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All quasi-affine connected Generalized Dynkin Diagram with rank $= 3$ and $2$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 3$ and $2$ are also found., Comment: 338 pages

All quasi-affine connected Generalized Dynkin Diagrams with rank $> 5$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $> 5$ are also found., Comment: 77 pages. arXiv admin note: substantial text overlap with arXiv:2202.09940

All quasi-affine connected Generalized Dynkin Diagram with rank $= 5$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 5$ are also found., Comment: 79 pages

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1307.8227

In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras., Comment: 6 pages. arXiv admin note: substantial text overlap with arXiv:1704.06810

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak L(V)$ of arithmetic root systems and the dimension for $\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak B(V) = F\oplus \mathfrak L^-(V)$ and $\mathfrak L^-(V)= \mathfrak L(V)$. We obtain an explicit basis of $\mathfrak L^ - (V)$ over quantum linear space $V$ with $\dim V=2$., Comment: 23 pages. Version to appear in Journal of Lie Theory

Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.

It is shown that if $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim (\mathfrak L^-(V)) = \infty$ $($resp. $ \dim (\mathfrak L(V)) = \infty $$)$ $($ resp. $ \dim (\mathfrak B(V)) = \infty $$)$ if and only if $\Delta(\mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $\mathfrak B(V)$ equal to zero, viz. $\overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_j\in \mathfrak B(V)$, are given., Comment: 24 pages

We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$ and {\rm (iii)} if $\Delta (\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V'$, which is twisting equivalent to $V$, such that $ \dim (\mathfrak L^ - (V')) = \infty.$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals., Comment: 29 Pages; Substantially revised version; To appear in International Journal of Mathematics

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations., Comment: LeTex 18 pages, need JOLT-macros to compile. To appear in Journal of Lie Theory

In this paper, we use braiding diagrams to present rules of shapes and designs. That is, we design colour, design size, design brightness, design codes by means of braiding., Comment: page 13. arXiv admin note: text overlap with arXiv:math/0511251

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra $\mathcal E_n$ and Nichols algebra $\mathfrak{B} ({\mathcal O}_{{(1, 2)}} , \epsilon \otimes {\rm sgn})$ of transposition over symmetry group by means of quiver Hopf algebras. We generalize {\rm FK } algebra. The characteristic of finiteness of Nichols algebras in thirteen ways and of {\rm FK } algebras ${\mathcal E}_n$ in nine ways is given. All irreducible representations of finite dimensional Nichols algebras %({\rm FK } algebras ${\mathcal E}_n$) and a complete set of hard super- letters of Nichols algebras of finite Cartan types are found. The sufficient and necessary condition for Nichols algebra $\mathfrak B(M)$ of reducible {\rm YD} module $M$ over $A \rtimes \mathbb{S}_n$ with ${\rm supp } (M) \subseteq A$ to be finite dimensional is given. % Some conditions for a braided vector space to become a {\rm YD} module over finite commutative group are obtained. It is shown that hard braided Lie Lyndon word, standard Lyndon word, Lyndon basis path, hard Lie Lyndon word and standard Lie Lyndon word are the same with respect to $ \mathfrak B(V)$, Cartan matrix $A_c$ and $U(L^+)$, respectively, where $V$ and $L$ correspond to the same finite Cartan matrix $A_c$., Comment: 45pages

All quasi-affine connected Generalized Dynkin Diagram with rank $= 4$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 4$ are also found., Comment: 325 pages

All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group $\mathbb Z_n$ are found. It is proved that finite dimensional Nichols algebra over $\mathbb Z_2$ is a quantum linear space and Nichols algebra of connected Yetter-Drinfeld module $V$ over $\mathbb Z_n$ with $\dim V >3$ is infinite dimensional., Comment: 22 pages, substantially shortened and reference added. Version to appear in Journal of Lie Theory

Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words., Comment: 9pages

We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional., Comment: Combined with arXiv:0902.4748 plus substantial changes. To appear International Journal of Mathematics

To classify the finite dimensional pointed Hopf algebras with $G= {\rm McL}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}., Comment: 339pages

Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups ${\rm HS}$ and ${\rm Co3}$ are found., Comment: 9pages

To classify the finite dimensional pointed Hopf algebras with $G= {\rm HS}$ or ${\rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}., Comment: 409 pages

The general theory of the radicals of Lie algebras are established. Baer radicals of untwisted affine Lie algebras are found., Comment: 14pages

It is an important open problem whether the dimension of the Nichols algebra B(O,\rho) is finite when O is the class of the transpositions and \rho is the sign representation, with m>= 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs (O,\rho) might give rise to finite-dimensional Nichols algebras., Comment: 14 pages; typos added in abstract

All -1-type pointed Hopf algebras and central quantum linear spaces with Weyl groups of exceptional type are found. It is proved that every non -1-type pointed Hopf algebra with real $G(H)$ is infinite dimensional and every central quantum linear space over finite group is finite dimensional. It is proved that except a few cases Nichols algebras of reducible Yetter-Drinfeld modules over Weyl groups of exceptional type are infinite dimensional., Comment: 44pages

To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 95--112., Comment: 268pages

To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 65--94., Comment: 380pages

To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_6$, $E_7$, $F_4$, $G_2$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP., Comment: 505pages

The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one., Comment: 19 pages; To appear in Bulletin of the Australian Mathematical Society

Blatter-Montgomery duality theorem is generalized into braided tensor categories. It is shown that $Hom(V,W)$ is a braided Yetter-Drinfeld module for any two braided Yetter-Drinfeld modules $V$ and $W$., Comment: 27Pages

We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneider's Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algebra of rooted trees are two important examples of Hopf algebras. We give their representation and show that they have not any nonzero integrals. We structure their graded Drinfeld doubles and show that they are local quasitriangular Hopf algebras., Comment: 7 pages. arXiv admin note: text overlap with arXiv:math/0105210 by other authors

The condition for double bicrosssum to be a braided Lie bialgebra is given. The result generalizes quantum double, bicrosssum, bicrosscosum, bisum. The quantum double of braided Lie bialgebras is constructed. The relation between double crosssum of Lie algebras and double crossproduct of Hopf algebras is given., Comment: 25pages. Improve some words

We obtain the formula computing the number of isomorphic classes of ramification systems with characters over group $S_n$ ($n \not=6$) and their representatives., Comment: 18pages

We obtain the formula computing the number of isomorphic classes of element systems with characters over finite commutative group $G$., Comment: 21pages; Proceeding 2011 International Conference on Business Management and Electronic Information, Vol.05, Shanghai,China

We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors., Comment: 12pages

Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\ ],\Delta_{r}, r) $ is a coboundary (or triangular) Lie bialgebra are given., Comment: 7pages

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category., Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided Hopf algebras; In symmetric tensor categories, we show the duality theorem and construct quantum double; In ordinary vector space category with ordinary twist, we obtain the relation between the global dimension, the weak dimension, the Jacbson radical, Baer radical of algebra $R$ and its crossed product $R #_\sigma H$. We also give the relation between the decompositions of comodules and coalgebras. We classify quiver Hopf algebras. We obtain all solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra $L$ with dim $L \le 3$. We also give the sufficient and necessary conditions for $(L, \hbox {[ ]}, \Delta_r, r)$ to be a coboundary (or triangular) Lie bialgebra., Comment: 318pages