Dmytro R. Popovych, Serhii D. Koval, Roman O. Popovych
Using an original method, we find the algebra of generalized symmetries of a remarkable (1+2)-dimensional ultraparabolic Fokker-Planck equation, which is also called the Kolmogorov equation and is singled out within the entire class of ultraparabolic linear second-order partial differential equations with three independent variables by its wonderful symmetry properties. It turns out that the essential part of this algebra is generated by the recursion operators associated with the nilradical of the essential Lie invariance algebra of the Kolmogorov equation, and the Casimir operator of the Levi factor of the latter algebra unexpectedly arises in the consideration.
{"title":"Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation","authors":"Dmytro R. Popovych, Serhii D. Koval, Roman O. Popovych","doi":"arxiv-2409.10348","DOIUrl":"https://doi.org/arxiv-2409.10348","url":null,"abstract":"Using an original method, we find the algebra of generalized symmetries of a\u0000remarkable (1+2)-dimensional ultraparabolic Fokker-Planck equation, which is\u0000also called the Kolmogorov equation and is singled out within the entire class\u0000of ultraparabolic linear second-order partial differential equations with three\u0000independent variables by its wonderful symmetry properties. It turns out that\u0000the essential part of this algebra is generated by the recursion operators\u0000associated with the nilradical of the essential Lie invariance algebra of the\u0000Kolmogorov equation, and the Casimir operator of the Levi factor of the latter\u0000algebra unexpectedly arises in the consideration.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the properties of the autonomous superposition operator on the space of formal power series, including those with nonzero constant term. We prove its continuity and smoothness with respect to the topology of pointwise convergence and a natural Fr'echet manifold structure. A necessary and sufficient condition for the left composition inverse of a formal power series to exist is provided. We also present some properties of the Fr'echet-Lie group structures on the set of nonunit formal power series.
{"title":"Topology and geometry of the general composition of formal power series - towards Fréchet-Lie group-like formalism","authors":"Dawid Bugajewski","doi":"arxiv-2409.09853","DOIUrl":"https://doi.org/arxiv-2409.09853","url":null,"abstract":"In this article, we study the properties of the autonomous superposition\u0000operator on the space of formal power series, including those with nonzero\u0000constant term. We prove its continuity and smoothness with respect to the\u0000topology of pointwise convergence and a natural Fr'echet manifold structure. A\u0000necessary and sufficient condition for the left composition inverse of a formal\u0000power series to exist is provided. We also present some properties of the\u0000Fr'echet-Lie group structures on the set of nonunit formal power series.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this note is to study some algebraic properties of irreducible ideals of monoids. We establish relations between irreducible, prime, and semiprime ideals. We explore some properties of irreducible ideals in local, Noetherian, and Laskerian monoids.
{"title":"Some results on irreducible ideals of monoids","authors":"Amartya Goswami","doi":"arxiv-2409.09757","DOIUrl":"https://doi.org/arxiv-2409.09757","url":null,"abstract":"The purpose of this note is to study some algebraic properties of irreducible\u0000ideals of monoids. We establish relations between irreducible, prime, and\u0000semiprime ideals. We explore some properties of irreducible ideals in local,\u0000Noetherian, and Laskerian monoids.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $H_n(boldsymbol{u})$ be the degenerate cyclotomic Hecke algebra with parameter $boldsymbol{u}=(u_1, ldots, u_m)$ over $mathbb{C}(boldsymbol{u})$. We define and construct the (non-)normalized Markov traces on the sequence ${H_n(boldsymbol{u})}_{n=1}^{infty}$. This allows us to provide a canonical symmetrizing form on $H_n(boldsymbol{u})$ and show that the Brudan--Kleshchev trace on $H_n(boldsymbol{u})$ is a specialization of the non-normalized Markov traces.
{"title":"Markov traces on degenerate cyclotomic Hecke algebras","authors":"Deke Zhao","doi":"arxiv-2409.09372","DOIUrl":"https://doi.org/arxiv-2409.09372","url":null,"abstract":"Let $H_n(boldsymbol{u})$ be the degenerate cyclotomic Hecke algebra with\u0000parameter $boldsymbol{u}=(u_1, ldots, u_m)$ over\u0000$mathbb{C}(boldsymbol{u})$. We define and construct the (non-)normalized\u0000Markov traces on the sequence ${H_n(boldsymbol{u})}_{n=1}^{infty}$. This\u0000allows us to provide a canonical symmetrizing form on $H_n(boldsymbol{u})$ and\u0000show that the Brudan--Kleshchev trace on $H_n(boldsymbol{u})$ is a\u0000specialization of the non-normalized Markov traces.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove Lie's Third Theorem for Lie $infty$-algebras: Every finite-type, homologically and non-negatively graded $L_infty$-algebra over $mathbb{R}$ integrates to a finite-dimensional Lie $infty$-group.
{"title":"Lie's Third Theorem for Lie $infty$-Algebras","authors":"Christopher L. Rogers, Jesse Wolfson","doi":"arxiv-2409.08957","DOIUrl":"https://doi.org/arxiv-2409.08957","url":null,"abstract":"We prove Lie's Third Theorem for Lie $infty$-algebras: Every finite-type,\u0000homologically and non-negatively graded $L_infty$-algebra over $mathbb{R}$\u0000integrates to a finite-dimensional Lie $infty$-group.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A description of group automorphisms of all two-dimensional algebras, considered up to isomorphism, over any basic field is provided.
它描述了在任何基本域上的所有二维代数的群自形性（考虑到同构）。
{"title":"A description of automorphism groups of all two-dimensional algebras over any basic field","authors":"Eshmirzayev Sh., Bekbaev U","doi":"arxiv-2409.08814","DOIUrl":"https://doi.org/arxiv-2409.08814","url":null,"abstract":"A description of group automorphisms of all two-dimensional algebras,\u0000considered up to isomorphism, over any basic field is provided.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We state criteria for a nilpotent Lie algebra $g$ to admit an invariant metric. We use that $g$ possesses two canonical abelian ideals $ide(g) subset mathfrak{J}(g)$ to decompose the underlying vector space of $g$ and then we state sufficient conditions for $g$ to admit an invariant metric. The properties of the ideal $mathfrak{J}(g)$ allows to prove that if a current Lie algebra $g otimes Sa$ admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from $Sa times Sa$ into the space of centroids of $g/mathfrak{J}(g)$. We also prove that in any nilpotent Lie algebra $g$ there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct $g$ by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on $g$.
{"title":"Invariant Metrics on Nilpotent Lie algebras","authors":"R. García-Delgado","doi":"arxiv-2409.09017","DOIUrl":"https://doi.org/arxiv-2409.09017","url":null,"abstract":"We state criteria for a nilpotent Lie algebra $g$ to admit an invariant\u0000metric. We use that $g$ possesses two canonical abelian ideals $ide(g)\u0000subset mathfrak{J}(g)$ to decompose the underlying vector space of $g$ and\u0000then we state sufficient conditions for $g$ to admit an invariant metric. The\u0000properties of the ideal $mathfrak{J}(g)$ allows to prove that if a current\u0000Lie algebra $g otimes Sa$ admits an invariant metric, then there must be an\u0000invariant and non-degenerate bilinear map from $Sa times Sa$ into the space\u0000of centroids of $g/mathfrak{J}(g)$. We also prove that in any nilpotent Lie\u0000algebra $g$ there exists a non-zero, symmetric and invariant bilinear form.\u0000This bilinear form allows to reconstruct $g$ by means of an algebra with unit.\u0000We prove that this algebra is simple if and only if the bilinear form is an\u0000invariant metric on $g$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To a submodular function we define a class of preorders, conforming preorders. A submodular function $z$ corresponds to a generalized permutahedron $Pi(z)$. We show the faces of $Pi(z)$ are in bijection with the conforming preorders. The face poset structure of $Pi(z)$ induces two order relations $lhd$ and $blacktriangleleft$ on conforming preorder, and we investigate their properties. Ardila and Aguiar introduced a Hopf monoid of submodular functions/generalized permutahedra. We show there is a cointeracting bimonoid of modular functions. By recent theory of L.Foissy this associates a canonical polynomial to any submodular function.
{"title":"Submodular functions, generalized permutahedra, conforming preorders, and cointeracting bialgebras","authors":"Gunnar Fløystad, Dominique Manchon","doi":"arxiv-2409.08200","DOIUrl":"https://doi.org/arxiv-2409.08200","url":null,"abstract":"To a submodular function we define a class of preorders, conforming\u0000preorders. A submodular function $z$ corresponds to a generalized permutahedron\u0000$Pi(z)$. We show the faces of $Pi(z)$ are in bijection with the conforming\u0000preorders. The face poset structure of $Pi(z)$ induces two order relations\u0000$lhd$ and $blacktriangleleft$ on conforming preorder, and we investigate\u0000their properties. Ardila and Aguiar introduced a Hopf monoid of submodular\u0000functions/generalized permutahedra. We show there is a cointeracting bimonoid\u0000of modular functions. By recent theory of L.Foissy this associates a canonical\u0000polynomial to any submodular function.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a ring. In cite{MD4} Mao and Ding defined an special class of $R$-modules that they called ( FP_n )-projective $R$-modules. In this paper, we give some new characterizations of ( FP_n )-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the ( FP_n )-injective global dimension in terms of ( FP_n )-projective $R$-modules are obtained. Using the ( FP_n )-projective dimension of an $R$-module defined by Ouyang, Duan and Li in cite{Ouy} we introduce a slightly different ( FP_n )-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of cite{Ouy} when the ring in question is strong $n$-coherent.
{"title":"Some remarks about $FP_{n}$-projectives modules","authors":"Viviana Gubitosi, Rafael Parra","doi":"arxiv-2409.08334","DOIUrl":"https://doi.org/arxiv-2409.08334","url":null,"abstract":"Let $R$ be a ring. In cite{MD4} Mao and Ding defined an special class of\u0000$R$-modules that they called ( FP_n )-projective $R$-modules. In this paper,\u0000we give some new characterizations of ( FP_n )-projective $R$-modules and\u0000strong $n$-coherent rings. Some known results are extended and some new\u0000characterizations of the ( FP_n )-injective global dimension in terms of (\u0000FP_n )-projective $R$-modules are obtained. Using the ( FP_n )-projective\u0000dimension of an $R$-module defined by Ouyang, Duan and Li in cite{Ouy} we\u0000introduce a slightly different ( FP_n )-projective global dimension over the\u0000ring $R$ which measures how far away the ring is from being Noetherian. This\u0000dimension agrees with the $(n,0)$-projective global dimension of cite{Ouy}\u0000when the ring in question is strong $n$-coherent.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we shall describe all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Firstly, we give the matrix characterization of the Rota-Baxter operator on split semi-quaternion algebra. Then we give the corresponding matrix representations of all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Finally, we shall prove that the Ma et al. results about the Rota-Baxter operators on Sweedler algebra are just special cases of our results.
{"title":"The Rota-Baxter algebra structures on split semi-quaternion algebra","authors":"Chen Quanguo, Deng Yong","doi":"arxiv-2409.07699","DOIUrl":"https://doi.org/arxiv-2409.07699","url":null,"abstract":"In this paper, we shall describe all the Rota-Baxter operators with any\u0000weight on split semi-quaternion algebra. Firstly, we give the matrix\u0000characterization of the Rota-Baxter operator on split semi-quaternion algebra.\u0000Then we give the corresponding matrix representations of all the Rota-Baxter\u0000operators with any weight on split semi-quaternion algebra. Finally, we shall\u0000prove that the Ma et al. results about the Rota-Baxter operators on Sweedler\u0000algebra are just special cases of our results.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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