Related papers: Kummer-Witt-Jackson algebras
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
A Hom-type generalization of non-commutative Poisson algebras, called non-commutative Hom-Poisson algebras, are studied. They are closed under twisting by suitable self-maps. Hom-Poisson algebras, in which the Hom-associative product is…
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras called here…
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. Let $P$ be a non-commutative Poisson algebra over some algebraically closed field of…
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by…
We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
We introduce and study flipped non-associative polynomial rings. In particular, we show that all Cayley-Dickson algebras naturally appear as quotients of a certain type of such rings; this extends the classical construction of the complex…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over…
In this work, we introduce a new class of algebras called skew-Brauer graph algebras, which generalize the well-known Brauer graph algebras. We establish that skew-Brauer graph algebras are symmetric and can be defined using a Brauer graph…
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions $n<7$ over algebraically closed fields of…
In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at…
In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlev\'{e} transcendent models and missing label problems. In this paper we treat all of these algebras as…
We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by…
We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes.…
We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to…