【摘要】：Yetter-Drinfeld范疇是代數(shù)學的重要研究對象之一,在數(shù)學,物理,拓撲學等領域有著廣泛的應用.Majid給出了Radford雙積的一個范疇的解釋:B是Yetter-Drinfeld范疇HHyD中的Hopf代數(shù)當且僅當B×#H是Radford雙積Hopf代數(shù).近年來,在Hopf代數(shù)理論中Yetter-Drinfeld范疇的研究吸引了許多學者.本文對Yetter-Drinfeld范疇進行了相關研究.主要內容如下:(1)我們延伸雙邊smash余積結構到雙邊交義余積C×αHβ×D.則我們可得到smash積代數(shù)C#H#D和雙邊交叉余積余代數(shù)C× αHβ×D 成為雙代數(shù)的充要條件,這推廣了Majid double雙積.在雙邊交叉余積余代數(shù)C×αHβ× D中當取C = K或D=K時即為右或左交叉余積.(2)我們從Hom-Hopf模代數(shù)(余代數(shù))著手給出Rota-Baxter monoidal Hom-代數(shù)(余代數(shù))的結構,然后引入Rota-Baxter monoidal Hom-雙代數(shù)的概念,并且Radford雙積monoidal Hom-Hopf代數(shù)為Rota-Baxter monoidal Hom-雙代數(shù)提供例子.進一步,我們考慮Rota-Baxter monoidal Hom-系統(tǒng)和monoidal Hom-dendriform代數(shù)之間的關系,并且通過不同權的Rota-Baxter monoidal Hom-代數(shù)(余代數(shù))得到pre-Lie Hom-代數(shù)(余代數(shù))的結構.(3)研究在(m, n)-Hom-Yetter-Drinfeld范疇H(HHyD(Z))中的Lie代數(shù).首先引入(m,n)-Hom Lie代數(shù)的概念,進而我們證明當辮子τ在(m,n)-Hom-Yetter-Drinfeld范疇H(HHyD(Z))中是對稱且(A,α)具有適當?shù)腖ie括號時,(A,α)就能構造出一個(m,n)-Hom Lie代數(shù).我們還將證明如果(A,α)還是兩個(H,β)可交換的Hom-子代數(shù)的和時,就有[A,A][A, A] =0.(4)首先引入(lazy)Horn-2-余循環(huán)在Hom-Hopf代數(shù)上的概念,并且研究它們的一些性質.進一步我們延伸在Hom-Yetter-Drinfeld范疇中的(lazy) Hom-2-余循環(huán)到Radford雙積Hom-Hopf代數(shù).(5)首先給出smash積monoidal BiHom-代數(shù)(B#H,αB(?)αH,βB(?)βH)和smash余積mon-oidal BiHom-余代數(shù)(B×H,αB(?)αH,βB(?)βH),進而得到(B#H,αB(?)αH,βB(?)βH))和(B×H,αB(?)αH,βB(?)βH)構成Radford雙積monoidal BiHom-Hopf代數(shù)的充分必要條件.這也為構造新的辮子張量范疇(即Yetter-Drinfeld范疇的廣義形式)提供了條件.
[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 偽 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 偽 H 尾 脳 D to be bialgebras, which generalizes the Majid double biproduct. C 脳 偽 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, 偽) has appropriate Lie brackets, (A, 偽) can construct a (mtn) -Hom Lie algebra. We will also prove that if (A, 偽) 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. 榪涗竴姝ユ垜浠歡浼稿湪Hom-Yetter-Drinfeld鑼冪暣涓殑(lazy) Hom-2-浣欏驚鐜埌Radford鍙岀НHom-Hopf浠ｆ暟.(5)棣栧厛緇欏嚭smash縐痬onoidal BiHom-浠ｆ暟(B#H,偽B(?)偽H,尾B(?)尾H)鍜宻mash浣欑Нmon-oidal BiHom-浣欎唬鏁，

本文編號：2245531