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 Schneiders 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.
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 $p_{uu} ot= 1$ 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.
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.
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.
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 $A_c$.
All affine connected Generalized Dynkin Diagram with rank 4 are found.
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.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
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}.
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}.
