### A free associative algebra as a free module over a Specht subalgebra.

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We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring ${A}_{0}$ and of the Weyl algebra ${A}_{1}$, both over a field of characteristic zero, by establishing the same type of results for the family of algebras $${A}_{h}=\langle x,y\mid yx-xy=h\left(x\right)\rangle \phantom{\rule{0.166667em}{0ex}},$$ where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $\mathbb{F}$ of prime characteristic and study $\mathbb{F}\left[t\right]$comodule algebra structures on ${A}_{h}$. We also compute the Makar-Limanov invariant of absolute constants...

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra $s{l}_{2}$, the Verma module over a Kac-Moody algebra, the Verma module...

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

To a commutative ring K, and a family of K-algebras indexed by the vertex set of a graph, we associate a K-algebra obtained by a mixture of coproduct and tensor product constructions. For this, and related constructions, we give exact sequences and deduce homological properties.

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.