环与代数

In this paper, we introduce the notion of an $\mathbb{N}^p$-graded birack and construct its isotope. Every involutive $\mathbb{N}^p$-graded birack gives rise to an $\mathbb{N}^p$-graded Yang-Baxter algebra. We study the relation between…

Let $M$ be a finitely generated skew field over a ground field $k$, and let $G$ be a finite group of $k$-linear automorphisms of $M$. This paper investigates finite generation of the skew subfield $M^G$ of $G$-invariants in $M$, and…

In this paper, we consider the relatively free algebra of rank $n$, $F_n(\mathfrak{N}_p)$, in the variety of Lie nilpotent associative algebras of index $p$, denoted by $\mathfrak{N}_p$, over a field of characteristic zero. We describe an…

We introduce a broader class of nonassociative Ore extensions that unifies and generalizes several earlier constructions. We prove generalizations of Hilbert's Basis Theorem for this class, showing that they arise immediately from the…

We thoroughly explore the class of k-step nilpotent Lie algebras associated with a simple graph looking for k-step nilpotent Lie algebras which are rigid in the variety of at most k-step nilpotent Lie algebras. We find out that, besides the…

We investigate the transposed Poisson structures on both the $q$-analog Virasoro-like algebra and $q$-quantum torus Lie algebra considering the cases where $q$ is generic and where $q$ is a primitive root of unity, respectively. We…

The problem of determining whether a diagonally dominant matrix is singular or nonsingular is a classical topic in matrix theory. This paper develops necessary and sufficient conditions for the singularity or nonsingularity of diagonally…

In this paper, we first study Rota-Baxter Hopf algebras of weight $-1$ and construct a matched pair of Hopf algebras on every Rota-Baxter Hopf algebra of weight $-1$. Then we propose the notion of projection homomorphism pairs on a matched…

We develop the geometric and homological framework for non-commutative $n$-ary $\Gamma$-semirings by constructing a sheaf and derived theory over their non-commutative $\Gamma$-spectrum. Starting with a non-commutative $n$-ary…

We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$, $l \in \{1,...,p\}$, where $A_l$ and $B_l$ are…

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincar\'e series of the invariants and concomitants of the general linear Lie supergroup…

In this short note we count the finite semirings up to isomorphism, and up to isomorphism and anti-isomorphism for some small values of $n$; for which we utilise the existing library of small semigroups in the GAP package Smallsemi.

For von Neumann *-regular rings R of endomorphisms (the involution given by taking adjoints) of inner product spaces we provide a condition on r in R (in terms of action of r on finite dimensional subspaces) for r being a unit. It remains…

Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the…

We give an alternative description of the grammic monoid in terms of weakly increasing subsequences. Specifically, we show that words $u,v$ in the generators $\{1,\ldots, n\}$ determine the same element of the grammic monoid of rank $n$ if…

We construct a family of connected Hopf algebras with finite Gelfand-Kirillov dimension, none of which is an iterated Hopf Ore extension of the universal enveloping algebra of its primitive part. This provides a negative answer to a…

This paper develops the homological backbone of the theory of non-commutative $n$-ary $\Gamma$-semirings. Starting from an $n$-ary $\Gamma$-semiring $(T,+,\tilde{\mu})$ and its $\Gamma$-ideals, we work in the slot-sensitive categories of…

Recently, the concept of Aut-stable subspaces has played an important role in the characterization of polynomial rings, a topic that remains a challenging problem in algebraic geometry (see [8]). It turns out that polynomial rings with more…

In this article, we prove that in a PI-ring (or polynomial identity ring) $S$, for an element $A \in \mathbb{M}_m(S)$ if $A^n= A^{n+1}X$ for some $n \in \mathbb{N}$ and $X \in \mathbb{M}_m(S)$, then there exists an element $Y\in…

In this paper, we introduce the concepts of crossed modules of post-Lie algebras and cat$^1$-post-Lie algebras. It is proved that these two concepts are equivalent to each other. Secondly, we construct a non-abelian cohomology for post-Lie…