Gelfand Kirillov Dimensions Research Articles

Polynomial log-volume growth and the GK-dimensions of twisted homogeneous coordinate rings

Let f be a zero entropy automorphism of a compact Kähler manifold X . We study the polynomial log-volume growth \mathrm{Plov}(f) of f in light of the dynamical filtrations introduced in our previous work with T.-C. Dinh. We obtain new upper bounds and lower bounds of \mathrm{Plov}(f) . As a corollary, we completely determine \mathrm{Plov}(f) when \dim X = 3 , extending a result of Artin–Van den Bergh for surfaces. When X is projective, \mathrm{Plov}(f) + 1 coincides with the Gelfand–Kirillov dimensions \mathrm{GKdim}(X,f) of the twisted homogeneous coordinate rings associated to (X,f) . Reformulating these results for \mathrm{GKdim}(X,f) , we improve Keeler’s bounds of \mathrm{GKdim}(X,f) and provide effective upper bounds of \mathrm{GKdim}(X,f) which only depend on \dim X .

Free bicommutative superalgebras

We introduce the variety Bsup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z2-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativityx(yz)=(−1)x‾y‾y(xz) and (xy)z=(−1)y‾z‾(xz)y, where u‾∈{0,1} is the parity of the homogeneous element u.We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.

Residue representations - The rank one case

With each resonance of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type, one can associate a residue representation. The purpose of this paper is to study them. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We give an algorithm that aims at determining if these representations are irreducible, finding their Langlands parameters, their Gelfand-Kirillov dimensions and wave front sets. As an example, we apply this algorithm to the Laplacian of the p-forms in the cases of all the classical real rank-one Lie groups.

Gelfand–Kirillov Dimensions and Associated Varieties of Highest Weight Modules

Abstract In this paper, we present a uniform formula of Lusztig’s $ \textbf {a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand–Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux associated with Weyl group elements by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish–Chandra modules are explicitly determined, including the exceptional cases.

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

For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ 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 dimensions of simple modules. The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$ and $\Lambda_{\mathfrak q}$ is simple, studied in A. Gupta, $\mathrm{GK}$-dimensions of simple modules over $K[X^{\pm 1}, \sigma]$, Comm. Algebra, 41(7) (2013), 2593-2597, is considered without assuming the simplicity, and it is shown that a dichotomy still holds for the GK dimension of simple modules.

Clover nil restricted Lie algebras of quasi-linear growth

The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra 302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS) 10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras [Formula: see text], where a field of positive characteristic is arbitrary and [Formula: see text] an infinite tuple of positive integers. All these algebras have a nil [Formula: see text]-mapping. We prove that [Formula: see text]. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [Formula: see text]. We construct a subfamily of nil restricted Lie algebras [Formula: see text], with parameters [Formula: see text], [Formula: see text], having extremely slow quasi-linear growth of type: [Formula: see text], as [Formula: see text]. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2) 177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is “almost dying” by having a “quasi-linear” growth as above, for infinitely many [Formula: see text] it has a rather fast intermediate growth of type [Formula: see text], for such periods the algebra is “resuscitating”. The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction.

Finite dimensional Hopf algebras over the Kac-Paljutkin algebra $H_8$.

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method.

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

ABSTRACTIn 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,ℂ)-modules and 𝔰𝔭(2n,ℂ)-modules that appeared in the ℤ-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.

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

ABSTRACTGiven 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.

Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

Abstract Let $(G,K)$ be an irreducible Hermitian symmetric pair of non-compact type with $G={SU}(p,q)$, and let $\lambda$ be an integral weight such that the simple highest weight module $L(\lambda)$ is a Harish-Chandra $({\mathfrak{g}},K)$-module. We give a combinatorial algorithm for the Gelfand–Kirillov (GK) dimension of $L(\lambda)$. This enables us to prove that the GK dimension of $L(\lambda)$ decreases as the integer $\langle{\lambda+\rho},{\beta}^{\vee} \rangle$ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of $L(\lambda)$.

A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

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.

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-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

Representations of the n-Dimensional Quantum Torus

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 whenwhere 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 orwhere 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.

Leavitt path algebras with finitely presented irreducible representations

Let E be an arbitrary graph, K be any field and let L=LK(E) be the corresponding Leavitt path algebra. Necessary and sufficient conditions (both graphical and algebraic) 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 relation ≥. When the graph is E is finite, the above graphical conditions were shown in [6] to be equivalent to LK(E) having finite Gelfand–Kirillov dimension. Examples are constructed showing that this equivalence no longer holds for infinite graphs and a complete description is obtained of Leavitt path algebras over arbitrary graphs having finite Gelfand–Kirillov dimensions. The “building blocks” for these algebras seem to be von Neumann rings and the Laurent polynomial ring K[x,x−1].

Cohomological comparison theorem

If f is an idempotent in a ring Λ, then we find sufficient conditions which imply that the cohomology rings ⊕n≥0ExtΛn(Λ/r,Λ/r) and ⊕n≥0ExtfΛfn(fΛf/frf,fΛf/frf) are eventually isomorphic. This result allows us to compare finite generation and Gelfand–Kirillov dimensions of the cohomology rings of Λ and fΛf. We are also able to compare the global dimensions of Λ and fΛf.

Gelfand-Kirillov dimensions of the ℤ2-graded oscillator representations of $$\mathfrak{s}\mathfrak{l}$$ (n)

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.

GROWTH OF REES QUOTIENTS OF FREE INVERSE SEMIGROUPS DEFINED BY SMALL NUMBERS OF RELATORS

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.

Dimensions of quantum determinantal rings

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.

On growth of Lie algebras, generalized partitions, and analytic functions

In this paper we discuss some recent results on two different types of growth of Lie algebras that lead to some combinatorial problems. First, we study the growth of finitely generated Lie algebras (Sections 1–4). This problem leads to a study of generalized partitions. Recently the author has suggested a series of q-dimensions of algebras Dim q, q∈ N which includes, as first terms, dimensions of vector spaces, Gelfand–Kirillov dimensions, and superdimensions. These dimensions enabled us to describe the change of a growth in transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper we give some results on asymptotics for those generalized partitions. As a main application, we obtain an asymptotical result for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on growth of complex functions which are analytic in the unit circle. By growth of such functions we mean their growth when the variable tends to 1. Also we discuss for all levels q=2,3,… what numbers α>0 can be a q-dimension of some Lie (associative) algebra. Second, we discuss a ‘codimension growth’ for varieties of Lie algebras (Sections 5 and 6). It is useful to consider some exponential generating functions called complexity functions. Those functions are entire functions of a complex variable provided the varieties of Lie algebras are nontrivial. We compute the complexity functions for some varieties. The growth of a complexity function for an arbitrary polynilpotent variety is evaluated. Here we need to study the connection between the growth of a fast increasing entire function and the behavior of its Taylor coefficients. As a result we obtain a result for the asymptotics of the codimension growth of a polynilpotent variety of Lie algebras. Also we obtain an upper bound for a growth of an arbitrary nontrivial variety of Lie algebras.

Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

We study rational actions of a linear algebraic groupGon an algebraV, and the induced actions on Rat(V), the spectrum of rational ideals ofV(a subset of Spec(V) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions onV. In particular, we relate properties of a rational idealJand itsorb, the ideal (J:G)=⋂γ∈Gγ(J). The rational ideals ofVcontaining the orb ofJare precisely those in the Zariski-closureXof the orbit ofJin Rat(V). TheG-stratumofJconsists of all rational ideals inXwhose orbit is dense inX(i.e., whose orb is equal to the orb ofJ). We show that theG-stratum of a rational ideal consists of exactly oneG-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit ofJis open in its closure in Rat(V), provided thatJis locally closed. Among other results, we show that the semiprime ideal (J:G) is Goldie, and we relate the uniform and Gelfand–Kirillov dimensions ofV/JandV/(J:G).