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 a new proof of the quantum Gelfand-Kirillov conjecture for sl_N, thus recovering the result of Fauquant-Millet. Moreover, we provide an explicit description of skew fields of fractions for quantized gl_N and sl_N generalizing Alev and Dumas.
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.
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 $n$.
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.
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).
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 $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.
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…
We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p>2, and we compute its Z_2-graded Gelfand-Kirillov dimension as a Z_2-graded PI-algebra.
The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfa…
We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Groebner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Groebner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer.
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.
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.
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.
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.
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 $K$.
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…
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.
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…
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.
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.
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…
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.
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 …
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…
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…
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.
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 …
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.
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 …
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.
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.
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…
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.
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.
CIEM-CONICET；Facultad de Matemática, Física, Astronomía y Computación, Universidad Nacional de Córdoba（科尔多瓦国立大学数学、物理、天文学和计算学院）；Department of Mathematics and Data Science, Vrije Universiteit Brussel（布鲁塞尔自由大学数学与数据科学系）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳国际数学中心）；Philipps-Universität Marburg, FB Mathematik und Informatik（马尔堡菲利普斯大学数学与信息学系）
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 …
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.
We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in arXiv:1512.09271.
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 $4$.
A short proof is given to Dixmier's 6'th problem for the Weyl algebra (and other algebras of Gelfand-Kirillov dimension which is less than 3 like rings of differential operators on smooth irreducible algebraic curves).
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.
We solve the noncommutative Noether's problem for the reflection groups by showing that the skew field of the invariants of the Weyl algebra under the action of any reection group is a Weyl field, that is isomorphic to a skew field of some Weyl algebra over a transcendental extension of the ground field. We also extend this result to the invariants of the ring of differential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.
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.
Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand-Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac-Moody algebras.
We investigate homological and ring-theoretic properties of universal quantum linear groups that coact on Artin-Schelter regular algebras A(n) of global dimension 2, especially with central homological codeterminant (or central quantum determinant). As classified by Zhang, the algebras A(n) are connected \N-graded algebras that are finitely generated by n indeterminants of degree 1, subject to one quadratic relation. In the case when the homological codeterminant of the coaction is trivial, we show that the quantum group of interest, defined independently by Manin and by Dubois-Violette and La…
We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.
The m x n quantum grassmannian, G_q(m,n), is the subalgebra of the algebra of m x n quantum matrices that is generated by the maximal m x m quantum minors. Several properties of G_q(m,n) are established. In particular, a basis of G_q(m,n) is obtained, and it is shown that G_q(m,n) is a noetherian domain of Gelfand-Kirillov dimension m(n-m)+1. The algebra G_q(m,n) is identified as the subalgebra of coinvariants of a natural left coaction of the m x m quantum special linear group on the algebra of m x n quantum matrices and it is shown that G_q(m,n) is a maximal order.
