coalgebra via the action ⇀ and the 4-tuples (H, K T , ⇀ , ↼) is a matched pair of Hopf algebras by Theorem 4.2.Moreover, the Hopf algebra Conversely, let H and K be two cocommutative Hopf algebras such that K is an H-module bialgebra via an action ⇀.Let T : K → H be a coalgebra homomorphism, and (K ⊗ H, • T , 1 T , ∆ T , ε T , S T ) the Hopf algebra obtained from the smash product K ⋊ H via the linear isomorphism Φ T given in (34).
T is a relative Rota-Baxter operator with respect to the H-module bialgebra (K, ⇀).
Applying m H (T ⊗ id) and T ⊗ ε H to it respectively, we obtain that Namely, (17) holds, and T is a relative Rota-Baxter operator.
Let (H, ⊲) be a post-Hopf algebra and H ⊲ ≔ (H, * ⊲ , 1, ∆, ε, S ⊲ ) the subadjacent Hopf algebra given in Theorem 2.4.By Proposition 3.2, the identity map id : H → H ⊲ is a relative Rota-Baxter operator.By Theorem 4.3, we have Corollary 4.5.Let (H, ⊲) be a cocommutative post-Hopf algebra.Then the 4-tuple (H ⊲ , H ⊲ , ⊲, ⊳) is a matched pair of cocommutative Hopf algebras, where ⊳ is given by Moreover, we have the compatibility condition Proof.We only need to check the stated compatibility condition, which follows from At the end of this section, we show that post-Hopf algebras and relative Rota-Baxter operators on cocommutative Hopf algebras give rise to solutions of the Yang-Baxter equation.Definition 4.6.A solution of the Yang-Baxter equation on a vector space V is an invertible linear endomorphism R : where ⊳ is defined by (33), is a coalgebra isomorphism and a solution of the Yang-Baxter equation on the vector space H.

Proof. Denote by H l
⊲ and H r ⊲ two copies of the Hopf algebra H ⊲ .By Corollary 4.5, (H l ⊲ , H r ⊲ , ⊲, ⊳) is a matched pair of cocommutative Hopf algebras.Thus, A = H l ⊲ ⊲⊳ H r ⊲ is a Hopf algebra that factorized into Hopf algebras H l ⊲ and H r ⊲ .By Theorem 4.2, there is a coalgebra isomorphism We consider the coalgebra homomorphism ⊲ is a cocommutative Hopf algebra, we deduce that Ψ is a coalgebra isomorphism.Moreover, Ψ satisfies the following equations: For all x, y ∈ H, we have By (37), we have m H ⊲ = m H ⊲ • Ψ.Thus, we deduce that R = Ψ is a braiding operator on the cocommutative Hopf algebra H ⊲ ≔ (H, * ⊲ , 1, ∆, ε, S ⊲ ).By [23,Theorem 4.11], we obtain that R is a solution of the Yang-Baxter equation on the vector space H.
is a coalgebra isomorphism and a solution of the Yang-Baxter equation on the vector space T k{OT }.More precisely, we have Example 4.9.Consider the pre-Hopf algebra (S k{T }, ∆ cosh , ⊲) given in Example 2.18.Then R : is a coalgebra isomorphism and a solution of the Yang-Baxter equation on the vector space S k{T }.More precisely, for forests X, Y ∈ S k{T }, we have Let T : K → H be a relative Rota-Baxter operator on H with respect to a commutative H-module bialgebra (K, ⇀).By Theorem 3.3, (K, ⊲ T ) is a commutative post-Hopf algebra.By Corollary 3.4, there is a descendent Hopf algebra K T = (K, * T , ∆, ε, S T ), such that K is a K T -module bialgebra via the action ⊲ T defined in (20).By Corollary 4.5, we have is a coalgebra isomorphism and a solution of the Yang-Baxter equation on the vector space K, where ⊲ T and ⊳ T are defined by (20) and (38) respectively.

Equivalent characterizations of relative Rota-Baxter operators
In this section, we give some alternative characterizations of relative Rota-Baxter operators using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.

5.1.
Restrictions and extensions of relative Rota-Baxter operators.Let K be a cocommutative H-module bialgebra via an action ⇀.It is obvious that via the restrictions of the action ⇀, we obtain actions of G(H) on G(K) and of P(H) on P(K), for which we use the same notations.As expected, a relative Rota-Baxter operator with respect to a cocommutative H-module bialgebra (K, ⇀) will naturally induces a relative Rota-Baxter operator on the group G(H) and on the Lie algebra P(H) respectively.Theorem 5.1.Let T : K → H be a relative Rota-Baxter operator with respect to a cocommutative H-module bialgebra (K, ⇀).
(i) T | G(K) is a relative Rota-Baxter operator on the group G(H) with respect to the action (G(K), ⇀); (ii) T | P(K) is a relative Rota-Baxter operator on the Lie algebra P(H) with respect to the action (P(K), ⇀).
Proof.Since T is a coalgebra homomorphism, it follows that T | G(K) is a map from G(K) to G(H), and T | P(K) is a map from P(K) to P(H).Hence, T | P(K) is a relative Rota-Baxter operator on the Lie algebra P(H) with respect to the action (P(K), ⇀).
Let φ : h → Der(k) be an action of a Lie algebra (h, [•, •] h ) on (k, [•, •] k ).Then φ can be extended to a module bialgebra action φ : where T k (k) is the tensor k-algebra of k, x ∈ h and y 1 , . . ., y r ∈ k, r ≥ 1.As h acts on k by derivations, it induces a module bialgebra action φ of U(h) on U(k).
The following extension theorem of relative Rota-Baxter operators from Lie algebras to their universal enveloping algebras generalizes the case of Rota-Baxter operators given in [21, Theorem 2].Theorem 5.2.Any relative Rota-Baxter operator T : k → h on a Lie algebra h with respect to an action (k; φ) can be extended to a unique relative Rota-Baxter operator T : U(k) → U(h) with respect to the extended U(h)-module bialgebra (U(k), φ) by where those T (y k )'s left to T are interpreted as the left multiplication by them.
Furthermore, the post-Hopf algebra (U(k), ⊲ T ) induced by the relative Rota-Baxter operator T : U(k) → U(h) as in Theorem 3.3 coincides with the extended post-Hopf algebra (U(k), ⊲T ) from (k, ⊲ T ) given in Theorem 2.8.Namely, we have the following diagram Then it is straightforward to deduce that T (J k ) = 0 and we have the induced linear map T : Next we prove that T : U(k) → U(h) is a relative Rota-Baxter operator.Namely, It can be done by induction on m.The case when m = 1 is due to the recursive definition (41) of T .For yu ∈ U(k) m+1 , since φ is a module bialgebra action, we have T (yu) T which implies that T : U(k) → U(h) is a relative Rota-Baxter operator.The above procedure also implies that the extension from T : k → h to T : U(k) → U(h) is unique.By (19), the induced post-Lie product ⊲ T on k is given by Then by Theorem 2.8, the extended post-Hopf product ⊲T on U(k) is recursively defined by On the other hand, by (20), we know that In particular, y ⊲ T 1 = 0, 1 Therefore, the two post-Hopf products on U(k) coincide, and we get the desired diagram.
Theorem 5.4.A coalgebra homomorphism T : K → H is a relative Rota-Baxter operator with respect to a cocommutative H-module bialgebra (K, ⇀) if and only if the graph Gr T is a Hopf subalgebra of the smash product Hopf algebra K ⋊ H and isomorphic to K T .
Proof.Let T : K → H be a relative Rota-Baxter operator.Then for all a, b ∈ K, we have as the binary operation * T on K defined in ( 23) is a coalgebra homomorphism by the cocommutativity of K, which implies that Gr T is a subalgebra of K ⋊ H with unit 1#1 = 1#T (1).Also, as T is a coalgebra homomorphism and K is cocommutative, ∆(a 1 #T (a 2 )) = (a 1 #T (a 3 )) ⊗ (a 2 #T (a 4 )) = (a 1 #T (a 2 )) ⊗ (a 3 #T (a 4 )) ∈ Gr T ⊗ Gr T , S (a 1 #T (a 2 )) = (S H (T (a 1 )) ⇀ S K (a 2 ))#S H (T (a 3 )) It is well known that T : k → h is a relative Rota-Baxter operator if and only if the graph of T , Gr T := {(u, T (u)) | u ∈ k} is a subalgebra of k ⋊ h.Now we consider the lifted relative Rota-Baxter operator T : U(k) → U(h) of the relative Rota-Baxter operator T : k → h.It turns out that the Hopf algebra Gr T can serve as the universal enveloping algebra of the Lie algebra Gr T .Proposition 5.5.Let T : U(k) → U(h) be the lifted relative Rota-Baxter operator of the relative Rota-Baxter operator T : k → h.Then Gr T ≃ U(Gr T ), i.e. the universal enveloping algebra of the graph Gr T of the relative Rota-Baxter operator T : k → h is isomorphic to the Hopf algebra Gr T , which is the graph of the relative Rota-Baxter operator T : U(k) → U(h).
Also, note that Im ψ generates Gr T as an algebra.Hence, ψ induces a Hopf algebra isomorphism ψ : U(Gr T ) → Gr T by the Theorem of Heyneman and Radford [32, Theorem 5.3.1], as U(Gr T ) 1 = k ⊕ Gr T and ψ| U(Gr T ) 1 is also injective.5.3.Module and module bialgebra characterization.Next we give another characterization of relative Rota-Baxter operators on Hopf algebras using new module structures and new module bialgebra structures.Let H and K be Hopf algebras such that K is a cocommutative H-module bialgebra via an action ⇀.Namely, (17) holds, and T : K → H is a relative Rota-Baxter operator.
The following result is straightforward to obtain.
Lemma 5.7.Let H and K be two cocommutative Hopf algebras such that K is an H-module bialgebra via an action ⇀.Then K is a K ⋊ H-module bialgebra defined by Proof.Let T : K → H be a relative Rota-Baxter operator.By Theorem 5.4, the graph Gr T is a Hopf algebra inside K ⋊ H. Therefore K becomes a Gr T -module bialgebra by Lemma 5.7.Furthermore, pulled back by the Hopf algebra isomorphism Ψ : K T → Gr T given in (42), K becomes a K T -module bialgebra via the desired action ad T .

Corollary 4 . 10 .
The 4-tuples (K T , K T , ⊲ T , ⊳ T ) is a matched pair of cocommutative Hopf algebras, here ⊲ T is given by(20) and ⊳ T is given bya ⊳ T b = S T a 1 ⊲ T b 1 * T a 2 * T b 2 .(38)Moreover, we have the compatibility conditiona * T b = (a 1 ⊲ T b 1 ) * T (a 2 ⊳ T b 2 ).(39)By Theorem 4.7, we have Corollary 4.11.Let T : K → H be a relative Rota-Baxter operator with respect to a commutative H-module bialgebra (K, ⇀).Then R : K For any a, b ∈ G(K), we have T (a)T (b) = T a(T (a) ⇀ b) , which implies that T | G(K) is a relative Rota-Baxter operator on the group G(H) with respect to the action (G(K), ⇀).For any a, b ∈ P(K), we have T (a)T (b) = T (ab) + T T (a) ⇀ b , and thus [T (a), T (b)] = T T (a) ⇀ b − T T (b) ⇀ a + T ([a, b]).

5. 2 .Definition 5 . 3 .
Graph characterization.Now we use graphs to characterize relative Rota-Baxter operators on Hopf algebras.Given any coalgebra homomorphism f : K → H, we define the graph of f , which is denoted by Gr f , as the subspace im(

Theorem 5 . 6 .
A coalgebra homomorphism T : K → H is a relative Rota-Baxter operator if and only if K endowed with the binary operation * T in(23) is an algebra, denoted by K T = (K, * T ), and H is a K T -module via the action ⋆ T defined bya ⋆ T x ≔ T (a)x, ∀x ∈ H, a ∈ K.Proof.If T : K → H is a relative Rota-Baxter operator, then by Corollary 3.4, K T = (K, * T ) is an algebra with unit 1. Also,1 ⋆ T x = T (1)x = 1x = x, (a * T b) ⋆ T x = T (a * T b)x = T (a)T (b)x = a ⋆ T (b ⋆ T x), for any x ∈ H, a, b ∈ K.That is, H is a K T -module.Conversely, if K T = (K, * T ) isan algebra and H is a K T -module via the stated action ⋆ T , then particularly T (a 1 (T (a 2 ) ⇀ b)) = T (a * T b) = T (a * T b)1 = (a * T b) ⋆ T 1 = a ⋆ T (b ⋆ T 1) = T (a)(T (b)1) = T (a)T (b).

(Proposition 5 . 8 .
a#x).b ≔ ad a (x ⇀ b), ∀x ∈ H, a, b ∈ K. Let T : K → H be a relative Rota-Baxter operator.Then K has a cocommutative K T -module bialgebra structure via the following action, ad T,a b ≔ ad a 1 (T (a 2 ) ⇀ b), ∀a, b ∈ K.
then (g, g) is a homomorphism from the relative Rota-Baxter operator id H :H → H ⊲ to id H ′ : H ′ → H ′ ⊲ ′ .Consequently, we obtain a functor Υ : cocPH → cocrRB from the category of cocommutative post-Hopf algebras to the category of relative Rota-Baxter operators with respect to cocommutative left module bialgebras.
Proposition 3.2.Let (H, ⊲) be a cocommutative post-Hopf algebra and H ⊲ the subadjacent Hopf algebra.Then the identity map id H : H → H ⊲ is a relative Rota-Baxter operator with respect to the left H ⊲ -module bialgebra (H, ⊲).Moreover, if g : H → H ′ is a post-Hopf algebra homomorphism from (H, ⊲) to (H ′ , ⊲ ′ ), Proposition 21.6], we have Theorem 4.2.With above notations, (H, K, ⇀, ↼) is a matched pair of Hopf algebras if and only if there exist a Hopf algebra A and injective Hopf algebra homomorphisms i K : K → A, i H : H → A such that the map