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The Gelfand–Kirillov dimension has gained importance since its introduction as a 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. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.

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 Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

In this article, by using the method of calculating the Gelfand-Kirillov dimension given in [1] and the Groebner-Shirshov basis for quantized enveloping algebra of $U_{q}(B_{2})$ given in [2], we study the problem of computing the Gelfand-Kirillov dimension of quantized enveloping algebra of $U_{q}(B_{2})$. The main conclusion we get in this paper is that the Gelfand-Kirillov dimension of quantized enveloping algebra of $U_{q}(B_{2})$ is 10. We hope this result will provide some ideas to calculate the Gelfand-Kirillov dimension of quantized enveloping algebra of $U_{q}(B_{n})$.

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won., Comment: 39 pages

The Gelfand–Kirillov dimension of Hecke–Kiselman algebras defined by oriented graphs is studied. It is shown that the dimension is infinite if and only if the underlying graph contains two cycles connected by an (oriented) path. Moreover, in this case, the Hecke–Kiselman monoid contains a free noncommutative submonoid. The dimension is finite if and only if the monoid algebra satisfies a polynomial identity.

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: Gr\"obner-Shirshov basis method. We develop the Gr\"obner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gr\"obner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras., Comment: 20 pages

The n-dimensional quantum torus 𝒪 q((F×)n) is defined as the associative F-algebra generated by x1,…, xn together with their inverses satisfying the relations xixj = qijxjxi, where q = (qij). We show that the modules that are finitely generated over certain commutative sub-algebras ℬ are ℬ-torsion-free and have finite length. We determine the Gelfand–Kirillov dimensions of simple modules in the case when where K.dim stands for the Krull dimension. In this case, if M is a simple 𝒪 q((F×)n)-module, then 𝒢𝒦-dim(M) = 1 or where 𝒵(C) stands for the center of an algebra C. We also show that there always exists a simple F*A-module satisfying the above inequality.

We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely generated algebras of quadratic growth over arbitrary fields., Comment: 19 pages

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. Examples are constructed showing that this equivalence no longer holds if the graph is infinite and a complete description is obtained of Leavitt path algebras over arbitrary graphs having finite Gelfand-Kirillov dimension, Comment: 23 pages

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 $n$., Comment: 21 pages

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible $$\mathfrak{s}\mathfrak{l}$$ (n, $$\mathbb{F}$$ )-modules that appeared in the ℤ2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.

Let R be a finitely generated algebra over a field of characteristic 0 with a locally nilpotent derivation δ ≠ 0. We show that if {ie313-1}, where the invariants {ie313-2} are prime and satisfy a polynomial identity, then {ie313-3}. Furthermore, when R is a domain, the same conclusion holds without the assumption that R is finitely generated. This enables us to obtain a result on skew polynomial rings. These results extend work of Bell and Smoktunowicz on domains with GK dimension in the interval [2, 3).

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

During the last decade, a great deal of activity has been devoted to the calculation of the Hilbert-Poincare series of unitary highest weight representations and related modules in algebraic geometry. However, uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring (in the dual pair setting) and is inspired by Wang’s formula for the dimension of a minimal nilpotent orbit.

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

We survey some recent results on graded Gelfand-Kirillov dimension of PI-algebras over a field F of characteristic 0. In particular, we focus on verbally prime algebras with the grading inherited by that of Vasilovsky and upper triangular matrices, i.e., UTn(F), UTn(E) and UTa,b(E), where E is the infinite dimensional Grassmann algebra., 2010 Mathematics Subject Classification: 16R10, 16W55, 15A75.

We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its ℤ2-graded Gelfand–Kirillov (GK) dimension as a ℤ2-graded PI-algebra.

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8]., Comment: arXiv admin note: text overlap with arXiv:1207.6503

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., Comment: 12 pages

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra $\mathcal{B}$ of a Yetter-Drinfeld module $V$ on which a Lie algebra $\mathfrak g$ acts by biderivations. Specializing to Nichols algebras of diagonal type, we find unrolled versions of the small, the De Concini-Procesi and the Lusztig divided power quantum group, respectively., 11 pages

We determine the Gelfand–Kirillov dimension of a weighted Leavitt path algebra [Formula: see text] where [Formula: see text] is a field and [Formula: see text] a row-finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over [Formula: see text] is isomorphic to a finite product of matrix rings over [Formula: see text].

In this paper, the growth and Gelfand-Kirillov dimension of some primitive abundant semigroups are investigated. It is shown that for certain primitive abundant (regular) semigroup S, S as well as the semigroup algebra K [S] has polynomial growth if and only if all of its cancellative submonoids (subgroups) T as well as K[T] have polynomial growth. As applications, it is shown that if S is a finitely generated primitive inverse monoid having the permutational property, then clK dim K[S] = GK dim K[S] = rk(S).

We show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

We study the asymptotic behavior of a finitely presented Rees quotient S = Inv 〈A|ci = 0(i = 1, …, k)〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterized, yielding a sequence of two-generated three relator semigroups whose Gelfand–Kirillov dimensions form an infinite set, namely {4, 5, 6, …}. The results are applied to give a best possible lower bound, in terms of the size of the generating set, on the number of relators required to guarantee polynomial growth of a finitely presented Rees quotient, assuming no generator is nilpotent. A natural operator is introduced, from the class of all finitely presented inverse semigroups to the class of finitely presented Rees quotients of free inverse semigroups, and applied to deduce information about inverse semigroup presentations with one or many relations. It follows quickly from Magnus' Freiheitssatz for one relator groups that every inverse semigroup Π = Inv 〈a1, …, an|C = D 〉 has exponential growth if n > 2. It is shown that the growth of Π is also exponential if n = 2 and the Munn trees of both defining words C and D contain more than one edge.

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., Changed the title and corrected some typos

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

Bavula states [V. Bavula, The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic, J. Pure Appl. Algebra 212 (2008), pp. 2320–233...

For an arbitrary countable field, we construct an associative algebra that is graded, generated by two 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. The Jacobson radical is very important for the study of noncommutative algebras. For a given ring R one usually studies the Jacobson radical J(R) of R, and the semiprimitive part R/J(R). As related evidence of a connection between these notions, a result of Amitsur says that the Jacobson radical of a finitely generated algebra over an uncountable field is nil, and it is known that all nil rings are Jacobson radical. It is important to know when Jacobson radical are nil because nil rings have interesting properties. For example subalgebras of nil algebras are nil, which does not hold in general for Jacobson radical rings. The Jacobson radical is important for determining the structure of rings and is a generalization of the Wedderburn radical for finitely dimensional algebras.

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., 20 pages