Rings and Algebras

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New submissions for Fri, 10 May 24

[1]  arXiv:2405.05504 [pdf, ps, other]

Title: The standard generators of the tetrahedron algebra and their look-alikes

Authors: Jae-Ho Lee

Comments: 30 pages

Subjects: Rings and Algebras (math.RA)

The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $\boxtimes$ that "looks like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2, 3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $\xi \in \boxtimes$ is called $x_{ij}$-like whenever both (i) $\xi$ commutes with $x_{ij}$; (ii) $\xi$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive basis for $\boxtimes$ with the property that every basis element is either $x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from multiple points of view.

[2]  arXiv:2405.05859 [pdf, ps, other]

Title: Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras

Authors: Amir Fernández Ouaridi, Rosa María Navarro, David A. Towers

Subjects: Rings and Algebras (math.RA)

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras $\mathcal{P}$ of dimension $n$. We introduce the invariants $\alpha$ and $\beta$ for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if $\alpha(\mathcal{P}) = n-1$. We characterize the Poisson algebras with $\alpha(\mathcal{P}) = n-2$ over arbitrary fields. In particular, we characterize Lie algebras $L$ with $\alpha(L) = n-2$. We also show that $\alpha(\mathcal{P}) = n-2$ for nilpotent Poisson algebras implies $\beta(\mathcal{P})=n-2$. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.

Replacements for Fri, 10 May 24

[3]  arXiv:2211.06978 (replaced) [pdf, ps, other]

Title: A periodicity theorem for extensions of Weyl modules

Authors: Mihalis Maliakas, Dimitra-Dionysia Stergiopoulou

Comments: Referee's comments incorporated. To appear in Mathematische Zeitschrift

Subjects: Representation Theory (math.RT); Rings and Algebras (math.RA)

[4]  arXiv:2311.03647 (replaced) [pdf, ps, other]

Title: An algebraic formulation of nonassociative quantum mechanics

Authors: Peter Schupp, Richard J. Szabo

Comments: 35 pages; v2: minor corrections; v3: minor changes, Conclusions section added; Final version to appear in Journal of Physics A

Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Rings and Algebras (math.RA)

[5]  arXiv:2405.04683 (replaced) [pdf, ps, other]

Title: Multicomplex Ideals, Modules and Hilbert Spaces

Authors: Derek Courchesne, Sébastien Tremblay

Comments: 28 pages

Subjects: Mathematical Physics (math-ph); Rings and Algebras (math.RA)

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