Journal of the London Mathematical Society-Second Series最新文献

{"title":"Mean-field limit of 2D stationary particle systems with signed Coulombian interactions","authors":"Jan Peszek, Rémy Rodiac","doi":"10.1112/jlms.70068","DOIUrl":"https://doi.org/10.1112/jlms.70068","url":null,"abstract":"<p>We study the mean-field limits of critical points of interaction energies with Coulombian singularity. An important feature of our setting is that we allow interaction between particles of opposite signs. Particles of opposite signs attract each other whereas particles of the same signs repel each other. In two dimensional, we prove that the associated empirical measures converge to a limiting measure <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math> that satisfies a two-fold criticality condition: in velocity form or in vorticity form. Our setting includes the stationary attraction–repulsion problem with Coulombian singularity and the stationary system of point vortices in fluid mechanics. In this last context, in the case where the limiting measure is in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>H</mi>\u0000 <mtext>loc</mtext>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H^{-1}_{text{loc}}(mathbb {R}^2)$</annotation>\u0000 </semantics></math>, we recover the classical criticality condition stating that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mo>∇</mo>\u0000 <mo>⊥</mo>\u0000 </msup>\u0000 <mi>g</mi>\u0000 <mo>*</mo>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation>$nabla ^perp g ast mu$</annotation>\u0000 </semantics></math>, with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>−</mo>\u0000 <mi>log</mi>\u0000 <mo>|</mo>\u0000 <mi>x</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$g(x)=-log |x|$</annotation>\u0000 </semantics></math>, is a stationary solution of the incompressible Euler equation. This result, is, to the best of our knowledge, new in the case of particles with different signs (for particles of the positive sign, it was obtained by Schochet in 1996). In order to derive the limiting criticality condition in the velocity form, we follow an approach devised by Sandier–Serfaty in the context of Ginzburg–Landau vortices. This consists of passing to the limit in the stress-energy tensor associated with the velocity field. On the oth","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143114067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rational approximation of holomorphic semigroups revisited","authors":"Charles Batty,&nbsp;Alexander Gomilko,&nbsp;Yuri Tomilov","doi":"10.1112/jlms.70066","DOIUrl":"https://doi.org/10.1112/jlms.70066","url":null,"abstract":"<p>Using a recently developed <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math>-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143114066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homology of spectral minimal partitions","authors":"Gregory Berkolaiko,&nbsp;Yaiza Canzani,&nbsp;Graham Cox,&nbsp;Jeremy L. Marzuola","doi":"10.1112/jlms.70065","DOIUrl":"https://doi.org/10.1112/jlms.70065","url":null,"abstract":"<p>A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the long-time behavior of solutions to the Navier–Stokes–Fourier system on unbounded domains","authors":"Elisabetta Chiodaroli,&nbsp;Eduard Feireisl","doi":"10.1112/jlms.70067","DOIUrl":"https://doi.org/10.1112/jlms.70067","url":null,"abstract":"<p>We consider the Navier–Stokes–Fourier (NSF) system on an unbounded domain in the Euclidean space <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$R^3$</annotation>\u0000 </semantics></math>, supplemented by the far-field conditions for the phase variables, specifically: <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ϱ</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>ϑ</mi>\u0000 <mo>→</mo>\u0000 <msub>\u0000 <mi>ϑ</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>u</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varrho rightarrow 0, vartheta rightarrow vartheta _infty, {bm u}rightarrow 0$</annotation>\u0000 </semantics></math> as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mspace></mspace>\u0000 <mo>|</mo>\u0000 <mi>x</mi>\u0000 <mo>|</mo>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$ |x| rightarrow infty$</annotation>\u0000 </semantics></math>. We study the long-time behavior of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ϱ</mi>\u0000 <mi>s</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <msub>\u0000 <mi>ϑ</mi>\u0000 <mi>s</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>ϑ</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <msub>\u0000 <mi>u</mi>\u0000 <mi>s</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varrho _s = 0, vartheta _s = vartheta _infty, {bm u}_s = 0$</annotation>\u0000 </semantics></math> in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143111977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weakly subnormal subgroups and variations of the Baer–Suzuki theorem","authors":"Robert M. Guralnick,&nbsp;Hung P. Tong-Viet,&nbsp;Gareth Tracey","doi":"10.1112/jlms.70057","DOIUrl":"https://doi.org/10.1112/jlms.70057","url":null,"abstract":"<p>A subgroup <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> of a finite group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is weakly subnormal in <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> is not subnormal in <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> but it is subnormal in every proper overgroup of <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> in <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>. In this paper, we first classify all finite groups <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> that contains a weakly subnormal <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-subgroup for some prime <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>. We then determine all finite groups containing a cyclic weakly subnormal <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-subgroup. As applications, we prove a number of variations of the Baer–Suzuki theorem using the orders of certain group elements.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A quantitative version of Northcott's theorem on points of bounded height: The function field case","authors":"Jeffrey Lin Thunder","doi":"10.1112/jlms.70059","DOIUrl":"https://doi.org/10.1112/jlms.70059","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;¯&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$overline{K}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denote an algebraic closure of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. For given integers &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mgeqslant 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ngeqslant 2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; we count points in projective space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;¯&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {P}^{n-1}(overline{K})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with absolute logarithmic height &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and generating an extension of degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$d&gt;2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; over &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Specifically, we derive an asymptotic estimate for the number of such points as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mrightarrow infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143119933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ℓ\u0000 p\u0000 \u0000 $ell ^p$\u0000 metrics on cell complexes","authors":"Thomas Haettel,&nbsp;Nima Hoda,&nbsp;Harry Petyt","doi":"10.1112/jlms.70062","DOIUrl":"https://doi.org/10.1112/jlms.70062","url":null,"abstract":"<p>Motivated by the observation that groups can be effectively studied using metric spaces modelled on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$ell ^1$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$ell ^2$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$ell ^infty$</annotation>\u0000 </semantics></math> geometry, we consider cell complexes equipped with an <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$ell ^p$</annotation>\u0000 </semantics></math> metric for arbitrary <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>. Under weak conditions that can be checked locally, we establish non-positive curvature properties of these complexes, such as Busemann-convexity and strong bolicity. We also provide detailed information on the geodesics of these metrics in the special case of CAT(0) cube complexes.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143119997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Lane–Emden conjecture with convolution part","authors":"Lele Du,&nbsp;Xiang Li,&nbsp;Minbo Yang","doi":"10.1112/jlms.70064","DOIUrl":"https://doi.org/10.1112/jlms.70064","url":null,"abstract":"<p>We study the Hartree type Lane–Emden conjecture, which states the nonexistence of the positive classical solutions for the following Hartree type system \u0000\u0000 </p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143119247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On noncommutative leapfrog map","authors":"Bao Wang,&nbsp;Shi-Hao Li","doi":"10.1112/jlms.70063","DOIUrl":"https://doi.org/10.1112/jlms.70063","url":null,"abstract":"<p>We investigate the integrability of the noncommutative leapfrog map in this paper. First, we derive the explicit formula for the noncommutative leapfrog map and corresponding discrete zero-curvature equation by employing the concept of noncommutative cross-ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of noncommutative Laurent bi-orthogonal polynomials. Finally, the Poisson structure for this discrete noncommutative map is formulated with the help of a noncommutative network. Through these constructions, we aim to enhance our understanding of the integrability properties of the noncommutative leapfrog map and its related mathematical structures.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Root of unity quantum cluster algebras and discriminants","authors":"Bach Nguyen,&nbsp;Kurt Trampel,&nbsp;Milen Yakimov","doi":"10.1112/jlms.70060","DOIUrl":"https://doi.org/10.1112/jlms.70060","url":null,"abstract":"<p>We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>[</mo>\u0000 <mi>ε</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>${mathbb {Z}}[varepsilon]$</annotation>\u0000 </semantics></math> of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math> is a root of unity.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}