We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.
This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.
Mixed spin-(1/2,1/2,1) trimer with two different Landé g factors and two different exchange couplings is considered. The main feature of the model is nonconserving magnetization. The Hamiltonian of the system is diagonalized analytically. We presented a detailed analysis of the ground-state properties, revealing several possible ground-state phase diagrams and magnetization profiles. The main focus is on how nonconserving magnetization affects quantum entanglement. We have found that nonconserving magnetization can bring a continuous dependence of the entanglement quantifying parameter (negativity) on the magnetic field within the same eigenstate, while for the case of uniform g factors it is a constant. The main result is an essential enhancement of the entanglement in the case of uniform couplings for one pair of spins caused by an arbitrary small difference in the values of g factors. This enhancement is robust and brings almost a sevenfold increase of the negativity. We have also found a weakening of entanglement for other cases. Thus, nonconserving magnetization offers a broad opportunity to manipulate the entanglement by means of a magnetic field. Published by the American Physical Society 2024
Vitor H. Fernandes, Jörg Koppitz, Tiwadee Musunthia
In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.
We revisit an archive submission by Denton et al. (Eigenvectors from eigenvalues: a survey of a basic identity in linear algebra. arXiv:1908.03795v3 [math.RA], 2019) on n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrödinger operators on a compact interval.
A subalgebra $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Likewise, $\mathcal{A}$ is said to be idempotent compressible if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. In this paper, a case-by-case analysis is used to classify the unital projection compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$, up to transposition and unitary equivalence. It is observed that every algebra shown to admit the projection compression property is, in fact, idempotent compressible. We therefore extend the findings of Cramer, Marcoux, and Radjavi (arXiv:1904.06803 [math.RA]) in the setting of $\mathbb{M}_3(\mathbb{C})$, proving that the two notions of compressibility agree for all unital matrix algebras.
We report on some computations with nilpotent orbits in simple Lie algebras of exceptional type within the SLA package of GAP4. Concerning reachable nilpotent orbits our computations firstly confirm the classification of such orbits in Lie algebras of exceptional type by Elashvili and Grelaud, secondly they answer a question by Panyushev, and thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types. The second topic of this note concerns abelianizations of centralizers of nilpotent elements. We give tables with their dimensions.
We show that the pair given by the power set and by the ''Grassmannian'' (set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize several results from preceding joint work with M. Kinyon (arxiv: math.RA/0903.5441), which concerned abelian groups, to the case of general non-abelian groups. Most notably, pairs of subgroups parametrize torsor and semitorsor structures on the power set. The role of associative algebras and -pairs from loc. cit. is now taken by analogs of near-rings.
The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.
In our paper arXiv: math.RA/0110333 v1 Oct 2001 we showed that the number of algebras defined by a binary operation satisfying a formally irreducible identity between two n-iterates is O( e^{-n/16}S_{n}^{2} for n --> infinity, S_{n} being the nth-Catalan number. This was proved by using exclusively the series of tableaux A_{n}. By using also the series of tableaux B_{n}, we now sharpen this result to O{(n+2)/n|e^{-n/16}-2/n)|S_{n}^{2}}
I present a construction "a` la Chevalley" of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This extends the similar work performed in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014 - arXiv:0808.0785v8 [math.RA]], where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in [F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (to appear), 92 pages - arXiv:1109.0626v5 [math.RA], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra.
Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation whose value at 0 is 0. Divisive meadows are inversive meadows with the multiplicative inverse operation replaced by a division operation. We give finite equational specifications of the class of all inversive meadows and the class of all divisive meadows. It depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. We show that inversive and divisive meadows of rational numbers can be obtained as initial algebras of finite equational specifications. In the spirit of Peacock's arithmetical algebra, we study variants of inversive and divisive meadows without an additive identity element and/or an additive inverse operation. We propose simple constructions of variants of inversive and divisive meadows with a partial multiplicative inverse or division operation from inversive and divisive meadows. Divisive meadows are more basic if these variants are considered as well. We give a simple account of how mathematicians deal with 1 / 0, in which meadows and a customary convention among mathematicians play prominent parts, and we make plausible that a convincing account, starting from the popular computer science viewpoint that 1 / 0 is undefined, by means of some logic of partial functions is not attainable.