GK dimension of some connected Hopf algebras

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.     

Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three

In this paper, we present some modules over the rank-three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.     

量子环面上斜导子李代数的表示

记L为量子环面上的斜导子李代数，本文构造了一族从sl2-模到L-模的函子F^αg，并对L-模F^αg（V）的结构进行了完全刻画，最后给出了L-模F^αg1（V）与F^βg2（W）同构的充分必要条件。     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

In this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,?)-modules and 𝔰𝔭(2n,?)-modules that appeared in the ?-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.     

Classification of Quasifinite Modules with Nonzero Central Charges for EALAs of Type A with Coordinates in Quantum Torus

The author first constructs a Lie algebra ∑ ：=∑（q, Wd） from rank 3 quantum torus, which is isomorphic to the core of EALAs of type Ad-1 with coordinates in quantum torus Cqd, and then gives the necessary and sufficient conditions for the highest weight modules to be quasifinite nonzero central charges are Finally the irreducible Z-graded quasifinite ∑-modules with classified.     

Tauvel’s height formula for quantum nilpotent algebras

Tauvel’s height formula, which provides a link between the height of a prime ideal and the Gelfand–Kirillov dimension of the corresponding factor algebra, is verified for quantum nilpotent algebras.     

The Ends of Algebras

We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

量子环面李代数的自同构群,泛中心扩张与导子

Cq=Cq[x1^±1,x2^±1]为复数域上的量子环面，其中q≠0是一个非单位根，D（Cq）为Cq的导子李代数．记Lq为Cq＋D（Cq）的导出子代数．该文研究李代数Lq的自同构群，泛中心扩张和导子李代数．     

一类量子环面李代数的单性(英文)

令C_q:=C_q[t₁^±1,t₂^±1,t₃^±1]为量子环面结合代数,L_q=C_q/C为量子环面李代数。本文定义了一个与q=（qij）_i,j=1³相关的指数方程体系,称之为C_q的特征方程组。通过这个特征方程组,证明了L_q是非单的当且仅当特征方程组在Z³中有非零解。对于|q|=0,证明了在C_q中存在一个极大交换子代数I,并且I严格包含中心Z（C_q）。同时文中也指出,对于|q|≠0,在C_q中不存在这样的子代数。     

Automorphism Group of q-Quantum Torus Lie Algebra with q a Root of Unity*

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x ^?1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x ^?1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _?(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.     

一类量子环面李代数的自同构群

=C_q[x_1~(±1),x_2~(±1)]为复数域上的非交换环面结合代数,A=\C,Der为的导子李代数．本文研究李代数L_q=DerA的自同构群Aut L_q．     

一类基于量子环面非交换结合代数的模

在量子环面[1]上构造一类非交换结合代数AQ-模M(a,b),我们还刻划了AQ-模的结构并揭示[2]一类商模序列:每个商模Mn(a)/Mn+1(a)都同构于M(a,0),每个商模的自同构群AutMn(a)/Mn+1(a)均与C*同构.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

Representations of a Class of Associative Algebras Related to the Quantum Torus

1 Introduction The mitivation of this paper comes from the work of[5].We kmow that vertex algebratheory is one of the importans parts in the study of infinte dimensional Lie theory,while thelattice vertex algebras form one of the most important and fundamental classes of vertexalgebras.In they studied the representation theory for certain” half lattice”vertex algebra     

Representations of Extended Affine Lie Algebras Coordinatized by Certain Quantum Tori

An irreducible representation of the extended affine Lie algebra of type A _n-1 coordinatized by a quantum torus of variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra .     

Conjugation in Representations of the Zassenhaus Algebra

Let F be an algebracially closed field of characteristic p > 2, and L be the p ⁿ -dimensional Zassenhaus algebra with the maximal invariant subalgebra L ₀ and the standard filtration {L _i }| ^pn−2 _{i =−1}. Then the number of isomorphism classes of simple L-modules is equal to that of simple L ₀-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p ⁿ . Received May 10, 1999, Accepted December 8, 1999     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.