Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23143

Generalized Mock-Poisson Algebras: Constructions and Structural Properties

Zafar Normatov
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce generalized mock-Poisson algebras as a mock analogue of generalized Poisson algebras. We prove that $A_\ast^3$ is an ideal annihilated by both operations and that the quotient $A/A_\ast^3$ is a mock-Poisson algebra. We also show that the dimension seven is the smallest dimension in which a generalized mock-Poisson algebra need not be a mock-Poisson algebra. We give several constructions using derivations, anti-Zinbiel algebras, tensor products, and one-dimensional extensions. We study polarization and depolarization and characterize the corresponding admissible algebras. We prove that every finite-dimensional generalized mock-Poisson algebra over a field of characteristic zero is nilpotent. Finally, we describe representations in terms of semidirect products.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zafar Normatov. 2026-10-16. Generalized Mock-Poisson Algebras: Constructions and Structural Properties. https://arxiv.org/abs/2610.23143

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Linear algebra over T-pairs

This is part of an ongoing project to find a general algebraic framework for semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of negation, and such basic properties such as unique factorization of polynomials, multiplicativity of determinants, and the characteristic polynomial of a matrix, all fail. (In fact in the max-plus algebra, the sum of two nonzero elements is never zero!) Consequently 0 is replaced by a distinguished T-submodule $ A_0$ of $ A,$ and $( A, A_0)$ is called a ``pair.'' This paper treats linear algebra over a (not necessarily distributive) semiring pair, with a range of applications to tropical algebra as well as related areas such as hyperrings and fuzzy rings. We turn to matrices and the question of whether the row rank, column rank, and submatrix rank of a matrix are equal. The submatrix rank is less than or equal to the row rank and the column rank in many cases, including ``metatangible pairs'' with unique negation, but there is a counterexample to equality, discovered some time ago by the second author, which we provide in a more general setting (``pairs of the second kind'') that includes the hyperfield of signs. We do find situations when equality holds, encompassing results by Akian, Gaubert, Guterman, Izhakian, Knebusch, and Rowen, including versions of Cramer's rule. We pay special attention to the question of Baker and Zhang whether $n+1$ vectors of length $n$ need be dependent. At the conclusion of the main part, we consider surpassing relations, which permit us to tighten our results. The categorical setting is given in the appendix.

math.RA↗
arXiv2026

A prime isoartinian ring that is not noetherian

A right module is isoartinian (resp., isonoetherian) if every descending (resp., ascending) chain of submodules is eventually constant up to isomorphism, and a ring R is right isoartinian (resp., isonoethe- rian) if RR is isoartinian (resp., isonoetherian). Facchini and Nazemian proved that a semiprime right isoartinian ring is right noetherian if and only if it is right isonoetherian if and only if it has finite Goldie dimen- sion and that every commutative semiprime isoartinian ring is noether- ian. We construct, over an arbitrary field K, a (non commutative) prime (right and left) isoartinian ring that is neither right nor left noetherian.

math.RA↗
arXiv2026

Derivations and quasi-derivations of Block Lie superalgebras $S(q)$

We compute the homogeneous derivations and the homogeneous quasi-derivations of the Block Lie superalgebras $S(q)$, whose even part is the Block Lie algebra $\mathcal{B}(q)$. For every $q\in\mathbb C$, the space $\mathfrak{Der}_θ(S(q))$ is the direct sum of the inner derivations and an explicitly given finite-dimensional space of outer derivations; in particular, for $q\neq0$ a new odd outer derivation appears if and only if $q\in2\mathbb Z^*$. For $q\in\mathbb C^*$ we prove that $\mathfrak{QDer}(S(q))=\mathfrak{Der}(S(q))\oplus\langle \id\rangle$ if $q\notin\mathbb Z$, and that for $q\in\mathbb Z^{*}$ the superalgebra $S(q)$ has an infinite-dimensional space of nontrivial quasi-derivations, namely all graded linear maps with values in the center. For $q=0$ we also describe all homogeneous quasi-derivations: modulo maps with values in the center, they are spanned by $\id$, three further maps and infinite families of maps shifting the $\mathbb Z^2$-grading; in particular, $\mathfrak{QDer}(S(0))/\mathfrak{Der}(S(0))$ is infinite-dimensional. As corollaries, we obtain the complete description of homogeneous $δ$-derivations of $S(q)$, $q\neq0$, and prove that, in contrast with the Block Lie algebras $\mathcal{B}(q)$, the superalgebras $S(q)$, $q\neq0$, do not admit nontrivial transposed $δ$-Poisson superalgebra structures for any $δ\notin\{0,1\}$.

math.RA↗