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In this article, we prove that every ultragraph Leavitt path algebra is a direct limit of Leavitt path algebras of finite graphs and determine the Gelfand-Kirillov dimension of an ultragraph Leavitt path algebra. We also characterize ultragraph Leavitt path algebras whose simple modules are finitely presented, and show that these algebras have finite Gelfand-Kirillov dimension. Moreover, we construct new classes of simple modules over ultragraph Leavitt path algebras associated with minimal infinite emitters and minimal sinks, which have not appeared in the context of Leavitt path algebras of graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Nichols algebra B (W X , r) associated to any non-degenerate involutive solution (X, r) of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension nm with m, n ∈ Z ≥ 2 . It turns out that the Nichols algebra B (W X , r) has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle. [ABSTRACT FROM AUTHOR]

We study the Frobenius skew polynomial ring over a polynomial ring in finitely many variables and show that growth function associated to the variables and the Frobenius satisfies a linear recurrence. We introduce the concept of Gelfand-Kirillov base for a finitely generated standard graded algebra over the prime field of characteristic p, where p is prime, and show its existence for the polynomial ring. [ABSTRACT FROM AUTHOR]

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 Gröbner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Gröbner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer. [ABSTRACT FROM AUTHOR]

The Gelfand-Kirillov dimension measures the asymptotic rate of growth of algebras. For every associative dialgebra D , the quotient A D := D / I d (S) , where I d (S) is the ideal of D generated by the set S := { x ⊢ y − x ⊣ y ∣ x , y ∈ D } , is called the associative algebra associated to D . We show that G K d i m (D) ≤ 2 G K d i m (A D). Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2. [ABSTRACT FROM AUTHOR]

Tauvel's height formula, which provides a link between the height of a prime ideal and the Gelfand–Kirillov dimension of the corresponding factor algebra, is verified for quantum nilpotent algebras. [ABSTRACT FROM AUTHOR]

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements. [ABSTRACT FROM AUTHOR]

Given 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. [ABSTRACT FROM AUTHOR]

In this paper, we give a method to compute the Gelfand-Kirillov dimensions of some polynomial-type weight modules. These modules are infinite-dimensional irreducible 픬(n,ℂ)-modules and 픭(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. [ABSTRACT FROM AUTHOR]

Let K be a field, and a finitely generated K-algebra with the PBW K-basis . It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( =  the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset with , where V(U) = K-span. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( =  the number of generators of A), thereby L has the elimination property. [ABSTRACT FROM AUTHOR]

Then-dimensional quantum torus 𝒪q((F×)n) is defined as the associativeF-algebra generated byx1,…,xntogether with their inverses satisfying the relationsxixj = 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, ifMis a simple 𝒪q((F×)n)-module, then 𝒢𝒦-dim(M) = 1 orwhere 𝒵(C) stands for the center of an algebraC. We also show that there always exists a simpleF*A-module satisfying the above inequality. [ABSTRACT FROM PUBLISHER]

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. 2320–2337] the following conjecture: (BC) Let A n be the Weyl algebra over a field K of characteristic p > 0. Then each K-algebra homomorphism is a monomorphism. The purpose of this article is to prove an analogue of BC for symplectic Poisson algebras, to check BC for A 1 and to show that BC is wrong for A n when n > 1. [ABSTRACT FROM PUBLISHER]

Let A have a locally finite and multiparameter indexed filtration F, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from F. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand-Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras. [ABSTRACT FROM AUTHOR]

We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology. [ABSTRACT FROM AUTHOR]

This paper gives a classification of the prime ideals, primitive ideals, and irreducible representations of where K is an algebraically closed field, n ≥ 3, and β ij &Element; K . The classification of the prime ideals of S proves that, under the Zariski topology, the topological dimension of spec S is no greater than the Gelfand-Kirillov dimension of S . Included is an appendix by A. Berliner, in which it is shown that the finite dimensional irreducible representations are all one-dimensional. [ABSTRACT FROM AUTHOR]

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponent. Recently, the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F (A q , k ), the latter being relatively free algebras with k generators and fixed solubility length q . Later, an application of generating functions allowed us to obtain a more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras. Our goal is to compute the growth for F (A q , m , k ), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is proved in generality of free polynilpotent Lie superalgebras. We study the growth for universal enveloping algebras of Lie superalgebras as well. Also, we study bases for free Lie superalgebras. [ABSTRACT FROM AUTHOR]

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand-Kirillov dimension equal to the Gelfand-Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand-Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand-Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the GelfandKirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so. [ABSTRACT FROM AUTHOR]