International Journal of Algebra and Computation

We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type $A_{\ell }$ and upper estimates of the width of principal congruence subgroups with respect to Tits–Vaserstein generators.

Let $n\ge 1$ be an integer. We introduce the notions of $n$-FP-gr-injective and $n$-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of $n$-gr-coherent rings in terms of $n$-FP-gr-injective and $n$-gr-flat modules. Moreover, we prove that the pairs (gr-${{ℱ}{ℐ}}_n$, gr-${{ℱ}}_n$) and (gr-${{ℱ}}_n$, gr-${{ℱ}{ℐ}}_n$) are duality pairs over left $n$-coherent rings, where gr-${{ℱ}{ℐ}}_n$ and gr-${{ℱ}}_n$ denote the subcategories of $n$-FP-gr-injective left $R$-modules and $n$-gr-flat right $R$-modules respectively. As applications, we obtain that any graded left (respectively, right) $R$-module admits an $n$-FP-gr-injective (respectively, $n$-gr-flat) cover and preenvelope.

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labeled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to ${coRNC}^2$, which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in ${coRNC}^2$; this result is even new in the one-dimensional case, where the straight-line programs produce words. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to ${coRNC}^2$. Using the Magnus embedding, it follows that the compressed word problem for a free metabelian group belongs to ${coRNC}^2$.

In 2011, Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper, he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal’cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal’cev term that have co-NP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth’s question is negative in general (assuming P$\ne$NP).

The automorphism group of a partially commutative group ${𝔾}_{\mathrm{\Gamma }}$ is generated by four types of automorphism: graph automorphisms, inversions, transvections and vertex conjugating automorphisms. We find a characterisation of the subgroup $\mathrm{\text{S}}t({𝒦})$ generated by inversions and transvections, in terms of stabilisers of subgroups of ${𝔾}_{\mathrm{\Gamma }}$ generated by the so-called “admissible” subsets of $V(\mathrm{\Gamma })$. This is used to give a decomposition of $\mathrm{\text{S}}t({𝒦})$ as a chain of semi-direct products of (mostly) tractable and recognisable subgroups; which in turn gives rise to a presentation of $\mathrm{\text{S}}t({𝒦})$.

We construct a torsion-free arithmetic lattice in ${\mathrm{\text{PGL}}}_2({𝔽}_2((t)))\times {\mathrm{\text{PGL}}}_2({𝔽}_2((t)))$ arising from a quaternion algebra over ${𝔽}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over ${𝔽}_2((t))$ by means of non-archimedean uniformization.

Let $L$ be a relatively free Lie algebra of finite rank $n$, with $n\ge 2$, $\widehat{L}$ be the completion of $L$ with respect to the topology defined by the lower central series $L={\gamma }_1(L)\supseteq {\gamma }_2(L)\supseteq \cdots$ of $L$ and $L_{(c)}=L/{\gamma }_c(L)$, with $c\ge 2$. We prove that, with respect to the formal power series topology, the automorphism group $\mathrm{\text{Aut}}(L)$ of $L$ is dense in the automorphism group $\mathrm{\text{Aut}}(\widehat{L})$ of $\widehat{L}$ if and only if $L$ is nilpotent. Furthermore, we show that there exists a dense subgroup of $\mathrm{\text{Aut}}(\widehat{L})$ generated by ${\mathrm{\text{GL}}}_n(K)$ and a finite set of IA-automorphisms if and only if $\mathrm{\text{Aut}}(L_{(c)})$ is generated by ${\mathrm{\text{GL}}}_n(K)$ and a finite set of IA-automorphisms independent upon $c$ for all $c$. We apply our study to several varieties of Lie algebras.

The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions, and we show that these bounds are often robust against the change of the basic operations of the structure.

We consider the quantum analog of the Lie commutator ${[x,y]}_q=xy-qyx$ for an invertible element $q$ of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form $[\dots [{[x_1,x_2]}_q,x_3{]}_q,\dots ,x_{s+1}{]}_q=0$.