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Shouchuan Zhang

Publications and source records attributed to Shouchuan Zhang.

At least 19 recordsLinked to original sources
arXiv2020

Nichols algebras over classical Weyl groups

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.

math.QA↗
arXiv2020

Basis of Nichols Braided Lie Algebras

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.

math.QA↗
arXiv2019

On the Hopf Algebra of Rooted Trees

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.

math-ph↗
arXiv2019

Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

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 $Δ(\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.

math.QA↗
arXiv2017

Structures of Nichols (braided) Lie algebras of diagonal type

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 $\dim V=2$.

math.QA↗
arXiv2017

Nichols algebras over classical Weyl groups, Fomin-Kirillov algebras and Lyndon basis

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 $\mathfrak{B} ({\mathcal O}_{(1, 2)} , ε\otimes {\rm sgn})$ 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 $A_c$.

math.QA↗
arXiv2015

On Nichols (braided) Lie 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 $Δ(\mathfrak B(V))$ 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.

math.QA↗
arXiv2014

Relationship between Nichols braided Lie algebras and Nichols algebras

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.

math.QA↗
arXiv2014

Quantum Theory in Design

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.

math.QA↗
arXiv2013

Finite dimensional Nichols algebras over finite cyclic groups

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.

math.NT↗