环与代数

We present a Hopf algebraic generalization of the Abe-Kanno theorem on a correspondence between subgroups of an algebraic group and invariant subfields of the field of rational functions. It applies to residually finite-dimensional Hopf…

We present recursive constructions of several families of generalized Cartan matrices associated with Kac-Moody algebras, whose sizes and coranks grow exponentially. The constructions are encoded by connected multigraphs and by…

The question of whether a ring is additively generated by its units has been studied from several perspectives in ring theory and algebraic graph theory. In this paper, we investigate this problem for finite rings, not necessarily…

In this paper, we study local (anti-)superderivations on finite-dimensional nilpotent Lie superalgebras. Firstly, we prove that every finite-dimensional 2-step nilpotent Lie superalgebra over a field $\mathbb{F}$ with…

In this paper, we establish degree obstructions to the equivalence of generalized Airy operators of the same type. As an application, we answer a question posed by Nicholas M. Katz in Inventiones Mathematicae (87, pp. 13-61,1987). The main…

In this paper, we develop the theory of nilpotency and solvability for transposed Novikov-Poisson algebras. We first establish several equivalent conditions for a dialgebra to be nilpotent, and show that the lower central series of a…

We study a family of Hopf algebras arising as liftings of the Jordan plane over the infinite cyclic group. We determine their centres, prime and primitive spectra, and automorphism groups. We show that every prime ideal is completely prime…

M.V. Zaicev and S.K. Segal, as well as S. D\u{a}sc\u{a}lescu, B. Ion, C. N\u{a}st\u{a}sescu, and D. Raios Montes studied certain gradings on matrix rings and algebras - 'elementary' gradings. However, examples of gradings on a matrix ring…

We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity,…

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a…

We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal…

We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z = x(yz) = y(zx)$. This cohomology, denoted $H^\bullet_{\mathrm{cyc}}(A, M)$, is a subtheory of…

Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity…

We extend the construction of Steinberg algebras of ample groupoids to \'etale semicategories. We also relate ample semicategories to Boolean restriction semigroups via a representation result extending previously known results for…

Axial algebras are non-associative algebras generated by idempotents, called axes, whose adjoint action satisfies a fusion law. When this fusion law is graded, axes naturally lead to automorphisms of the algebra, and so such axial algebras…

In this paper, we study pure-projective tilting modules and related classes of rings. We introduce the notion of a pure-tilting hereditary ring, namely, a ring over which every ideal is pure-projective tilting, and investigate its…

In this article we study the polynomial identity (PI) property of skew PBW extensions. We show that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows us to determine, from the known…

In this paper, we develop a unified approach for various operators on Lie conformal algebras. Given a quasi-twilled Lie conformal algebra $(\Ep,\Vs,\Ws)$, we introduce two dual families of operators: \emph{right deformation maps}…

We determine all involutions in the Cayley-Dickson construction that extend the involution of the original $*$-algebra. We also find all algebra isomorphisms between the resulting Cayley doubles that extend the identity automorphism of the…

We introduce and study $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{3})^2 - (x^{2})^3 = 0$. This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree…