scholarly journals Growth of étale groupoids and simple algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 375-397 ◽  
Author(s):  
Volodymyr Nekrashevych
Keyword(s):  
Quadratic Growth ◽  
Finitely Generated ◽  
Dimension Formula ◽  
Convolution Algebras ◽  
Simple Algebras ◽  
Kirillov Dimension

We study growth and complexity of étale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand–Kirillov dimension [Formula: see text] and simple finitely generated algebras of quadratic growth over arbitrary fields.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

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.

scholarly journals Standard prime ideals and lying over for finite extensions of Noetherian algebras

1995 ◽  
Vol 37 (3) ◽  
pp. 311-326
Author(s):  
Günter Krause
Keyword(s):  
Prime Factor ◽  
Finite Sequence ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Associated Prime ◽  
Kirillov Dimension

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂…⊂ Ni−1 ⊂ Ni ⊂.… ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.

Gelfand-Kirillov Dimension is Exact for Noetherian PI Algebras

1984 ◽  
Vol 27 (2) ◽  
pp. 247-250 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Finitely Generated ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

AbstractIf O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

scholarly journals Gröbner-Shirshov Bases Theory for Trialgebras

2021 ◽  
Author(s):  
Juwei Huang ◽  
Yuqun Chen
Keyword(s):  
Normal Forms ◽  
Finitely Generated ◽  
Kirillov Dimension

We establish a Gröbner-Shirshov bases theory for trialgebras and show that every ideal of a free trialgebra has a unique reduced Gröbner-Shirshov basis. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand-Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated respectively.

Cofiniteness of Local Cohomology Modules

2014 ◽  
Vol 21 (04) ◽  
pp. 605-614 ◽  
Author(s):  
Kamal Bahmanpour ◽  
Reza Naghipour ◽  
Monireh Sedghi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Partial Answer ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Local Ring ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and [Formula: see text], then [Formula: see text] is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then [Formula: see text] is finitely generated if and only if 0 ≤ n ∉ W, where [Formula: see text]. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then [Formula: see text] is J-cominimax, and [Formula: see text] is finitely generated (resp., minimax) if and only if [Formula: see text] is finitely generated for all [Formula: see text] (resp., [Formula: see text]). Moreover, the concept of the J-cofiniteness dimension [Formula: see text] of M relative to I is introduced, and we explore an interrelation between [Formula: see text] and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then [Formula: see text].

Finitely generated simple algebras: A question of B. I. Plotkin

2004 ◽  
Vol 143 (1) ◽  
pp. 341-359 ◽  
Author(s):  
A. I. Lichtman ◽  
D. S. Passman
Keyword(s):  
Finitely Generated ◽  
Simple Algebras

GROWTH OF FINITELY GENERATED POLYNILPOTENT LIE ALGEBRAS AND GROUPS, GENERALIZED PARTITIONS, AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE

1999 ◽  
Vol 09 (02) ◽  
pp. 179-212 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Unit Circle ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Precise Asymptotics ◽  
Finitely Generated ◽  
Asymptotic Growth ◽  
Enveloping Algebra ◽  
Finitely Generated Groups ◽  
Kirillov Dimension

Recently, the author has suggested a series of dimensions of algebras which includes as first terms dimension of a vector space, Gelfand–Kirillov dimension, and superdimension. These dimensions enabled us to describe the change of the growth in the transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper, we obtain more precise asymptotics for generalized partitions. As a main application, we obtain more precise asymptotics for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of the lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on the growth of functions analytic in the unit circle. By the growth of such functions, we mean their growth when the variable tends to 1. Finally, we study two kinds of p-central series for free polynilpotent finitely generated groups. We obtain asymptotics for the ranks of these series, in one case we have an example of a polynomial, but not rational growth.

scholarly journals Gröbner–Shirshov Bases Theory for Trialgebras

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1207
Author(s):  
Juwei Huang ◽  
Yuqun Chen
Keyword(s):  
Normal Forms ◽  
Finitely Generated ◽  
Kirillov Dimension

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.

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