Narrow Results

Let be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from .

This includes the following classical structures in algebra:

• Boolean algebras.

• Varieties of rings generated by finitely many finite fields.

• Varieties of Heyting algebras generated by an n–element chain.

We classify nilpotent associative algebras of dimensions up to $4$ over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

In this paper, we introduce and study differential graded (DG) down–up algebras. In brief, a DG down–up algebra ${𝒜}$ is a connected cochain DG algebra such that its underlying graded algebra ${{𝒜}}^{\#}$ is a graded down–up algebra. We describe all possible differential structures on Noetherian DG down–up algebras. For those Noetherian DG down-up algebras with nonzero differential, we compute their DG automorphism groups; study their isomorphism problems; and show that they are all Calabi–Yau DG algebras.

We study a family of “symmetric” multiparameter quantized Weyl algebras ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ and some related algebras. We compute the Nakayama automorphism of ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$, give a necessary and sufficient condition for ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ to be Calabi-Yau, and prove that ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ is cancellative. We study the automorphisms and isomorphism problem for ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ and ${\mathcal{A}}_n^{\overline{q},\text{Λ}}\left(\mathbb{K}\right[t\left]\right)$. Similar results are established for the Maltsiniotis multiparameter quantized Weyl algebra ${\mathcal{A}}_n^{\overline{q},\text{Γ}}\left(\mathbb{K}\right)$ and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization ${\left({\mathcal{A}}_n^{\overline{q},\text{Λ}}\right(\mathbb{K}\left)\right)}_{Ƶ}$ and determine its automorphism group.

In 1985 Kevin Walker in his study of topology of polygon spaces [15] raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R³. We also prove that for planar polygon spaces the conjecture holds in several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to Gubeladze.

Let $G_2$ be a group acting on an abelian group $G_1$ via a homomorphism $\alpha \in \mathrm{\text{Hom}}(G_2;\mathrm{\text{Aut}}(G_1))$ and let a 2-cocycle $𝜀\in Z^2(G_2,G_1,\alpha )$. By Schreier’s theorem, the pair $(\alpha ,𝜀)$ determines a group $G_{(\alpha ,𝜀)}$ which can arise as a non-split extension of $G_1$ by $G_2$, denoted by $G_1\underset{(\alpha ,𝜀)}{\times }{G}_2$ and called the perturbed semidirect product of $G_1$ by $G_2$ under $(\alpha ,𝜀)$. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.

The object of this note is to give a concise proof of the following theorem: Let G = B(m,n) = 〈a, b; a^-1b^ma = bⁿ|m ≠ 0, n ≠ 0, m, n ∈ ℤ〉 and H = B(m′,n′) = 〈x,y; x^-1y^m′ x = y^n′|m′ ≠ 0, n′ ≠ 0, m′, n′ ∈ ℤ〉. Then G ≅ H if and only if m = m′ and n = n′.