Related papers: Group algebras whose units satisfy a Laurent Polyn…

In this article, we extend some results about algebra $A$ with the group of units $U(A)$ having a special polynomial identity, Laurent polynomial. And we present a new version of B. Hartley Conjecture with these identities.

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields…

In this article we shows some results about algebra with the group of units having special polynomial identity.

Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ over the infinite field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$, and let $\circledast:\mathbb{F}G\rightarrow \mathbb{F}G$ denote the involution defined by…

An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically…

Let K be a field of positive characteristic p, let R be either a group algebra K[G] or a restricted enveloping algebra u(L), and let I be the augmentation ideal of R. We first characterize those R for which I satisfies a polynomial identity…

Let $U(KG)$ be the group of units of the group ring $KG$ of the group $G$ over a commutative ring $K$. The anti-automorphism $g\mapsto g\m1$ of $G$ can be extended linearly to an anti-automorphism $a\mapsto a^*$ of $KG$. Let $S_*(KG)=\{x\in…

Let K be a field of characteristic 2 and G a nonabelian locally finite 2-group. Let V(KG) be the group of units with augmentation 1 in the group algebra KG. An explicit list of groups is given, and it is proved that all involutions in V(KG)…

Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that, if $C$ is a finite subset…

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring $\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$).…

In this article, we show that if $KG$ is Lie nilpotent group algebra of a group $G$ over a field $K$ of characteristic $p>0$, then $t_{L}(KG)=k$ if and only if $t^{L}(KG)=k$, for $k\in\{5p-3, 6p-4\}$, where $t_{L}(KG)$ and $t^{L}(KG)$ are…

We prove that the group algebra $KG$ of a group $G$ over a field $K$ is primitive, provided that $G$ has a free subgroup with the same cardinality as $G$, and that $G$ satisfies the following condition $(\ast)$: for each subset $M$ of $G$…

We show that an arithmetic lattice $\Gamma$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $\delta(v)$ where $v = \mu (G/\Gamma)$ is the co-volume of the lattice. We prove that $\delta$ is polynomial in general and…

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group…

We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.

Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let $U(KG)$ denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras $KG$, where the size of $KG$…

A group pair $(G, X)$ consists of a group $G$ together with a $G$-set $X$. Such a pair encodes properties of $G$ relative to the stabilisers of points in $X$. In this paper, we show how to combine properties of group pairs and their…

We prove that a finitely generated Lie algebra $L$ such that (i) every commutator in generators is ad-nilpotent, and (ii) $ L$ satisfies a polynomial identity, is nilpotent. As a corollary we get that a finitely generated residually-$p$…

Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is…

Let $G$ be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological dimension…