Yetter-Drinfeld范畴的相关研究
发布时间：2018-09-17 10:07
【摘要】：Yetter-Drinfeld范畴是代数学的重要研究对象之一,在数学,物理,拓扑学等领域有着广泛的应用.Majid给出了Radford双积的一个范畴的解释:B是Yetter-Drinfeld范畴HHyD中的Hopf代数当且仅当B×#H是Radford双积Hopf代数.近年来,在Hopf代数理论中Yetter-Drinfeld范畴的研究吸引了许多学者.本文对Yetter-Drinfeld范畴进行了相关研究.主要内容如下:(1)我们延伸双边smash余积结构到双边交义余积C×αHβ×D.则我们可得到smash积代数C#H#D和双边交叉余积余代数C× αHβ×D 成为双代数的充要条件,这推广了Majid double双积.在双边交叉余积余代数C×αHβ× D中当取C = K或D=K时即为右或左交叉余积.(2)我们从Hom-Hopf模代数(余代数)着手给出Rota-Baxter monoidal Hom-代数(余代数)的结构,然后引入Rota-Baxter monoidal Hom-双代数的概念,并且Radford双积monoidal Hom-Hopf代数为Rota-Baxter monoidal Hom-双代数提供例子.进一步,我们考虑Rota-Baxter monoidal Hom-系统和monoidal Hom-dendriform代数之间的关系,并且通过不同权的Rota-Baxter monoidal Hom-代数(余代数)得到pre-Lie Hom-代数(余代数)的结构.(3)研究在(m, n)-Hom-Yetter-Drinfeld范畴H(HHyD(Z))中的Lie代数.首先引入(m,n)-Hom Lie代数的概念,进而我们证明当辫子τ在(m,n)-Hom-Yetter-Drinfeld范畴H(HHyD(Z))中是对称且(A,α)具有适当的Lie括号时,(A,α)就能构造出一个(m,n)-Hom Lie代数.我们还将证明如果(A,α)还是两个(H,β)可交换的Hom-子代数的和时,就有[A,A][A, A] =0.(4)首先引入(lazy)Horn-2-余循环在Hom-Hopf代数上的概念,并且研究它们的一些性质.进一步我们延伸在Hom-Yetter-Drinfeld范畴中的(lazy) Hom-2-余循环到Radford双积Hom-Hopf代数.(5)首先给出smash积monoidal BiHom-代数(B#H,αB(?)αH,βB(?)βH)和smash余积mon-oidal BiHom-余代数(B×H,αB(?)αH,βB(?)βH),进而得到(B#H,αB(?)αH,βB(?)βH))和(B×H,αB(?)αH,βB(?)βH)构成Radford双积monoidal BiHom-Hopf代数的充分必要条件.这也为构造新的辫子张量范畴(即Yetter-Drinfeld范畴的广义形式)提供了条件.
[Abstract]:Yetter-Drinfeld category is one of the important research objects in algebra. It has been widely used in mathematics, physics, topology and other fields. Majid has given an explanation that a category of Radford double product is Hopf algebra in Yetter-Drinfeld category HHyD if and only if B 脳 #H is Radford biproduct Hopf algebra. In recent years, the study of Yetter-Drinfeld category in the theory of Hopf algebra has attracted many scholars. This paper studies the category of Yetter-Drinfeld. The main contents are as follows: (1) We extend the structure of bilateral smash coproduct to C 脳 伪 H 尾 脳 D. Then we obtain the necessary and sufficient conditions for the smash product algebra C#H#D and the bilateral cross coproduct coalgebra C 脳 伪 H 尾 脳 D to be bialgebras, which generalizes the Majid double biproduct. C 脳 伪 H 尾 脳 D is right or left cross coproduct when C = K or D = K. (2) We give the structure of Rota-Baxter monoidal Hom- algebra (coalgebra) from Hom-Hopf module algebra (coalgebra), and then introduce the concept of Rota-Baxter monoidal Hom- bialgebra. And Radford biproduct monoidal Hom-Hopf algebra provides an example for Rota-Baxter monoidal Hom- bialgebra. Furthermore, we consider the relationship between Rota-Baxter monoidal Hom- systems and monoidal Hom-dendriform algebras, and obtain the structure of pre-Lie Hom- algebras (coalgebras) by Rota-Baxter monoidal Hom- algebras with different weights. (3) We study Lie algebras in the (m, n) -Hom-Yetter-Drinfeld category H (HHyD (Z). First, we introduce the concept of (mtn) -Hom Lie algebra, and then we prove that when the braid 蟿 is symmetric in (mtn) -Hometter-Drinfeld category H (HHyD (Z) and (A, 伪) has appropriate Lie brackets, (A, 伪) can construct a (mtn) -Hom Lie algebra. We will also prove that if (A, 伪) is the sum of two (H, 尾) commutative Hom- subalgebras, there will be [A] [A, A] 0. (4) the concept of (lazy) Horn-2- cocycles on Hom-Hopf algebras is first introduced, and some properties of them are studied. 杩涗竴姝ユ垜浠欢浼稿湪Hom-Yetter-Drinfeld鑼冪暣涓殑(lazy) Hom-2-浣欏惊鐜埌Radford鍙岀НHom-Hopf浠ｆ暟.(5)棣栧厛缁欏嚭smash绉痬onoidal BiHom-浠ｆ暟(B#H,伪B(?)伪H,尾B(?)尾H)鍜宻mash浣欑Нmon-oidal BiHom-浣欎唬鏁，
本文编号：2245531
[Abstract]:Yetter-Drinfeld category is one of the important research objects in algebra. It has been widely used in mathematics, physics, topology and other fields. Majid has given an explanation that a category of Radford double product is Hopf algebra in Yetter-Drinfeld category HHyD if and only if B 脳 #H is Radford biproduct Hopf algebra. In recent years, the study of Yetter-Drinfeld category in the theory of Hopf algebra has attracted many scholars. This paper studies the category of Yetter-Drinfeld. The main contents are as follows: (1) We extend the structure of bilateral smash coproduct to C 脳 伪 H 尾 脳 D. Then we obtain the necessary and sufficient conditions for the smash product algebra C#H#D and the bilateral cross coproduct coalgebra C 脳 伪 H 尾 脳 D to be bialgebras, which generalizes the Majid double biproduct. C 脳 伪 H 尾 脳 D is right or left cross coproduct when C = K or D = K. (2) We give the structure of Rota-Baxter monoidal Hom- algebra (coalgebra) from Hom-Hopf module algebra (coalgebra), and then introduce the concept of Rota-Baxter monoidal Hom- bialgebra. And Radford biproduct monoidal Hom-Hopf algebra provides an example for Rota-Baxter monoidal Hom- bialgebra. Furthermore, we consider the relationship between Rota-Baxter monoidal Hom- systems and monoidal Hom-dendriform algebras, and obtain the structure of pre-Lie Hom- algebras (coalgebras) by Rota-Baxter monoidal Hom- algebras with different weights. (3) We study Lie algebras in the (m, n) -Hom-Yetter-Drinfeld category H (HHyD (Z). First, we introduce the concept of (mtn) -Hom Lie algebra, and then we prove that when the braid 蟿 is symmetric in (mtn) -Hometter-Drinfeld category H (HHyD (Z) and (A, 伪) has appropriate Lie brackets, (A, 伪) can construct a (mtn) -Hom Lie algebra. We will also prove that if (A, 伪) is the sum of two (H, 尾) commutative Hom- subalgebras, there will be [A] [A, A] 0. (4) the concept of (lazy) Horn-2- cocycles on Hom-Hopf algebras is first introduced, and some properties of them are studied. 杩涗竴姝ユ垜浠欢浼稿湪Hom-Yetter-Drinfeld鑼冪暣涓殑(lazy) Hom-2-浣欏惊鐜埌Radford鍙岀НHom-Hopf浠ｆ暟.(5)棣栧厛缁欏嚭smash绉痬onoidal BiHom-浠ｆ暟(B#H,伪B(?)伪H,尾B(?)尾H)鍜宻mash浣欑Нmon-oidal BiHom-浣欎唬鏁，
本文编号：2245531
本文链接：https://www.wllwen.com/kejilunwen/yysx/2245531.html
