Simple algebra with arbitrary odd Gelfand-Kirillov dimension

Gelfand-Kirillov dimension of differential difference algebras

Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more.

Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two

Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension

Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a connected Hopf algebra is either infinity or a positive integer. We also classify connected Hopf algebras of GK-dimension three over an algebraically closed field of characteristic zero.

Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two

We classify all non-affine Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero that are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\text{Ext}^1_H(k, k) \neq 0$. The affine ones were classified by the authors in 2010.

Weighted Leavitt path algebras of finite Gelfand-Kirillov dimension

We determine the Gelfand-Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over a field $K$ is isomorphic to a finite product of matrix rings over $K$.

Gelfand-Kirillov dimension for rings

The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered-graded rings, skew $PBW$ extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand-Kirillov dimension for modules and the Gel…

On a conjecture of Goodearl: Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2

For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.

Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-K…

A dichotomy result for prime algebras of Gelfand-Kirillov dimension two

Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.

Centralizers in Domains of Finite Gelfand-Kirillov Dimension

We study centralizers of elements in domains. We generalize a result of the author and Small, showing that if $A$ is a finitely generated noetherian domain and $a\in A$ is not algebraic over the extended centre of $A$, then the centralizer of $a$ has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of $A$. In the case that $A$ is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a $A$ that is not algebraic over the extended centre of $A$ satisfies a polynomial identity.

The Gelfand-Kirillov dimension of Hecke-Kiselman algebras

Hecke-Kiselman algebras $A_Θ$, over a field $k$, associated to finite oriented graphs $Θ$ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand-Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph $Θ$ that determines the dimension of $A_Θ$ is found. Namely, we prove that the Gelfand-Kirillov dimension of $A_Θ$ is the sum of the number of cyclic subgraphs of $Θ$ and the number of oriented paths of a special type in the graph, each counted certain specific number of …

Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension two

Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive qu…

Gelfand-Kirillov dimension and Jordan algebras

Let A be any associative algebra graded by a finite abelian group G, then if we denote by GKdim_k(A) and GKdim^G_k (A) the Gelfand-Kirillov dimension of its relatively free algebra and its relatively free G-graded algebra in k variables respectively, then GKdim_k(A)\leq GKdim^G_k (A). We show a counterexample of the previous result for Jordan algebras (hence non-associative). In particular, there exists a $Z_2$-grading on $UJ_n$, the Jordan algebra of $n\times n$ upper triangular matrices, n equal to 2 or 3, such that the previous inequality does not hold.

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.

Gelfand-Kirillov dimensions of simple modules over twisted group algebras $k \ast A$

For the $n$-dimensional multiparameter quantum torus algebra $Λ_{\mathfrak q}$ over a field $k$ defined by a multiplicatively antisymmetric matrix $\mathfrak q = (q_{ij})$ we show that in the case when the torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$ is large enough there is a characteristic set of values (possibly with gaps) from $0$ to $n$ that can occur as the Gelfand--Kirillov dimension of simple modules. The special case when $\mathrm{K}.\dim(Λ_{\mathfrak q}) = n - 1$ and $Λ_{\mathfrak q}$ is simple studied in A.~Gupta, {\it $\uppercase{\mbox{GK}}$-dimensions …

On Nichols algebras of infinite rank with finite Gelfand-Kirillov dimension

We classify infinite-dimensional decomposable braided vector spaces arising from abelian groups whose components are either points or blocks such that the corresponding Nichols algebras have finite Gelfand-Kirillov dimension. In particular we exhibit examples with $\operatorname{GKdim} = n$ for any natural number $n$.

Simple algebras of Gelfand-Kirillov dimension two

Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.

Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two

We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\Ext^1_H(k,k)\neq 0$. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.

Gelfand-Kirillov dimension of cosemisimple Hopf algebras

In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl-Zhang (2007), of Banica-Vergnioux (2009), and of D'Andrea-Pinzari-Rossi (2017).

Division algebras of Gelfand-Kirillov dimension two

Let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension less than 3, and let $Q(A)$ denote the quotient division algebra of $A$. We show that if $D$ is a division subalgebra of $Q(A)$ of GK dimension at least 2 then $Q(A)$ is finite dimensional as a left $D$-vector space. We use this to show that if $A$ is a finitely generated domain of GK dimension less than 3 over an algebraically closed field $K$ then any division subalgebra $D$ of $Q(A)$ is either a finitely generated field extension of $K$ of transcendence degree at most one, or $Q(A)$ is finite dimensional as a left …

On Gelfand-Kirillov conjecture for some W-algebras

Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a W-algebra then it holds for the universal enveloping algebra $\mathrm U(\frak g)$. This together with a result of A. Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for some $W$-algebras attached to some nilpotent orbits in Lie algebras of types $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.

Gelfand-Kirillov dimension of commutative subalgebras of simple infinite dimensional algebras and their quotient division algebras

For central simple finitely generated algebras of finite Gelfand-Kirillov dimension and for their division algebras upper bounds are obtained for the transcendence degree of their commutative subalgebras and subfields respectively. In the formulae for the upper bounds are used only two dimensions of the algebras: the Gelfand-Kirillov and the filter dimension. In the case of the ring of differential operators on a smooth irreducible affine algebraic variety the upper bounds are the exact upper bounds. A similar upper bound is given for the transcendence degree of isotropic subalgebras of strong…

Gelfand-Kirillov 维数 <4 的 Leavitt 路代数在 Morita 等价意义下的分类

Classification of Leavitt Path Algebras with Gelfand-Kirillov Dimension <4 up to Morita Equivalence

本文利用约化算法与 Serre 子范畴滤过，引入了有向图的加权 Hasse 图作为 Morita 不变量，并给出了 Gelfand-Kirillov 维数小于 4 的 Leavitt 路代数在 Morita 等价意义下的完备分类。

Gelfand-Kirillov Conjecture and Harish-Chandra Modules for Finite W-Algebras

We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classif…

Williams猜想对Gelfand-Kirillov维数为3的图成立

Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

该研究针对Gelfand-Kirillov维数为3的图，给出其强移位等价的数论准则，证明移位等价与强移位等价、Leavitt路代数分次Morita等价、分次K理论保序模同构等价，并得出相关代数与范畴等价的推论，验证Williams猜想成立。

The amenability and non-amenability of skew fields

We investigate the amenability of skew filed extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable skew fields of infinite Gelfand-Kirillov transcendence degree. For example, the free field is non-amenable.

有限Gelfand-Kirillov维数的余交换Hopf代数的Dixmier-Moeglin等价性

The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension

本文证明具有多项式有界增长的诺特余交换Hopf代数满足Dixmier-Moeglin等价性，并指出Lorenz的反例说明该假设不可去除，进而猜想该等价性对所有多项式有界增长的有限生成复诺特Hopf代数成立。

GK维数为一的对称运算代数

Symmetric operads of GK-dimension one

论文证明了不存在GK维数严格介于1与2之间的有限生成对称运算代数，同时分类了GK维数为1的有限生成素对称运算代数。

Leavitt path algebras satisfying a polynomial identity

Leavitt path algebras L of an arbitrary graph E over a field K satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When E is a finite graph, L satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of L being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph E, the Leavitt path algebra L has Gelfand-Kirillov dimension zero if and only if E has no cycles. Likewise, L has Gelfand-Kirillov dimension one if and only if E contains at least one cycle, bu…

A Dichotomy for GK dimensions of simple modules over simple differential rings

The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.

Existence of Hopf subalgebras of GK-dimension two

Let $H$ be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If $H$ is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then $H$ contains a Hopf subalgebra of Gelfand-Kirillov dimension two.

Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_N

It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the noncommutative Noether problem for the symmetric group. It is also shown that the quantum Gelfand-Kirillov conjecture for gl_N (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type D_n. The proof is based on the theory of Galois rings developed by the first author and Ovsienko. From here we obtain …

Algebraic entropy and a complete classification of path algebras over finite graphs by growth

The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand-Kirillov dimension and the entropy. We give a complete classification of path algebras over finite graphs by dimension, Gelfand-Kirillov dimension and algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic …

Gelfand-Kirillov conjecture for symplectic reflection algebras

We construct functorially a class of algebras using the formalism of double derivations. These algebras extend to higher dimensions Crawley-Boevey and Holland's construction of deformed preprojective algebras and encompass symplectic reflection algebras associated to wreath products. We use this construction to show that the quotient field of a symplectic reflection algebra is "rational", confirming a pair of conjectures of Etingof and Ginzburg.

作为一阶公式的盖尔范德-基里洛夫猜想

Gelfand-Kirillov conjecture as a first-order formula

针对特征零代数闭域上半单李代数的盖尔范德-基里洛夫猜想，研究证明其成立性等价于环语言中某一阶语句在$ACF_0$中的可证性，为该猜想提供了模型论刻画。

On the growth of graded polynomial identities of sl_n

Let K be a field of characteristic 0 and L be a G-graded Lie PI-algebra, where G is a finite group. We define the graded Gelfand-Kirillov dimension of L. Then we measure the growth of the Z_n-graded polynomial identities of the Lie algebra of n x n traceless matrices sl_n(K) giving an exact value of its Z_n-graded Gelfand-Kirillov dimension.

Leavitt path algebras of hypergraphs

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute their Gelfand-Kirillov dimension, obtain some results on ring-theoretic properties like simplicity, von Neumann regularity and Noetherianess and investigate their K-theory and graded K-theory. By doing so we obtain new results on the Gelfand-Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of weigh…

Enveloping algebras with just infinite Gelfand-Kirillov dimension

Let $\mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mf g)$ is {\em just infinite} in the sense that any proper quotient of $U(\mf g)$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(\mf g)$ by proper Poisson ideals. In fact, we prove more generally that any cent…

Transcendence Degree of Division Algebras

We define a transcendence degree for division algebras, by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let $k$ be a field, let $A$ be a finitely generated $k$-algebra that is an Ore domain, and let $D$ denote the quotient division algebra of $A$. If $A$ does not satisfy a polynomial identity then the Gelfand-Kirillov dimension of $K$ is at most th…

On finite GK-dimensional Nichols algebras of diagonal type

It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show that a Nichols algebra of affine Cartan type has infinite Gelfand-Kirillov dimension.

Ore extensions of commutative rings and the Dixmier-Moeglin equivalence

We consider Ore extensions of the form $T:=R[x;σ,δ]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.

2-unitary Operads of GK-dimension 3

We study and classify the 2-unitary operads of Gelfand-Kirillov dimension three.

Leavitt path algebras with finitely presented irreducible representations

Let E be an arbitrary graph, K be any field and let L be the corresponding Leavitt path algebra. Necessary and sufficient conditions (which are both algebraic and graphical) are given under which all the irreducible representations of L are finitely presented. In this case, the graph E turns out to be row finite and the cycles in E form an artinian partial ordered set under a defined preorder. When the graph E is finite, the above graphical conditions were shown to be equivalent to the algebra L having finite Gelfand-Kirillov dimension in a paper by Alahmadi, Alsulami, Jain and Zelmanov. Examp…

代数的增长函数及其在Leavitt路径代数中的应用

Growth functions of algebras and an application to Leavitt path algebras

本文研究代数的三种增长函数（Gelfand-Kirillov维数、超维数和熵），精确刻画其适用场景，并应用于Leavitt路径代数，给出其为PI的新判据。

Nil algebras with restricted growth

It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.

Hilbert series of modules of GK-dimension two over elliptic algebras

We characterize the Hilbert functions and minimal resolutions of (critical) Cohen-Macaulay graded right modules of Gelfand-Kirillov dimension two over generic quadratic and cubic three dimensional Artin-Schelter regular algebras.

非对称 operad 的增长

Growth of nonsymmetric operads

该研究证明非对称 operad 的 Gelfand–Kirillov 维数不存在 1 到 2 之间的间隙，并构造了覆盖所有允许维数的一元生成例子，同时反驳了关于生成级数的两个猜想。

Modules over quantum Laurent polynomials I

It is shown that the Gelfand--Kirillov dimension for modules over quantum Laurent polynomials is tensor-minimal. The Brookes--Groves invariant associated with a tensor product of modules is determined. It is also shown that there can be nonholonmic simple modules.

Hopf algebras of GK-dimension two with vanishing Ext-group

We construct and study a family of finitely generated Hopf algebra domains $H$ of Gelfand-Kirillov dimension two such that $\Ext^1_H(k,k)=0$. Consequently, we answer a question of Goodearl and the second-named author.

关于诺特点态霍普夫代数

On Noetherian pointed Hopf algebras

本文针对点态霍普夫代数相关猜想，提出多循环-有限群上后Nichols代数的诺特性等价于仿射性与有限Gelfand-Kirillov维数，并给出部分支持结果及相关分析。

Primitive algebraic algebras of polynomially bounded growth

We show that if $k$ is a countable field, then there exists a finitely generated, infinite-dimensional, primitive algebraic $k$-algebra $A$ whose Gelfand-Kirillov dimension is at most six. In addition to this we construct a two-generated primitive algebraic $k$-algebra. We also pose many open problems.

Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the c…

Affine semiprime algebras of GK dimension one are (still) pi

In this note, we give a new proof of the fact that an affine semiprime algebra R of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound for the pi degree of R.

A note on affine representable algebras

We consider affine representable algebras, that is, finitely generated algebras over a field that can be embedded into some matrix algebra over a commutative algebra. We show that this algebra can in fact be chosen to be a polynomial algebra. We also prove a refined version of a theorem of V.T. Markov stating that the Gelfand-Kirillov dimension of any affine representable algebra is an integer.