Symmetries in connected graded algebras and their PBW-deformations

Yongjun Xu; Xin Zhang

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Certain partitions on a set and their applications to different classes of graded algebras

Antonio J. Calderón Martín, Boubacar Dieme (2021)

Communications in Mathematics

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Let $(𝔄, ϵ_{u})$ and $(𝔅, ϵ_{b})$ be two pointed sets. Given a family of three maps $ℱ = {f_{1} : 𝔄 \to 𝔄; f_{2} : 𝔄 \times 𝔄 \to 𝔄; f_{3} : 𝔄 \times 𝔄 \to 𝔅}$ , this family provides an adequate decomposition of $𝔄 ∖ {ϵ_{u}}$ as the orthogonal disjoint union of well-described $ℱ$ -invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$ -algebras.

On generalized Beurling’s theorem and symmetry of $L_{1}$ -group algebras

Zenobia Anusiak (1971)

Colloquium Mathematicae

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Quantised ${𝔰𝔩}_{2}$ -differential algebras

Andrey Krutov, Pavle Pandžić (2024)

Archivum Mathematicum

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We propose a definition of a quantised ${𝔰𝔩}_{2}$ -differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${𝔰𝔩}_{2}$ are natural examples of such algebras.

Solvable Leibniz algebras with NF n ⊕ [...] F m 1 $\begin{matrix} F_{m}^{1} \end{matrix}$ nilradical

L.M. Camacho, B.A. Omirov, K.K. Masutova, I.M. Rikhsiboev (2017)

Open Mathematics

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All finite-dimensional solvable Leibniz algebras L, having N = NFn⊕ [...] Fm1 $\begin{matrix} F_{m}^{1} \end{matrix}$ as the nilradical and the dimension of L equal to n+m+3 (the maximal dimension) are described. NFn and [...] Fm1 $\begin{matrix} F_{m}^{1} \end{matrix}$ are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m, respectively. Moreover, we show that these algebras are rigid.

A representation theorem for tense $n \times m$ -valued Łukasiewicz-Moisil algebras

Aldo Victorio Figallo, Gustavo Pelaitay (2015)

Mathematica Bohemica

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In 2000, Figallo and Sanza introduced $n \times m$ -valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$ -valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class $_{n \times m}$ of tense $n \times m$ -valued Łukasiewicz-Moisil algebras (or tense LM $_{n \times m}$ -algebras), namely $n \times m$ -valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense...

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

On cardinalities of algebras of formulas for $ω_{0}$ -categorical theories

Jan Waszkiewicz (1973)

Colloquium Mathematicae

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Matrix representation of finite effect algebras

Grzegorz Bińczak, Joanna Kaleta, Andrzej Zembrzuski (2023)

Kybernetika

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In this paper we present representation of finite effect algebras by matrices. For each non-trivial finite effect algebra $E$ we construct set of matrices $M (E)$ in such a way that effect algebras $E_{1}$ and $E_{2}$ are isomorphic if and only if $M (E_{1}) = M (E_{2})$ . The paper also contains the full list of matrices representing all nontrivial finite effect algebras of cardinality at most $8$ .

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

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We investigate derived equivalences between subalgebras of some $Φ$ -Auslander-Yoneda algebras from a class of $n$ -angles in weakly $n$ -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$ -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to $n$ -angle cases. Finally, we give an explicit example to illustrate our result.

Quantized semisimple Lie groups

Rita Fioresi, Robert Yuncken (2024)

Archivum Mathematicum

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The goal of this expository paper is to give a quick introduction to $q$ -deformations of semisimple Lie groups. We discuss principally the rank one examples of $𝒰_{q} ({𝔰𝔩}_{2})$ , $𝒪 ({SU}_{q} (2))$ , $𝒟 ({SL}_{q} (2, ℂ))$ and related algebras. We treat quantized enveloping algebras, representations of $𝒰_{q} ({𝔰𝔩}_{2})$ , generalities on Hopf algebras and quantum groups, $*$ -structures, quantized algebras of functions on $q$ -deformed compact semisimple groups, the Peter-Weyl theorem, $*$ -Hopf algebras associated to complex semisimple Lie groups and the Drinfeld...

Commutative graded- $S$ -coherent rings

Anass Assarrar, Najib Mahdou, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

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Recently, motivated by Anderson, Dumitrescu’s $S$ -finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$ -coherent rings, which is the $S$ -version of coherent rings. Let $R = ⨁_{α \in G} R_{α}$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$ , and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$ . We define $R$ to be graded- $S$ -coherent ring if every finitely generated homogeneous ideal of $R$ is $S$ -finitely presented. The purpose of this paper is to give...

The structures of Hopf $*$ -algebra on Radford algebras

Hassan Suleman Esmael Mohammed, Hui-Xiang Chen (2019)

Czechoslovak Mathematical Journal

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We investigate the structures of Hopf $*$ -algebra on the Radford algebras over $ℂ$ . All the $*$ -structures on $H$ are explicitly given. Moreover, these Hopf $*$ -algebra structures are classified up to equivalence.

$ℵ_{0}$ -categoricity of products of 1-unary algebras

Philip Olin (1975)

Colloquium Mathematicae

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