Startseite Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
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Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras

  • Wei Liu und Xiaoli Fang EMAIL logo
Veröffentlicht/Copyright: 18. November 2021
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 19 Heft 1

Abstract

In this paper, we investigate a more general category of Θ -Yetter-Drinfeld modules ( Θ ∈ Aut H ( H ) ) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ -Yetter-Drinfeld modules with a bijective antipode S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ -Yetter-Drinfeld category over a Hom-Hopf algebra H is symmetric, then H is trivial. Finally, we find a sufficient and necessary condition for a Θ -Yetter-Drinfeld category to be pseudosymmetric.

Keywords: Hom-Hopf algebra; Θ-Yetter-Drinfeld module; symmetric pair
MSC 2010: 16T05; 16T15

1 Introduction

The genesis of Hom-structures may be found in the physics literature from the 1990, concerning quantum deformations of algebras of vector fields, especially Witt and Virasoro algebras. These classes of examples led to the development of Hom-Lie algebras; see [1,2, 3,4]. Recently, various Hom-Lie structures have been studied further by many scholars; see [5,6,7, 8,9,10, 11,12]. The idea of tailoring associativity-like conditions by linear maps was migrated to other algebraic structures. The concepts of Hom-algebras, Hom-coalgebras, Hom-Hopf algebras and quasi-triangular Hom-bialgebras were introduced and further developed; see [13,14, 15,16,17, 18,19,20, 21,22].

Yetter-Drinfeld modules are known to be at the origin of a very vast family of solutions to the Yang-Baxter equation. Therefore, it is very natural to extend the notion of Yetter-Drinfeld module into the Hom-setting. The main purpose to study Yetter-Drinfeld module in the Hom-setting is to prove that the category of Yetter-Drinfeld modules over a Hom-Hopf algebra with a bijective antipode is also a braided tensor category. Furthermore, we can construct some solutions of the classical Yang-Baxter equation and the Hom-Yang-Baxter equation introduced by Yau [12]. Two directions of the study were developed, one considering the category of Yetter-Drinfeld modules under monoidal Hom-Hopf algebras introduced by Caenepeel and Goyvaerts [23]; see also [24,25,26] and others, discussing the category of Yetter-Drinfeld modules under Hom-Hopf algebras in [14,27]. For a monoidal Hom-Hopf algebra, Caenepeel and Goyvaerts explained it from a categorical point of view. Therefore, we can write the corresponding conditions by the categorical method. Different from the monoidal Hom-Hopf algebra, the conditions in Hom-Hopf algebra are obtained more unpredictable than the monoidal version. For a Hom-Hopf algebra, there exists two definitions of Yetter-Drinfeld modules in the studies of Makhlouf and Panaite [14] and Li and Ma [27], respectively. It is obvious that these two definitions of Yetter-Drinfeld modules are not equivalent. Surprisingly, such two categories of Yetter-Drinfeld modules are both braided tensor categories and provide solutions of the Yang-Baxter equation and the Hom-Yang-Baxter equation. The motivation of the definition of Yetter-Drinfeld modules by Makhlouf and Panaite relies on the main tool called “twisting principle,” which was introduced by Yau [11]. But for the definition by Li and Ma, they gave it by massive calculations to satisfy the corresponding conditions. Naturally inspired by the above two definitions, the authors in [28] gave the definition of v -Yetter-Drinfeld modules, v ∈ Z ; see (2.15). And they showed that every category of v -Yetter-Drinfeld modules is also a braided tensor category and provides some solutions of the Yang-Baxter equation and the Hom-Yang-Baxter equation. Therefore, the v -Yetter-Drinfeld module unifies the above two definitions of Yetter-Drinfeld modules. Observing the compatibility of v -Yetter-Drinfeld modules, we find that the condition (2.16) is equivalent to the condition (2.15). In other words, we have proven that if p = 2 v − 2 is even, the definition of v -Yetter-Drinfeld is favorable. One natural question to ask is whether the definition of v is favorable or not if p is odd. Furthermore, there exist more general favorable definitions of Yetter-Drinfeld module over a Hom-Hopf algebra. In this paper, we give the definition of Θ -Yetter-Drinfeld modules, Θ ∈ Aut H ( H ) where we denote the group of all Hom-Hopf algebra automorphisms of H by Aut H ( H ) and prove that even if p is odd, the definition of Yetter-Drinfeld is in the same way favorable. Meanwhile, all the definitions of Yetter-Drinfeld modules over a Hom-Hopf algebra that appeared in [14,27,28] can be regarded as special cases of the Θ -Yetter-Drinfeld module.

This paper is organized as follows. In Section 1, we recall some definitions and results which will be used later. In Section 2, after introducing the concept of Θ -Yetter-Drinfeld category ℋY D Θ H H , we prove that every Θ -Yetter Drinfeld category over a Hom-Hopf algebra with a bijective antipode is a braided tensor category and that every Θ -Yetter Drinfeld category over a Hom-Hopf algebra provides a solution of the Hom-Yang-Baxter equation and the Yang-Baxter equation. In Section 3, by the method of symmetric pairs, we show that symmetric Θ -Yetter-Drinfeld categories ℋY D Θ H H over a Hom-Hopf algebra are all trivial. The results obtained in this section generalize the corresponding results in [28,29, 30,31]. In Section 4, we find a necessary and sufficient condition for a Θ -Yetter-Drinfeld category ℋY D Θ H H over a Hom-Hopf algebra H to be pseudosymmetric.

2 Preliminaries

We work over a base field k . All algebras, linear spaces etc. will be over k , and unadorned ⊗ means ⊗ k . For a comultiplication Δ : C → C ⊗ C on a vector space C , we use a Sweedler-type notation Δ ( c ) = c 1 ⊗ c 2 for all c ∈ C . In what follows, we assume that k -linear maps α − are bijective, although some notions are not supposed to be bijective.

Definition 2.1

[14,32] A Hom-associative algebra is a quadruple ( A , μ , 1 A , α ) (abbr. ( A , α ) ) where A is a k -linear space, μ : A ⊗ A → A and α : A → A are k -linear maps such that

(2.1) α ( a a ′ ) = α ( a ) α ( a ′ ) , α ( 1 A ) = 1 A ,

(2.2) α ( a ) ( a ′ a ″ ) = ( a a ′ ) α ( a ″ ) , a 1 A = 1 A a = α ( a )

are satisfied for a , a ′ , a ″ ∈ A . Here we use the notation μ ( a ⊗ a ′ ) = a a ′ .

A morphism f : ( A , μ A , 1 A , α A ) → ( B , μ B , 1 B , α B ) of Hom-associative algebra is a linear map f : A → B such that α B ∘ f = f ∘ α A , f ( 1 A ) = 1 B and f ∘ μ A = μ B ∘ ( f ⊗ f ) .

Let ( A , α ) and ( B , β ) be two Hom-algebras. Then ( A ⊗ B , α ⊗ β ) in a Hom-associative algebra (called a tensor product Hom-associative algebra) with the multiplication ( a ⊗ b ) ( a ′ ⊗ b ′ ) = a a ′ ⊗ b b ′ and its unit 1 A ⊗ 1 B .

Definition 2.2

[14,18,33] A Hom-coassociative coalgebra is a quadruple ( C , Δ , ε C , β ) (abbr. ( C , β ) ), where C is a k -linear space, Δ : C → C ⊗ C , ε C : C → k and β : C → C are k -linear maps such that

(2.3) β ( c ) 1 ⊗ β ( c ) 2 = β ( c 1 ) ⊗ β ( c 2 ) , ε C ∘ β = ε C ,

(2.4) β ( c 1 ) ⊗ c 21 ⊗ c 22 = c 11 ⊗ c 12 ⊗ β ( c 2 ) , ε C ( c 1 ) c 2 = c 1 ε C ( c 2 ) = β ( c )

are satisfied for c ∈ C .

A morphism g : ( C , Δ C , α C ) → ( D , Δ D , α D ) of Hom-coassociative coalgebras is a linear map g : C → D such that α D ∘ g = g ∘ α C , ε C = ε D ∘ g and ( g ⊗ g ) ∘ Δ C = Δ D ∘ g .

Let ( C , α ) and ( D , β ) be two Hom-coassociative coalgebras. Then ( C ⊗ D , α ⊗ β ) is a Hom-coassociative coalgebra (called a tensor product Hom-coassociative coalgebra) with the comultiplication Δ ( c ⊗ d ) = c 1 ⊗ d 1 ⊗ c 2 ⊗ d 2 and its counit ε C ⊗ ε D .

Definition 2.3

[18,19] A Hom-bialgebra is a sextuple ( H , μ , 1 H , Δ , ε , α ) (abbr. ( H , α ) ), where ( H , μ , 1 H , α ) is a Hom-associative algebra and ( H , Δ , ε , α ) is a Hom-coassociative coalgebra such that Δ and ε are morphisms of Hom-algebras, i.e.,

(2.5) Δ ( h h ′ ) = Δ ( h ) Δ ( h ′ ) , Δ ( 1 H ) = 1 H ⊗ 1 H ,

(2.6) ε ( h h ′ ) = ε ( h ) ε ( h ′ ) , ε ( 1 H ) = 1 .

Furthermore, if there exists a linear map S : H → H such that

(2.7) S ( h 1 ) h 2 = h 1 S ( h 2 ) = ε ( h ) 1 H , S ( α ( h ) ) = α ( S ( h ) ) ,

then we call ( H , μ , 1 H , Δ , ε , α , S ) (abbr. ( H , α , S ) ) a Hom-Hopf algebra.

Let ( H , α , S ) and ( H ′ , α ′ , S ′ ) be two Hom-Hopf algebras. The linear map f : H → H ′ is called a Hom-Hopf algebra map if f ∘ α = α ′ ∘ f , f ∘ S = S ′ ∘ f and f is a Hom-algebra map and a Hom-coalgebra map.

Definition 2.4

[18,32] Let ( A , μ , 1 A , α A ) be a Hom-associative algebra, M a linear space and α M : M → M a linear map. A left A-Hom-module ( M , α M ) consists of a linear map A ⊗ M → M , a ⊗ m ↦ a ⋅ m satisfying the following conditions:

(2.8) α M ( a ⋅ m ) = α A ( a ) ⋅ α M ( m ) ;

(2.9) α A ( a ) ⋅ ( a ′ ⋅ m ) = ( a a ′ ) ⋅ α M ( m ) , 1 H ⋅ m = α M ( m )

for all a , a ′ ∈ A and m ∈ M . If ( M , α M ) and ( N , α N ) are left A -Hom-modules (both are A -Hom-actions, denoted by ⋅ ), a morphism of left A -Hom-modules f : M → N is a linear map satisfying the following conditions:

α N ∘ f = f ∘ α M and f ( a ⋅ m ) = a ⋅ f ( m ) , a ∈ A , m ∈ M .

Definition 2.5

[18,32] Let ( C , Δ C , ε , α C ) be a Hom-coassociative coalgebra, M a linear space and α M : M → M a linear map. A left C-Hom-comodule ( M , α M ) consists of a linear map λ : M → C ⊗ M (usually denoted by λ ( m ) = m ( − 1 ) ⊗ m ( 0 ) ) satisfying the following conditions:

(2.10) ( α C ⊗ α M ) ∘ λ = λ ∘ α M ;

(2.11) ( Δ C ⊗ α M ) ∘ λ = ( α C ⊗ λ ) ∘ λ , ( ε ⊗ i d ) ∘ λ = α M .

If ( M , α M ) and ( N , α N ) are left C -Hom-comodules, with Hom-coactions λ M : M → C ⊗ M and λ N : N → C ⊗ N , a morphism of left C -Hom-comodules g : M → N is a linear map satisfying the conditions α N ∘ g = g ∘ α M and ( i d C ⊗ g ) ∘ λ M = λ N ∘ g .

Definition 2.6

[14] Let ( H , μ H , 1 H , Δ H , ε H , α H ) be a Hom-bialgebra, M a linear space and α M : M → M a linear map such that ( M , α M ) is a left H -Hom-module with Hom-action H ⊗ M → M , h ⊗ m ↦ h ⋅ m and a left H -Hom-comodule with Hom-coaction M → H ⊗ M , m ↦ m ( − 1 ) ⊗ m ( 0 ) . Then ( M , α M ) is called a left-left Yetter-Drinfel module over H if, for all h ∈ H and m ∈ M , there holds the following identity:

(2.12) ( h 1 ⋅ m ) ( − 1 ) α H 2 ( h 2 ) ⊗ ( h 1 ⋅ m ) ( 0 ) = α H 2 ( h 1 ) α H ( m ( − 1 ) ) ⊗ α H ( h 2 ) ⋅ m ( 0 ) .

Notice that the choice of the compatibility condition (2.12) due to [14] was motivated by the twisting principle, but the compatibility condition of Yetter-Drinfeld module over a Hom-bialgebra is not the only choice as in (2.12). The following definition of Yetter-Drinfeld module was due to Li and Ma [27].

Definition 2.7

[27] Let ( H , μ H , 1 H , Δ H , ε H , α H ) be a Hom-bialgebra, ( M , α M ) a left H -Hom-module with Hom-action H ⊗ M → M denoted by h ⊗ m ↦ h ⋅ m and ( M , α M ) a left H -Hom-comodule with Hom-coaction M → H ⊗ M denoted by m ↦ m ( − 1 ) ⊗ m ( 0 ) . Then we call ( M , α M ) a left-left Yetter-Drinfeld module over a Hom-bialgebra ( H , α H ) if, for all h ∈ H and m ∈ M , there holds the following condition:

(2.13) h 1 α H ( m ( − 1 ) ) ⊗ α H 3 ( h 2 ) ⋅ m ( 0 ) = ( α H 2 ( h 1 ) ⋅ m ) ( − 1 ) h 2 ⊗ ( α H 2 ( h 1 ) ⋅ m ) ( 0 ) .

Remark 2.8

Obviously, the compatibility condition (2.12) is different from the condition (2.13). It is not hard to show that the definition of Yetter-Drinfeld module in Definition 2.7 does not satisfy the twisting principle. Indeed, by a direct computation, we have

( α H 2 ( h ( 1 ) ) ⊳ m ) ⟨ − 1 ⟩ ∗ h ( 2 ) ⊗ ( α H 2 ( h ( 1 ) ) ⊳ m ) ⟨ 0 ⟩ ≠ h ( 1 ) ∗ α H ( m ⟨ − 1 ⟩ ) ⊗ α H 3 ( h ( 2 ) ) ∗ m ⟨ 0 ⟩ ,

where the multiplication and comultiplication of H α H were denoted by h ∗ h ′ = α H ( h h ′ ) and h ( 1 ) ⊗ h ( 2 ) = α H ( h 1 ) ⊗ α H ( h 2 ) , respectively.

Remark 2.9

Since α is bijective, it is not hard to obtain that the condition (2.13) is equivalent to

(2.14) ( h 1 ⋅ m ) ( − 1 ) α H − 2 ( h 2 ) ⊗ ( h 1 ⋅ m ) ( 0 ) = α H − 2 ( h 1 ) α H ( m ( − 1 ) ) ⊗ α H ( h 2 ) ⋅ m ( 0 ) .

In [28], the authors gave a more general compatibility condition including the definitions in [14,27] as the special cases.

Definition 2.10

[28] Let ( H , α H ) be a Hom-bialgebra, ( M , α M ) a left H -Hom-module with Hom-action H ⊗ M → M , h ⊗ m ↦ h ⋅ m and ( M , α M ) a left H -Hom-comodule with Hom-coaction M → H ⊗ M , m ↦ m ( − 1 ) ⊗ m ( 0 ) . Then we call ( M , α M ) a left-left v-Yetter-Drinfeld module over a Hom-bialgebra ( H , α H ) if, for all h ∈ H , m ∈ M and v ∈ Z , there holds the following condition:

(2.15) ( α H 2 − v ( h 1 ) ⋅ m ) ( − 1 ) α H v ( h 2 ) ⊗ ( α H 2 − v ( h 1 ) ⋅ m ) ( 0 ) = α H v ( h 1 ) α H ( m − 1 ) ⊗ α H 3 − v ( h 2 ) ⋅ m ( 0 ) .

Remark 2.11

Being similar to Remark 2.9, we readily see that the condition (2.15) is equivalent to

(2.16) ( h 1 ⋅ m ) ( − 1 ) α H 2 v − 2 ( h 2 ) ⊗ ( h 1 ⋅ m ) ( 0 ) = α H 2 v − 2 ( h 1 ) α H ( m − 1 ) ⊗ α H ( h 2 ) ⋅ m ( 0 ) , v ∈ Z .

3 Θ -Yetter-Drinfeld category

In this section, we introduce a more general Θ -Yetter-Drinfeld category ( Θ ∈ Aut H ( H ) ), which unifies two different definitions of Hom-Yetter-Drinfeld category of Makhlouf and Panaite [14] and Li and Ma [27], respectively. We show that the category of Θ -Yetter-Drinfeld modules with a bijective antipode is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category.

In case that ( H , μ H , Δ H , α H , S ) is a Hom-Hopf algebra, we denote the group of all Hom-Hopf algebra automorphisms of H by Aut H ( H ) . Inspired by the above three definitions of Hom-Yetter-Drinfeld modules, we can give a more general definition of Hom-Yetter-Drinfeld modules as follows.

Definition 3.1

Let ( H , α H ) be a Hom-bialgebra and Θ ∈ Aut H ( H ) , ( M , α M ) a left H -Hom-module with Hom-action H ⊗ M → M , h ⊗ m ↦ h ⋅ m and ( M , α M ) a left H -Hom-comodule with Hom-coaction M → H ⊗ M , m ↦ m ( − 1 ) ⊗ m ( 0 ) . Then we call ( M , α M ) a left-left Θ -Yetter-Drinfeld module over a Hom-bialgebra ( H , α H ) if, for all h ∈ H and m ∈ M , there holds the following condition:

(3.1) ( h 1 ⋅ m ) ( − 1 ) Θ ( h 2 ) ⊗ ( h 1 ⋅ m ) ( 0 ) = Θ ( h 1 ) α H ( m ( − 1 ) ) ⊗ α H ( h 2 ) ⋅ m ( 0 ) .

Remark 3.2

We find an interesting fact that every definition that appeared in references can be regarded as the special cases of Θ -Yetter-Drinfeld module.

  1. If Θ = α 2 v − 2 , v ∈ Z , the definition of Θ -Yetter-Drinfeld module is the same as Definition 2.1 in [28].

  2. If Θ = α − 2 , the definition of Θ -Yetter-Drinfeld module is the same as Definition 2.7;

  3. If Θ = α 2 , the definition of Θ -Yetter-Drinfeld module is just Definition 2.6;

  4. If α H = i d , a M = i d and Θ = i d , the definition of Θ -Yetter-Drinfeld module is exactly the usual Yetter-Drinfeld module;

  5. We should note that it is impossible to get the definition only by the twisting principle.

Example 3.3

Let ( H , S , α H ) be a Hom-Hopf algebra and Θ ∈ Aut H ( H ) . Then ( H , α H ) is a Θ -Yetter-Drinfeld module in Definition 3.1 with a left H -Hom-action

h ⊳ g = ( Θ α H − 4 ( h 1 ) α H − 1 ( g ) ) S ( Θ α H − 3 ( h 2 ) )

and a left H -Hom-coaction by the Hom-comultiplication Δ . Denote it by H 1 = ( H , ⊳ , Δ , α H ) . Dually, ( H , α ) is a Θ -Yetter-Drinfeld module in Definition 3.1 with a left H -Hom-action by the Hom-multiplication μ and a left H -Hom-coaction

λ ( h ) = Θ α H − 4 ( h 11 ) Θ α H − 3 S ( h 2 ) ⊗ α H − 1 ( h 12 ) ;

it will be denoted by H 2 = ( H , μ , λ , α H ) .

Proof

We only establish the first part of the above example here. First, it is easy to see that ( H 1 , Δ , α ) is a left ( H , α ) -comodule. Second, we show that ( H 1 , ⊳ , α ) is a left ( H , α ) -module. In fact, for any h , g , l ∈ H 1 , we have

α ( h ) ⊳ α ( g ) = ( Θ α − 3 ( h 1 ) g ) S ( Θ α − 2 ( h 2 ) ) = α ( Θ α − 4 ( h 1 ) α − 1 ( g ) ) S ( Θ α − 3 ( h 2 ) ) = α ( h ⊳ g ) ,

which proves the equality (2.8). For proving (2.9), we observe that 1 H ⊳ g = ( Θ α − 4 ( 1 ) α − 1 ( g ) ) S ( Θ α − 3 ( 1 ) ) = ( 1 ⋅ g ) ⋅ 1 = α ( g ) and

α ( h ) ⊳ ( g ⊳ l ) = α ( h ) ⊳ ( ( Θ α − 4 ( g 1 ) α − 1 ( l ) ) S ( Θ α − 3 ( g 2 ) ) ) = ( Θ α − 3 ( h 1 ) α − 1 ( ( Θ α − 4 ( g 1 ) α − 1 ( l ) ) S ( Θ α − 3 ( g 2 ) ) ) ) S ( Θ α − 2 ( h 2 ) ) = ( 2.2 ) ( Θ α − 3 ( h 1 ) ( Θ α − 4 ( g 1 ) α − 1 ( l 1 ) ) ) ( S ( Θ α − 3 ( g 2 ) ) S ( Θ α − 3 ( h 2 ) ) ) = ( ( Θ α − 4 ( h 1 ) Θ α − 4 ( g 1 ) ) l 1 ) ( S ( Θ α − 3 ( g 2 ) ) S ( Θ α − 3 ( h 2 ) ) ) = ( Θ α − 4 ( h 1 g 1 ) l ) S ( Θ α − 3 ( h 2 g 2 ) ) = ( h g ) ⊳ α ( l ) .

Finally, we show the compatible condition (3.1) of Θ -Yetter-Drinfeld module. For this end, the left-hand side of (3.1) can be expressed as

Θ ( h 1 ) α ( l 1 ) ⊗ α ( h 2 ) ⋅ l 2 = ( 2.3 ) Θ ( h 1 ) α ( l 1 ) ⊗ ( Θ α − 3 ( h 21 ) α − 1 ( l 2 ) ) S Θ α − 2 ( h 22 ) ,

and also its right-hand side can be given by

( h 1 ⋅ l ) ( − 1 ) Θ ( h 2 ) ⊗ ( h 1 ⋅ l ) ( 0 ) = ( 2.5 ) , ( 2.3 ) ( ( Θ α − 4 ( h 111 ) α − 1 ( l 1 ) ) S ( Θ α − 3 ( h 122 ) ) ) Θ ( h 2 ) ⊗ ( Θ α − 4 ( h 112 ) α − 1 ( l 2 ) ) S ( Θ α − 3 ( h 121 ) ) = ( 2.4 ) , ( 2.2 ) ( ( Θ α − 2 ( h 11 ) l 1 ) S ( Θ α − 3 ( h 221 ) ) Θ α − 3 ( h 222 ) ) ⊗ ( Θ α − 3 ( h 12 ) α − 1 ( l 2 ) ) S ( Θ α − 2 ( h 21 ) ) = ( 2.7 ) , ( 2.2 ) , ( 2.4 ) Θ α − 1 ( h 11 ) α ( l 1 ) ⊗ ( Θ α − 3 ( h 12 ) α − 1 ( l 2 ) ) S ( Θ α v − 1 ( h 2 ) ) = ( 2.4 ) Θ ( h 1 ) α ( l 1 ) ⊗ ( Θ α − 3 ( h 21 ) α − 1 ( l 2 ) ) S Θ α − 2 ( h 22 ) ,

yielding the compatible condition (3.1) of Θ -Yetter-Drinfeld module and the proof is complete.

Remark 3.4

If we take Θ = α 2 v − 2 , v ∈ Z in Example 3.3, we obtain Example 2.4 in [28]. Also, if we take Θ = α − 2 , Example 3.3 is just Lemmas 3.3 and 3.4 in [31].

Definition 3.5

Let ( H , α H ) be a Hom-bialgebra and Θ ∈ Aut H ( H ) . We denote by ℋY D Θ H H the category whose objects are Θ -Yetter-Drinfeld modules ( M , α M ) over H , and the morphisms in the category are morphisms of left H -Hom-modules and left H -Hom-comodules.

Proposition 3.6

Let ( H , S , α H ) be a Hom-Hopf algebra and Θ ∈ Aut H ( H ) . Let ( M , α M ) be both a left H -Hom-module and a left H -Hom-comodule with notations as in Definition 3.1. Then (3.1) in Definition 3.1 is equivalent to

(3.2) ( Θ α H − 4 ( h 11 ) α H − 1 ( m ( − 1 ) ) ) Θ α H − 2 S ( h 2 ) ⊗ α H − 1 ( h 12 ) m ( 0 ) = ( h ⋅ m ) ( − 1 ) ⊗ ( h ⋅ m ) ( 0 )

for all h ∈ H and m ∈ M .

Proof

(3.1) ⇒ (3.2). For all h ∈ H and m ∈ M , we compute

( Θ α H − 4 ( h 11 ) α H − 1 ( m ( − 1 ) ) ) Θ α H − 2 S ( h 2 ) ⊗ α H − 1 ( h 12 ) m ( 0 ) = ( 2.1 ) , ( 2.8 ) , ( 2.10 ) α H − 4 ( Θ ( h 11 ) α H ( ( α M 2 ( m ) ) ( − 1 ) ) ) α H − 2 S ( h 2 ) ⊗ α M − 2 ( α ( h 12 ) ⋅ ( α M 2 ( m ) ) ( 0 ) ) = ( 3.1 ) α H − 4 ( ( h 11 ⋅ α M 2 ( m ) ) ( − 1 ) Θ ( h 12 ) ) Θ α H − 2 S ( h 2 ) ⊗ α M − 2 ( ( h 11 ⋅ α M 2 ( m ) ) ( 0 ) ) ) = ( 2.1 ) , ( 2.2 ) , ( 2.4 ) α H − 3 ( ( α H ( h 1 ) ⋅ α M 2 ( m ) ) ( − 1 ) ) ( Θ α H − 4 ( h 21 ) Θ α H − 4 S ( h 22 ) ) ⊗ α M − 2 ( ( α H ( h 1 ) ⋅ α M 2 ( m ) ) ( − 0 ) ) = ( 2.7 ) , ( 2.3 ) , ( 2.4 ) ( h ⋅ m ) ( − 1 ) ⊗ ( h ⋅ m ) ( 0 ) .

(3.2) ⇒ (3.1). For all h ∈ H and m ∈ M , we have

( h 1 ⋅ m ) ( − 1 ) Θ ( h 2 ) ⊗ ( h 1 ⋅ m ) ( 0 ) = ( 3.2 ) ( ( Θ α H − 4 ( h 111 ) α H − 1 ( m ( − 1 ) ) ) Θ α H − 2 S ( h 12 ) ) Θ ( h 2 ) ⊗ α H − 1 ( h 112 ) ⋅ m ( 0 ) = ( 2.2 ) , ( 2.4 ) ( Θ α H − 2 ( h 11 ) m ( − 1 ) ) ( Θ α H − 2 S ( h 21 ) α H 2 Θ ( h 22 ) ) ⊗ h 21 ⋅ m ( 0 ) = ( 2.7 ) , ( 2.1 ) , ( 2.2 ) Θ ( h 1 ) α H ( m − 1 ) ⊗ α H ( h 2 ) ⋅ m ( 0 ) .

This completes the proof.□

In the following, we give solutions of the Hom-Yang-Baxter introduced and studied by Yau [12].

Proposition 3.7

Let ( H , α H ) be a Hom-bialgebra, Θ ∈ Aut H ( H ) and ( M , α M ) , ( N , α N ) ∈ ℋY D Θ H H . Define the linear map

(3.3) τ M , N : M ⊗ N → N ⊗ M , m ⊗ n ↦ Θ − 1 α H ( m ( − 1 ) ) ⋅ n ⊗ m ( 0 ) ,

where m ∈ M and n ∈ N . Then,

  1. τ M , N ∘ ( α M ⊗ α N ) = ( α N ⊗ α M ) ∘ τ M , N , and

  2. if ( P , α P ) ∈ ℋY D Θ H H , the maps τ _ , _ satisfy the Hom-Yang-Baxter equation:

    ( α P ⊗ τ M , N ) ∘ ( τ M , P ⊗ α N ) ∘ ( α M ⊗ τ N , P ) = ( τ N , P ⊗ α M ) ∘ ( α N ⊗ τ M , P ) ∘ ( τ M , N ⊗ α P ) .

Proof

The property (i) can be easily proven. Now we claim the property (ii): given m ∈ M , n ∈ N and p ∈ P , using the conditions (2.10), (2.1) and (2.2), we obtain

( α P ⊗ τ M , N ) ∘ ( τ M , P ⊗ α N ) ∘ ( α M ⊗ τ N , P ) ( m ⊗ n ⊗ p ) = Θ − 1 α H 2 ( m ( − 1 ) n ( − 1 ) ) ⋅ α P 2 ( p ) ⊗ Θ − 1 α H 2 ( m ( 0 ) ( − 1 ) ) ⋅ α N ( n ( 0 ) ) ⊗ α M ( m ( 0 ) ( 0 ) ) ( τ N , P ⊗ α M ) ∘ ( α N ⊗ τ M , P ) ∘ ( τ M , N ⊗ α P ) ( m ⊗ n ⊗ p ) = Θ − 1 α H ( ( α H 2 Θ − 1 ( m ( − 1 ) ) ⋅ α N ( n ) ) ( − 1 ) ) ⋅ ( Θ − 1 α H ( m ( 0 ) ( − 1 ) ) ⋅ α P ( p ) ) ⊗ ( α H 2 Θ − 1 ( m ( − 1 ) ) ⋅ α N ( n ) ) ( 0 ) ⊗ α M ( m ( 0 ) ( 0 ) ) = ( 2.9 ) , ( 2.11 ) ( Θ − 1 ( ( Θ − 1 α H ( m ( − 1 ) 1 ) ⋅ α N ( n ) ) ( − 1 ) ) Θ − 1 α H ( m ( − 1 ) 2 ) ) ⋅ α P 2 ( p ) ⊗ ( Θ − 1 α H ( m ( − 1 ) 1 ) ⋅ α N ( n ) ) ( 0 ) ⊗ α M 2 ( m ( 0 ) ) = ( 2.1 ) , ( 2.3 ) Θ − 1 ( ( Θ − 1 α H ( m ( − 1 ) ) 1 ⋅ α N ( n ) ) ( − 1 ) Θ ( Θ − 1 α H ( m ( − 1 ) ) 2 ) ) ⋅ α P 2 ( p ) ⊗ ( Θ − 1 α H ( m ( − 1 ) ) 1 ⋅ α N ( n ) ) ( 0 ) ⊗ α M 2 ( m ( 0 ) ) = ( 3.1 ) Θ − 1 ( Θ ( Θ − 1 α ( m ( − 1 ) ) 1 ) α H ( α N ( n ) ( − 1 ) ) ) ⋅ α P 2 ( p ) ⊗ α H ( Θ − 1 α H ( m ( − 1 ) ) 2 ) ⋅ α N ( n ) ( 0 ) ⊗ α M 2 ( m ( 0 ) ) = Θ − 1 α H 2 ( m ( − 1 ) n ( − 1 ) ) ⋅ α P 2 ( p ) ⊗ Θ − 1 α H 2 ( m ( 0 ) ( − 1 ) ) ⋅ α N ( n ( 0 ) ) ⊗ α M ( m ( 0 ) ( 0 ) ) by (2.3), (2.1) and (2.11) ,

and the proof is complete.□

Now we will state the main result in this section.

Theorem 3.8

Let ( H , α H ) be a Hom-Hopf algebra with a bijective antipode S and a bijective α H , Θ ∈ Aut H ( H ) . Then the Θ -Yetter-Drinfeld category H H ℋY D Θ is a braided tensor category, with tensor product ⊗ ˆ , associativity constraints a − , − , − , braiding c − , − and the unit defined as follows: for all ( M , α M ) , ( N , α N ) , ( P , α P ) ∈ ℋY D Θ H H ,

  1. The tensor product ⊗ ˆ = ⊗ , and the left H -Hom-module H ⊗ ( M ⊗ N ) → M ⊗ N and the left H -Hom-comodule M ⊗ N → H ⊗ ( M ⊗ N ) are defined, respectively, by

    h ⊗ ( m ⊗ n ) ↦ h 1 ⋅ m ⊗ h 2 ⋅ n , ρ ( m ⊗ n ) ↦ α H − 2 ( m ( − 1 ) n ( − 1 ) ) ⊗ m ( 0 ) ⊗ n ( 0 ) ;

  2. a M , N , P : ( M ⊗ ˆ N ) ⊗ ˆ P → M ⊗ ˆ ( N ⊗ ˆ P ) is given by

    ( m ⊗ n ) ⊗ p ↦ α M − 1 ( m ) ⊗ ( n ⊗ α P ( p ) ) ;

  3. c M , N : M ⊗ ˆ N → N ⊗ ˆ M is defined by

    m ⊗ n ↦ Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ⊗ α M − 1 ( m ( 0 ) ) ;

  4. c M , N − 1 : N ⊗ ˆ M → M ⊗ ˆ N is given by

    n ⊗ m ↦ α M − 1 ( m ( 0 ) ) ⊗ S − 1 ( Θ − 1 ( m ( − 1 ) ) ) ⋅ α N − 1 ( n ) ;

  5. the unit is defined by I = ( k , i d k ) .

Proof

(Sketch of the proof) By the definition of a braided tensor category, we need to investigate many conditions by a large number of computations. In fact, we have to prove that M ⊗ ˆ N is also a θ -Yetter-Drinfeld module, a M , N , P is an isomorphism of left H -Hom-modules and left H -Hom-comodules, c M , N is a morphism of left H -Hom-modules and left H -Hom-comodules, c M , N − 1 ∘ c M , N = i d M ⊗ ˆ N , the pentagon axiom and hexagonal relation are satisfied, and so on. We think that it is a good exercise to a reader. For example, now we will prove the Θ -Yetter-Drinfeld compatibility condition (3.1) for M ⊗ ˆ N . Indeed, given m ∈ M and n ∈ N , simple computations yield

( h 1 ⋅ ( m ⊗ n ) ) ( − 1 ) Θ ( h 2 ) ⊗ ( h 1 ⋅ ( m ⊗ n ) ) ( 0 ) = α H − 2 ( ( h 11 ⋅ m ) ( − 1 ) ( h 12 ⋅ n ) ( − 1 ) ) Θ ( h 2 ) ⊗ ( h 11 ⋅ m ) ( 0 ) ⊗ ( h 12 ⋅ n ) ( 0 ) = α H − 1 ( ( α H ( h 1 ) ⋅ m ) ( − 1 ) ) α H − 2 ( ( h 21 ⋅ n ) ( − 1 ) Θ ( h 22 ) ) ⊗ ( α ( h 1 ) ⋅ m ) ( 0 ) ⊗ ( h 21 ⋅ n ) ( 0 ) by (2.1), (2.2) and (2.4) = ( 3.1 ) α H − 1 ( ( α H ( h 1 ) ⋅ m ) ( − 1 ) ) α H − 2 ( Θ ( h 21 ) α H ( n ( − 1 ) ) ) ⊗ ( α H ( h 1 ) ⋅ m ) ( 0 ) ⊗ α H ( h 22 ) ⋅ n ( 0 ) = α H − 2 ( ( h 11 ⋅ m ) ( − 1 ) Θ ( h 12 ) ) n ( − 1 ) ⊗ ( h 11 ⋅ m ) ( 0 ) ⊗ α H 2 ( h 2 ) ⋅ n ( 0 ) by (2.1), (2.2) and (2.4) = ( 3.1 ) α H − 2 ( Θ ( h 11 ) α H ( m ( − 1 ) ) ) n ( − 1 ) ⊗ α H ( h 12 ) ⋅ m ( 0 ) ⊗ α H 2 ( h 2 ) ⋅ n ( 0 ) = Θ ( h 1 ) ( α H ( m ( − 1 ) ) α H ( n ( − 1 ) ) ) ⊗ α H ( h 21 ) ⋅ m ( 0 ) ⊗ α H ( h 22 ) ⋅ n ( 0 ) by (2.1), (2.2) and (2.4) = Θ ( h 1 ) α H ( m ( − 1 ) n ( − 1 ) ) ⊗ α H ( h 2 ) ⋅ ( m ( 0 ) ⊗ n ( 0 ) ) .

Let us choose another part of the proof: the H -linearity of c M , N can be obtained by

c M , N ( h ⋅ ( m ⊗ n ) ) = Θ − 1 ( ( h 1 ⋅ m ) ( − 1 ) ) ⋅ α N − 1 ( h 2 ⋅ n ) ⊗ α M − 1 ( ( h 1 ⋅ m ) ( 0 ) ) = ( 2.8 ) , ( 2.9 ) ( Θ − 1 α − 1 ( ( h 1 ⋅ m ) ( − 1 ) ) α H − 1 ( h 2 ) ) ⋅ n ⊗ α M − 1 ( ( h 1 ⋅ m ) ( 0 ) ) = ( 2.1 ) Θ − 1 α H − 1 ( ( h 1 ⋅ m ) ( − 1 ) Θ ( h 2 ) ) ⋅ n ⊗ α M − 1 ( ( h 1 ⋅ m ) ( 0 ) ) = ( 3.1 ) Θ − 1 α H − 1 ( Θ ( h 1 ) α H ( m ( − 1 ) ) ) ⋅ n ⊗ α M − 1 ( α H ( h 2 ) ⋅ m ( 0 ) ) = ( 2.1 ) , ( 2.2 ) h 1 ⋅ ( Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ) ⊗ ( h 2 ⋅ α M − 1 ( m ( 0 ) ) ) = h ⋅ c M , N ( m ⊗ n ) ,

for all h ∈ H , m ∈ M and n ∈ N .□

Remark 3.9

Theorem 4.4 in [14] can be regarded as a special case of Theorem 3.8 if we take Θ = α 2 . Also, if we take Θ = α − 2 , then Theorem 4.7 in [27] is just a special case of this theorem. Furthermore, if we take Θ = α 2 v − 2 , v ∈ Z , Theorem 3.8 is the same as Theorem 2.12 in [28].

By Theorem 3.8, we know that the Θ -Yetter-Drinfeld category ℋY D Θ H H is a braided tensor category. Therefore, we can obtain the following corollary.

Corollary 3.10

Let ( H , α H ) be a Hom-Hopf algebra with a bijective antipode S . Assume ( M , α M ) , ( N , α N ) and ( P , α P ) ∈ ℋY D v H H . Then the braiding c satisfies the braid relation

( i d P ⊗ c M , N ) ∘ ( c M , P ⊗ i d N ) ∘ ( i d M ⊗ c N , P ) = ( c N , P ⊗ i d M ) ∘ ( i d N ⊗ c M , P ) ∘ ( c M , N ⊗ i d P ) .

Remark 3.11

This implies that c M , M is a solution of the Yang-Baxter equation for any object M of a Θ -Yetter-Drinfeld category ℋY D Θ H H . By Proposition 3.6 and Theorem 3.8, we find an interesting fact that every Θ -Yetter-Drinfeld category ℋY D Θ H H can give not only a solution of the Hom-Yang-Baxter equation but also a solution of the classical Yang-Baxter equation.

4 Symmetric pairs of Θ -Yetter-Drinfeld categories over a Hom-Hopf algebra

By [30], we know that the Yetter-Drinfeld category over a Hopf algebra H is symmetric if and only if H = k . In this section, we extend some interesting results to Θ -Yetter-Drinfeld categories ℋY D Θ H H , e.g., symmetric Θ -Yetter-Drinfeld categories ℋY D Θ H H over a Hom-Hopf algebra are all trivial. The results obtained in this section generalize the corresponding results in [28,29, 30,31].

Definition 4.1

Let ( M , α M ) and ( N , α N ) be two objects in a Θ -Yetter-Drinfeld category ℋY D Θ H H . The pair ( M , N ) is called a symmetric pair if c N , M c M , N = i d M ⊗ N . Particularly, the category ℋY D Θ H H is called symmetric whenever any pair of objects M , N ∈ ℋY D Θ H H is a symmetric pair.

Theorem 4.2

Let ( H , α ) be a Hom-Hopf algebra, then ( H 2 , H 1 ) is a symmetric pair if and only if H = k .

Proof

By Example 3.3, H 1 = ( H , ⊳ , Δ , α H ) and H 2 = ( H , μ , λ , α H ) are two v -Yetter Drinfeld modules. On setting 1 H ⊗ h ∈ H 2 ⊗ H 1 , we have

c H 2 , H 1 ( 1 H ⊗ h ) = Θ − 1 ( 1 H ( − 1 ) ) ⊳ α H − 1 ( h ) ⊗ α H − 1 ( 1 H ( 0 ) ) = 1 H ⊳ α H − 1 ( h ) ⊗ 1 H = ( 2.9 ) h ⊗ 1 H .

Furthermore, we have

c H 1 , H 2 c H 2 , H 1 ( 1 H ⊗ h ) = c H 1 , H 2 ( h ⊗ 1 H ) = Θ − 1 ( h ( − 1 ) ) ⋅ α H − 1 ( 1 H ) ⊗ α H − 1 ( h ( 0 ) ) = Θ − 1 ( h 1 ) ⋅ α H − 1 ( 1 ) ⊗ α H − 1 ( h 2 ) = Θ − 1 α H ( h 1 ) ⊗ α H − 1 ( h 2 ) .

Since ( H 2 , H 1 ) is a symmetric pair, i.e., c H 1 , H 2 c H 2 , H 1 = i d H 2 ⊗ H 1 , it follows that 1 H ⊗ h = Θ − 1 α H ( h 1 ) ⊗ α H − 1 ( h 2 ) . Applying i d ⊗ ε to the above equality gives

1 H ⊗ ε ( h ) = 1 H ε ( h ) = Θ − 1 α H ( h 1 ) ⊗ ε ( α H − 1 ( h 2 ) ) = ( 2.3 ) Θ − 1 α H ( h 1 ) ⊗ ε ( h 2 ) = ( 2.4 ) Θ − 1 α 2 ( h )

and it then follows that h = Θ α − 2 ( ε ( h ) 1 H ) = ε ( h ) 1 H . This means H = k , as desired. The converse is straightforward.□

The following corollary is easily obtained from the above theorem.

Corollary 4.3

Let ( H , α ) be a Hom-Hopf algebra such that the Θ -Yetter-Drinfeld category ℋY D Θ H H is symmetric. Then H = k .

Remark 4.4

  1. If α = i d and Θ = i d , Theorem 4.2 is exactly the famous conclusion in [30], namely, the symmetric Yetter-Drinfeld category H H Y D over a Hopf algebra is trivial.

  2. If Θ = α − 2 , Theorem 4.2 is just Theorem 3.5 in [31]. Also, if Θ = α 2 v − 2 , v ∈ Z , Theorem 4.2 is just Theorem 3.2 in [28].

  3. The special case of Theorem 4.2 has been considered in the setting of monoidal Hom-Hopf algebras; see [26].

Theorem 4.5

Let ( H , α ) be a Hom-Hopf algebra. Then ( H 1 , H 2 ) is a symmetric pair if and only if H = k .

Proof

Let h ⊗ 1 H ∈ H 1 ⊗ H 2 . Then we obtain

c H 1 , H 2 ( h ⊗ 1 H ) = Θ − 1 ( h ( − 1 ) ) ⋅ α H − 1 ( 1 H ) ⊗ α H − 1 ( h ( 0 ) ) = Θ − 1 ( h 1 ) ⋅ 1 H ⊗ α H − 1 ( h 2 ) = Θ − 1 α H ( h 1 ) ⊗ α H − 1 ( h 2 )

and also

c H 2 , H 1 c H 1 , H 2 ( h ⊗ 1 H ) = ( Θ − 1 α H ( h 1 ) ⊗ α H − 1 ( h 2 ) ) = Θ − 1 ( Θ − 1 α ( h 1 ) ( − 1 ) ) ⊳ α H − 1 ( α H − 1 ( h 2 ) ) ⊗ α H − 1 ( Θ − 1 α H ( h 1 ) ( 0 ) ) = Θ − 1 ( Θ α H − 4 ( Θ − 1 α H ( h 1 ) 11 ) Θ α H − 3 S ( Θ − 1 α H ( h 1 ) 2 ) ) ⊳ α H − 2 ( h 2 ) ⊗ α H − 1 ( α H − 1 ( Θ − 1 α H ( h 1 ) 12 ) ) = ( 2.3 ) ( Θ − 1 α H − 3 ( h 111 ) S Θ − 1 α H − 2 ( h 12 ) ) ⊳ α H − 2 ( h 2 ) ⊗ Θ − 1 α H − 1 ( h 112 ) ) = ( Θ α H − 4 ( ( Θ − 1 α H − 3 ( h 111 ) S Θ − 1 α H − 2 ( h 12 ) ) 1 ) α H − 1 ( α H − 2 ( h 2 ) ) × S ( Θ α H − 3 ( Θ − 1 α H − 3 ( h 111 ) S Θ − 1 α H − 2 ( h 12 ) ) 2 ) ⊗ Θ − 1 α H − 1 ( h 112 ) = ( Θ α H − 4 ( Θ − 1 α H − 3 ( h 1111 ) S Θ − 1 α H − 2 ( h 122 ) ) α H − 1 ( α H − 2 ( h 2 ) ) ) × S ( Θ α H − 3 ( Θ − 1 α H − 3 ( h 1112 ) S Θ − 1 α H − 2 ( h 121 ) ) ) ⊗ Θ − 1 α H − 1 ( h 112 ) = ( ( α H − 7 ( h 1111 ) S α H − 6 ( h 122 ) ) α H − 3 ( h 2 ) ) S ( α H − 6 ( h 1112 ) S α H − 5 ( h 121 ) ) ⊗ Θ − 1 α H − 1 ( h 112 ) ,

where the sixth equality uses (2.3) and the property of the antipode S . Since ( H 1 , H 2 ) is a symmetric pair, i.e., c H 2 , H 1 c H 1 , H 2 = i d H 1 ⊗ H 2 , it follows that h ⊗ 1 H is equals to

( ( α H − 7 ( h 1111 ) S α H − 6 ( h 122 ) ) α H − 3 ( h 2 ) ) S ( α H − 6 ( h 1112 ) S α H − 5 ( h 121 ) ) ⊗ Θ − 1 α H − 1 ( h 112 ) .

Applying ε ⊗ i d to the above equality yields ε ( h ) 1 H = Θ − 1 α H 2 ( h ) . It then follows that h = Θ α H − 2 ( ε ( h ) 1 H ) = ε ( h ) 1 H . This means H = k , as required. The converse is obvious.□

Remark 4.6

Note that Corollary 4.3 can also be obtained by Theorem 4.5.

5 Pseudosymmetry of Θ -Yetter-Drinfeld categories over a Hom-Hopf algebra

In this section, we will find a necessary and sufficient condition for a Θ -Yetter-Drinfeld category ℋY D Θ H H over a Hom-Hopf algebra ( H , S , α ) to be pseudosymmetric. The main theorem obtained in this section generalizes the main conclusions in [28,30,31].

Definition 5.1

[34,35] Let C be a tensor category and c a braiding on C . The braiding c is called a pseudosymmetry if, for all M , N , P ∈ C , there holds the following condition:

(5.1) ( i d P ⊗ c M , N ) ( c P , M − 1 ⊗ i d N ) ( i d M ⊗ c N , P ) = ( c N , P ⊗ i d M ) ( i d N ⊗ c P , M − 1 ) ( c M , N ⊗ i d P ) .

In this case, C is called a pseudosymmetric braided tensor category.

Obviously, a symmetric braided tensor category must be a pseudosymmetric braided tensor category.

Lemma 5.2

Let ( H , S , α ) be a cocommutative Hom-Hopf algebra. Then, the canonical braiding of the Θ -Yetter-Drinfeld category c H 2 , H 1 is the usual flip map.

Proof

For all x ∈ H 2 and y ∈ H 1 , we have

c H 2 , H 1 ( x ⊗ y ) = Θ ( x ( − 1 ) ) ⊳ α H − 1 ( y ) ⊗ α H − 1 ( x ( 0 ) ) = Θ − 1 ( Θ α H − 4 ( x 11 ) Θ α H − 3 S ( x 2 ) ) ⊳ α H − 1 ( y ) ⊗ α H − 2 ( x 12 ) = ( α H − 4 ( x 11 ) α H − 3 S ( x 2 ) ) ⋅ α H − 1 ( y ) ⊗ α H − 2 ( x 12 ) = ( α H − 4 ( x 12 ) α H − 3 S ( x 2 ) ) ⋅ α H − 1 ( y ) ⊗ α H − 2 ( x 11 ) = ( 2.4 ) ( α H − 4 ( x 21 ) α H − 4 S ( x 22 ) ) ⋅ α H − 1 ( y ) ⊗ α H − 1 ( x 1 ) = ( 2.7 ) , ( 24 ) 1 ⋅ α H − 1 ( y ) ⊗ x = y ⊗ x ,

where the fourth equality uses cocommutativity.□

Lemma 5.3

Let ( H , S , α ) be a Hom-Hopf algebra. Then, the canonical braiding of Θ -Yetter-Drinfeld category satisfies ( ε ⊗ i d ) c H 2 , H 1 ( x ⊗ y ) = ε ( y ) x for all x ∈ H 2 and y ∈ H 1 .

Proof

For all x ∈ H 2 , y ∈ H 1 , we calculate

( ε ⊗ i d ) c H 2 , H 1 ( x ⊗ y ) = ( ε ⊗ i d ) ( α H − 2 v + 2 ( x ( − 1 ) ) ⋅ α H − 1 ( y ) ⊗ α H − 1 ( x ( 0 ) ) ) = Θ − 1 ( Θ α H − 4 ( x 11 ) Θ α H − 3 S ( x 2 ) ) ⊳ α H − 1 ( y ) ⊗ α H − 2 ( x 12 ) = ( ε ⊗ i d ) ( ( α H − 4 ( x 11 ) α H − 3 S ( x 2 ) ) ⋅ α H − 1 ( y ) ⊗ α H − 2 ( x 12 ) ) = ( ε ⊗ i d ) ( ( Θ α H − 4 ( ( α H − 4 ( x 11 ) α H − 3 S ( x 2 ) ) 1 ) α H − 2 ( y ) ) × S ( Θ α H − 3 ( ( α H − 4 ( x 11 ) α H − 3 S ( x 2 ) ) 2 ) ) ) ⊗ α H − 2 ( x 12 ) = ε ( x 11 ) ε ( x 2 ) ε ( y ) α H − 2 ( x 12 ) = ( 2.4 ) ε ( y ) x ,

completing the proof.□

Theorem 5.4

Let ( H , α H ) be a Hom-Hopf algebra with a bijective antipode S . Then, the canonical braiding of the Θ -Yetter-Drinfeld category ℋY D Θ H H is pseudosymmetric if and only if ( H , α H ) is commutative and cocommutative.

Proof

( ⇐ ) Assume that ( H , α H ) is commutative and cocommutative. We first prove that the compatibility (3.2) becomes the equality

(5.2) ( h ⋅ m ) ( − 1 ) ⊗ ( h ⋅ m ) ( 0 ) = α H ( m ( − 1 ) ) ⊗ α H ( h ) ⋅ m ( 0 )

for all h ∈ H and m ∈ M . Indeed, routine manipulations give

( h ⋅ m ) ( − 1 ) ⊗ ( h ⋅ m ) ( 0 ) = ( ( Θ α H − 4 ( h 11 ) ) α H − 1 ( m ( − 1 ) ) ) Θ α H − 2 S ( h 2 ) ⊗ α H − 1 ( h 12 ) m ( 0 ) = ( 2.4 ) ( ( Θ α H − 3 ( h 1 ) ) α H − 1 ( m ( − 1 ) ) ) Θ α H − 3 S ( h 22 ) ⊗ α H − 1 ( h 21 ) m ( 0 ) = ( α H − 1 ( m ( − 1 ) ) ( Θ α H − 3 ( h 1 ) ) ) Θ α H − 3 S ( h 21 ) ⊗ α H − 1 ( h 22 ) m ( 0 ) = ( 2.1 ) m ( − 1 ) ( ( Θ α H − 3 ( h 1 ) ) Θ α H − 4 S ( h 21 ) ) ⊗ α H − 1 ( h 22 ) m ( 0 ) = ( 2.4 ) m ( − 1 ) ( ( Θ α H − 4 ( h 11 ) ) Θ α H − 4 S ( h 12 ) ) ⊗ h 2 m ( 0 ) = ( 2.7 ) m ( − 1 ) ⋅ Θ α H − 4 ( 1 H ) ε ( h 1 ) ⊗ h 2 ⋅ m ( 0 ) = α H ( m ( − 1 ) ) ⊗ α H ( h ) ⋅ m ( 0 ) ,

where we used commutativity and cocommutativity of H in the third equality.

Now we claim the equality (5.1). For this end, let ( M , α M ) , ( N , α N ) and ( P , α P ) ∈ ℋY D Θ H H . Then, for all m ∈ M , n ∈ N and p ∈ P , we obtain that

( i d P ⊗ c M , N ) ( c P , M − 1 ⊗ i d N ) ( i d M ⊗ c N , P ) ( m ⊗ n ⊗ p ) = ( i d P ⊗ c M , N ) ( c P , M − 1 ⊗ i d N ) ( m ⊗ Θ − 1 ( n ( − 1 ) ) ⋅ α P − 1 ( p ) ⊗ α N − 1 ( n ( 0 ) ) ) = ( i d P ⊗ c M , N ) ( α H − 1 ( ( Θ − 1 ( n ( − 1 ) ) ⋅ α H − 1 ( p ) ) ( 0 ) ) ⊗ S − 1 Θ ( ( Θ − 1 ( n ( − 1 ) ) ⋅ α H − 1 ( p ) ) ( − 1 ) ) ⋅ α H − 1 ( m ) ⊗ α N − 1 ( n ( 0 ) ) ) = ( 5.2 ) ( i d P ⊗ c M , N ) ( α H − 1 ( α H Θ − 1 ( n ( − 1 ) ) ⋅ α P − 1 ( p ) ( 0 ) ) ⊗ S − 1 Θ − 1 ( α H ( α P − 1 ( p ) ) ( − 1 ) ) ⋅ α M − 1 ( m ) ⊗ α N − 1 ( n ( 0 ) ) ) = ( i d P ⊗ c M , N ) ( Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 1 ( m ) ⊗ α N − 1 ( n ( 0 ) ) ) by (2.10) and (2.8) = Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ Θ − 1 ( ( S − 1 Θ − 1 ( p ( − 1 ) ) α M − 1 ( m ) ) ( − 1 ) ) ⋅ α N − 2 ( n ( 0 ) ) ⊗ α H − 1 ( ( S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 1 ( m ) ) ( 0 ) ) = ( 5.2 ) Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ Θ − 1 α H ( ( α M − 1 ( m ) ) ( − 1 ) ) ⋅ α N − 2 ( n ( 0 ) ) ⊗ α M − 1 ( α H S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 1 ( m ) ( 0 ) ) = ( 2.8 ) Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ Θ − 1 α H ( α M − 1 ( m ) ( − 1 ) ) ⋅ α N − 2 ( n ( 0 ) ) ⊗ Θ − 1 S − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ) ( 0 ) = ( 2.10 ) Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ Θ − 1 ( m ( − 1 ) ) ⋅ α N − 2 ( n ( 0 ) ) ⊗ Θ − 1 S − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ( 0 ) ) ,

and also

( c N , P ⊗ i d M ) ( i d N ⊗ c P , M − 1 ) ( c M , N ⊗ i d P ) ( m ⊗ n ⊗ p ) = ( c N , P ⊗ i d M ) ( i d N ⊗ c P , M − 1 ) ( Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ⊗ α M − 1 ( m ( 0 ) ) ⊗ p ) = ( c N , P ⊗ i d M ) ( Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ⊗ α P − 1 ( p ( 0 ) ) ⊗ S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ( 0 ) ) ) = Θ − 1 ( ( Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ) ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ α N − 1 ( ( Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ) ( 0 ) ) ⊗ S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ( 0 ) ) = ( 5.2 ) Θ − 1 ( α H ( ( α N − 1 ( n ) ) ( − 1 ) ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ α N − 1 ( α H Θ − 1 ( m ( − 1 ) ) ⋅ α N − 1 ( n ) ( 0 ) ) ⊗ S − 1 Θ − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ( 0 ) ) = ( 2.8 ) Θ − 1 ( n ( − 1 ) ) ⋅ α P − 2 ( p ( 0 ) ) ⊗ Θ − 1 ( m ( − 1 ) ) ⋅ α N − 2 ( n ( 0 ) ) ⊗ Θ − 1 S − 1 ( p ( − 1 ) ) ⋅ α M − 2 ( m ( 0 ) ) ,

as required.

( ⇒ ) Assume that the canonical braiding c is pseudosymmetric. We will first show that ( H , α ) is cocommutative. For this end, let 1 ⊗ Θ α H − 2 ( x ) ⊗ 1 ∈ H 2 ⊗ H 1 ⊗ H 2 . Then we obtain

( i d ⊗ ε ⊗ i d ) ( c H 1 , H 2 ⊗ i d H 2 ) ( i d H 1 ⊗ c H 2 , H 2 − 1 ) ( c H 2 , H 1 ⊗ i d H 2 ) ( 1 ⊗ Θ α H − 2 ( x ) ⊗ 1 ) = ( i d ⊗ ε ⊗ i d ) ( c H 1 , H 2 ⊗ i d H 2 ) ( i d H 1 ⊗ c H 2 , H 2 − 1 ) ( Θ α H − 2 ( x ) ⊗ 1 ⊗ 1 ) = ( i d ⊗ ε ⊗ i d ) ( c H 1 , H 2 ⊗ i d H 2 ) ( Θ α H − 2 ( x ) ⊗ 1 ⊗ 1 ) = ( i d ⊗ ε ⊗ i d ) ( Θ − 1 ( Θ α H − 2 ( x ) ( − 1 ) ) ⋅ 1 ⊗ α H − 1 ( Θ α H − 2 ( x ) ( 0 ) ) ⊗ 1 ) = ( i d ⊗ ε ⊗ i d ) ( α H − 1 ( x 1 ) ⊗ Θ α H − 3 ( x 2 ) ⊗ 1 ) = ( 2.4 ) x ⊗ 1 ,

where the fourth equality uses (2.2) and (2.3). On the other hand, we have

( i d ⊗ ε ⊗ i d ) ( i d H 2 ⊗ c H 2 , H 1 ) ( c H 2 , H 2 − 1 ⊗ i d H 1 ) ( i d H 2 ⊗ c H 1 , H 2 ) ( 1 ⊗ Θ α H − 2 ( x ) ⊗ 1 ) = ( i d ⊗ ε ⊗ i d ) ( i d H 2 ⊗ c H 2 , H 1 ) ( c H 2 , H 2 − 1 ⊗ i d H 1 ) ( 1 ⊗ α H − 1 ( x 1 ) ⊗ Θ α H − 3 ( x ) ( x 2 ) ) = ( i d ⊗ ε ⊗ i d ) ( i d H 2 ⊗ c H 2 , H 1 ) ( α H − 1 ( α H − 1 ( x 1 ) ( 0 ) ) ⊗ S − 1 Θ − 1 ( α H − 1 ( x 1 ) ( − 1 ) ) ⋅ α H − 1 ( 1 ) ⊗ Θ α H − 3 ( x ) ( x 2 ) ) = ( i d ⊗ ε ⊗ i d ) ( i d H 2 ⊗ c H 2 , H 1 ) ( α H − 1 ( α H − 1 ( α H − 1 ( x 1 ) 12 ) ) ⊗ S − 1 Θ − 1 × ( Θ α H − 4 ( x ) ( α H − 1 ( x 1 ) 11 ) Θ α H − 3 ( x ) ( S α H − 1 ( x 1 ) 2 ) ) ⋅ 1 ⊗ Θ α H − 3 ( x ) ( x 2 ) ) = ( i d ⊗ ε ⊗ i d ) ( i d H 2 ⊗ c H 2 , H 1 ) ( α H − 3 ( x 112 ) ⊗ S − 1 ( α H − 4 ( x 111 ) S α H − 3 ( x 12 ) ) ⊗ Θ α H − 3 ( x ) ( x 2 ) ) by (2.2) and (2.3) = α H − 3 ( x 112 ) ⊗ ε ( Θ α H − 3 ( x ) ( x 2 ) ) S − 1 ( α H − 4 ( x 111 ) S α H − 3 ( x 12 ) ) = α H − 2 ( x 12 ) ⊗ S − 1 ( α H − 3 ( x 11 ) S α H − 2 ( x 2 ) ) by (2.4) and (2.3) = α H − 2 ( x 12 ) ⊗ α H − 2 ( x 2 ) S − 1 ( α H − 3 ( x 11 ) ) ,

where the fifth equality follows by Lemma 5.3. So, we get x ⊗ 1 = α H − 2 ( x 12 ) ⊗ α H − 2 ( x 2 ) S − 1 ( α H − 3 ( x 11 ) ) . Furthermore, we have

x 2 ⊗ x 1 = x 2 ⊗ α H − 1 ( 1 ⋅ x 1 ) = α H − 2 ( x 212 ) ⊗ α H − 1 ( ( α H − 2 ( x 22 ) S − 1 ( α H − 3 ( x 211 ) ) ) x 1 ) = ( 2.9 ) α H − 2 ( x 212 ) ⊗ α H − 1 ( α H − 1 ( x 22 ) ( S − 1 ( α H − 3 ( x 211 ) ) α H − 1 ( x 1 ) ) ) = ( 2.4 ) α H − 1 ( x 12 ) ⊗ α H − 1 ( x 2 ( S − 1 ( α H − 3 ( x 112 ) ) α H − 3 ( x 111 ) ) ) = ( 2.7 ) α H − 1 ( x 12 ) ⊗ α H − 1 ( x 2 ⋅ 1 ) ε ( x 11 ) = ( 2.4 ) , ( 2.2 ) x 1 ⊗ x 2 ,

as required. Next we claim that ( H , α H ) is commutative. Indeed, given 1 ⊗ Θ α H − 2 ( x ) ( x ) ⊗ Θ α H − 2 ( y ) ∈ H 2 ⊗ H 1 ⊗ H 1 , on one hand, we have

( c H 1 , H 1 ⊗ i d H 2 ) ( i d H 1 ⊗ c H 1 , H 2 − 1 ) ( c H 2 , H 1 ⊗ i d H 1 ) ( 1 ⊗ Θ α H − 2 ( x ) ⊗ Θ α H − 2 ( y ) ) = ( c H 1 , H 1 ⊗ i d H 2 ) ( i d H 1 ⊗ c H 1 , H 2 − 1 ) ( Θ α H − 2 ( x ) ⊗ 1 ⊗ Θ α H − 2 ( y ) ) = ( c H 1 , H 1 ⊗ i d H 2 ) ( Θ α H − 2 ( x ) ⊗ Θ α H − 3 ( y 2 ) ⊗ S − 1 ( α H − 1 ( y 1 ) ) ) = Θ − 1 ( Θ α H − 2 ( x ) ( − 1 ) ) ⋅ α H − 1 ( Θ α H − 3 ( y 2 ) ) ⊗ α H − 1 ( Θ α H − 2 ( x ) ( 0 ) ) ⊗ S − 1 ( α H − 1 ( y 1 ) )

= α H − 2 ( x 1 ) ) ⋅ Θ α H − 4 ( y 2 ) ⊗ Θ α H − 3 ( x 2 ) ⊗ S − 1 ( α H − 1 ( y 1 ) ) = ( Θ α H − 4 ( α H − 2 ( x 11 ) ) α H − 1 ( Θ α H − 4 ( y 2 ) ) ) S ( Θ α H − 3 ( α H − 2 ( x 12 ) ) ) ⊗ Θ α H − 3 ( x 2 ) ⊗ S − 1 ( α H − 1 ( y 1 ) ) = ( Θ α H − 6 ( x 11 ) Θ α H − 5 ( y 2 ) ) S ( Θ α H − 5 ( x 12 ) ) ⊗ Θ α H − 3 ( x 2 ) ⊗ S − 1 ( α H − 1 ( y 1 ) ) ,

where the first equality is followed by Lemma 5.2. On the other hand, we compute

( i d H 1 ⊗ c H 2 , H 1 ) ( c H 1 , H 2 − 1 ⊗ i d H 1 ) ( i d H 2 ⊗ c H 1 , H 1 ) ( 1 ⊗ Θ α H − 2 ( x ) ⊗ Θ α H − 2 ( y ) ) = ( i d H 1 ⊗ c H 2 , H 1 ) ( c H 1 , H 2 − 1 ⊗ i d H 1 ) ( 1 ⊗ α H − 2 ( x 1 ) ⋅ Θ α H − 3 ( y ) ⊗ Θ α H − 3 ( x 2 ) ) = ( i d H 1 ⊗ c H 2 , H 1 ) ( c H 1 , H 2 − 1 ⊗ i d H 1 ) ( 1 ⊗ ( Θ α H − 6 ( x 11 ) Θ α H − 4 ( y ) ) S ( Θ α H − 5 ( x 12 ) ) ⊗ Θ α H − 3 ( x 2 ) ) = ( i d H 1 ⊗ c H 2 , H 1 ) ( ( Θ α H − 7 ( x 112 ) Θ α H − 5 ( y 2 ) ) S ( Θ α H − 6 ( x 121 ) ) ⊗ α H − 4 ( x 122 ) S − 1 ( α H − 5 ( x 111 ) α H − 3 ( y 1 ) ) ⊗ Θ α H − 3 ( x 2 ) ) = ( Θ α H − 7 ( x 112 ) Θ α H − 5 ( y 2 ) ) S ( Θ α H − 6 ( x 121 ) ) ⊗ Θ α H − 3 ( x 2 ) ⊗ α H − 4 ( x 122 ) S − 1 ( α H − 5 ( x 111 ) α H − 3 ( y 1 ) ) ,

where the last equality relies on Lemma 5.2. Since ℋY D Θ H H is pseudosymmetric, it follows that

( Θ α H − 6 ( x 11 ) Θ α H − 5 ( y 2 ) ) S ( Θ α H − 5 ( x 12 ) ) ⊗ Θ α H − 3 ( x 2 ) ⊗ S − 1 ( α H − 1 ( y 1 ) ) = ( Θ α H − 7 ( x 112 ) Θ α H − 5 ( y 2 ) ) S ( Θ α H − 6 ( x 121 ) ) ⊗ Θ α H − 3 ( x 2 ) ⊗ α H − 4 ( x 122 ) S − 1 ( α H − 5 ( x 111 ) α H − 3 ( y 1 ) ) .

Now applying ε ⊗ Θ − 1 α H 2 ⊗ S to the above equality yields

x ⊗ y = α H − 1 ( x 2 ) ⊗ ( α H − 4 ( x 11 ) α H − 2 ( y ) ) S α H − 3 ( x 12 ) .

and then we obtain

y x = ( ( α H − 4 ( x 11 ) α H − 2 ( y ) ) S α H − 3 ( x 12 ) ) α H − 1 ( x 2 ) = ( 2.2 ) ( α H − 3 ( x 11 ) α H − 1 ( y ) ) ( S α H − 3 ( x 12 ) α H − 2 ( x 2 ) ) = ( 2.4 ) ( α H − 2 ( x 1 ) α H − 1 ( y ) ) ( S α H − 3 ( x 21 ) α H − 3 ( x 22 ) ) = ( 2.7 ) ( α H − 2 ( x 1 ) α H − 1 ( y ) ) ⋅ 1 ε ( x 2 ) = ( 2.2 ) , ( 2.4 ) x y .

Therefore H is commutative, as desired. The proof is complete.□

  1. Funding information: The first author was financially supported by The Research Project of Shaoxing University (No. 2020LG1009) and Research Project of Shaoxing University Yuanpei College (No. KY2020C01). The second author was financially supported by the Natural Science Foundation of Zhejiang Province (No. LY14A010006) and the National Natural Science Foundation of China (No. 11571239).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2021-05-10
Accepted: 2021-07-13
Published Online: 2021-11-18

© 2021 Wei Liu and Xiaoli Fang, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0083
Schlagwörter für diesen Artikel
Hom-Hopf algebra; Θ-Yetter-Drinfeld module; symmetric pair
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Artikel in diesem Heft

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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Heruntergeladen am 9.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2021-0083/html
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