The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.
{"title":"Quotient singularities by permutation actions are canonical","authors":"Takehiko Yasuda","doi":"arxiv-2408.13504","DOIUrl":"https://doi.org/arxiv-2408.13504","url":null,"abstract":"The quotient variety associated to a permutation representation of a finite\u0000group has only canonical singularities in arbitrary characteristic. Moreover,\u0000the log pair associated to such a representation is Kawamata log terminal\u0000except in characteristic two, and log canonical in arbitrary characteristic.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192739","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 last two decades, since the seminal work of Selig, has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair's axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra $mathrm{Cl}(V)$ is isomorphic to the twisted trivial extension $mathrm{Cl}(V/langle e_0rangle)ltimes_alphamathrm{Cl}(V/langle e_0rangle)$, where $e_0$ is a degenerate vector and $alpha$ is the grade-involution.
{"title":"Exploiting degeneracy in projective geometric algebra","authors":"John Bamberg, Jeff Saunders","doi":"arxiv-2408.13441","DOIUrl":"https://doi.org/arxiv-2408.13441","url":null,"abstract":"The last two decades, since the seminal work of Selig, has seen projective\u0000geometric algebra (PGA) gain popularity as a modern coordinate-free framework\u0000for doing classical Euclidean geometry and other Cayley-Klein geometries. This\u0000framework is based upon a degenerate Clifford algebra, and it is the purpose of\u0000this paper to delve deeper into its internal algebraic structure and extract\u0000meaningful information for the purposes of PGA. This includes exploiting the\u0000split extension structure to realise the natural decomposition of elements of\u0000this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful\u0000demonstration of how Playfair's axiom for affine geometry arises from the\u0000ambient degenerate quadratic space. The highlighted split extension property of\u0000the Clifford algebra also corresponds to a splitting of the group of units and\u0000the Lie algebra of bivectors. Central to these results is that the degenerate\u0000Clifford algebra $mathrm{Cl}(V)$ is isomorphic to the twisted trivial\u0000extension $mathrm{Cl}(V/langle e_0rangle)ltimes_alphamathrm{Cl}(V/langle\u0000e_0rangle)$, where $e_0$ is a degenerate vector and $alpha$ is the\u0000grade-involution.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"143 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192738","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 investigate the notion of textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean ring that is either NI or one-sided perfect, then $R$ is periodic. Additionally, we demonstrate that every group ring $RG$ of a nilpotent group $G$ over a weakly 2-primal or one-sided perfect ring $R$ is semi-nil clean if and only if $R$ is periodic and $G$ is locally finite. Moreover, we also study those rings in which every unit is a sum of a periodic and a nilpotent element, calling them textit{unit semi-nil clean} rings. As a remarkable result, we show that if $R$ is an algebraic algebra over a field, then $R$ is unit semi-nil clean if and only if $R$ is periodic. Besides, we explore those rings in which non-zero elements are a sum of a torsion element and a nilpotent element, naming them textit{t-fine} rings, which constitute a proper subclass of the class of all fine rings. One of the main results is that matrix rings over t-fine rings are again t-fine rings.
{"title":"On Semi-Nil Clean Rings with Applications","authors":"M. H. Bien, P. V. Danchev, M. Ramezan-Nassab","doi":"arxiv-2408.13164","DOIUrl":"https://doi.org/arxiv-2408.13164","url":null,"abstract":"We investigate the notion of textit{semi-nil clean} rings, defined as those\u0000rings in which each element can be expressed as a sum of a periodic and a\u0000nilpotent element. Among our results, we show that if $R$ is a semi-nil clean\u0000ring that is either NI or one-sided perfect, then $R$ is periodic.\u0000Additionally, we demonstrate that every group ring $RG$ of a nilpotent group\u0000$G$ over a weakly 2-primal or one-sided perfect ring $R$ is semi-nil clean if\u0000and only if $R$ is periodic and $G$ is locally finite. Moreover, we also study those rings in which every unit is a sum of a\u0000periodic and a nilpotent element, calling them textit{unit semi-nil clean}\u0000rings. As a remarkable result, we show that if $R$ is an algebraic algebra over\u0000a field, then $R$ is unit semi-nil clean if and only if $R$ is periodic. Besides, we explore those rings in which non-zero elements are a sum of a\u0000torsion element and a nilpotent element, naming them textit{t-fine} rings,\u0000which constitute a proper subclass of the class of all fine rings. One of the\u0000main results is that matrix rings over t-fine rings are again t-fine rings.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192740","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 study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $Resubseteq G$, where $G$ is the reduced Gr"obner basis of $(Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras.
{"title":"Quadratic algebras and idempotent braided sets","authors":"Tatiana Gateva-Ivanova, Shahn Majid","doi":"arxiv-2409.02939","DOIUrl":"https://doi.org/arxiv-2409.02939","url":null,"abstract":"We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite\u0000set-theoretic solutions $(X,r)$ of the braid relations. We introduce an\u0000equivalent set of quadratic relations $Resubseteq G$, where $G$ is the\u0000reduced Gr\"obner basis of $(Re)$. We show that if $(X,r)$ is\u0000left-nondegenerate and idempotent then $Re= G$ and the Yang-Baxter algebra is\u0000PBW. We use graphical methods to study the global dimension of PBW algebras in\u0000the $n$-generated case and apply this to Yang-Baxter algebras in the\u0000left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a\u0000class of quadratic algebras and use this to show that for $(X,r)$\u0000left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can\u0000be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all\u0000left-nondegenerate idempotent solutions. We determined the Segre product in the\u0000left-nondegenerate idempotent setting. Our results apply to a previously\u0000studied class of `permutation idempotent' solutions, where we show that all\u0000their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and\u0000are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we\u0000construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz\u0000algebra in the idempotent case, showing that the latter is quadratic. We also\u0000construct noncommutative differentials on some of these quadratic algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192742","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 first examples of noncrossed product division algebras were given by Amitsur in 1972. His method is based on two basic steps: (1) If the universal division algebra $U(k,n)$ is a $G$-crossed product then every division algebra of degree $n$ over $k$ should be a $G$-crossed product; (2) There are two division algebras over $k$ whose maximal subfields do not have a common Galois group. In this note, we give a short proof for the second step in the case where $chr knmid n$ and $p^3|n$.
{"title":"A proof for a part of noncrossed product theorem","authors":"Mehran Motiee","doi":"arxiv-2408.12711","DOIUrl":"https://doi.org/arxiv-2408.12711","url":null,"abstract":"The first examples of noncrossed product division algebras were given by\u0000Amitsur in 1972. His method is based on two basic steps: (1) If the universal\u0000division algebra $U(k,n)$ is a $G$-crossed product then every division algebra\u0000of degree $n$ over $k$ should be a $G$-crossed product; (2) There are two\u0000division algebras over $k$ whose maximal subfields do not have a common Galois\u0000group. In this note, we give a short proof for the second step in the case\u0000where $chr knmid n$ and $p^3|n$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192741","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}
Lie Yamaguti algebras appear naturally on the smooth sections of the tangent bundle of a reductive homogeneous space when we interpret the torsion and curvature as algebraic operators. In this article we present a description of the free Lie Yamaguti algebra.
{"title":"On the free Lie-Yamaguti algebra","authors":"Jonatan Stava","doi":"arxiv-2408.10815","DOIUrl":"https://doi.org/arxiv-2408.10815","url":null,"abstract":"Lie Yamaguti algebras appear naturally on the smooth sections of the tangent\u0000bundle of a reductive homogeneous space when we interpret the torsion and\u0000curvature as algebraic operators. In this article we present a description of\u0000the free Lie Yamaguti algebra.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192746","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}
M. C. ROdríguez-Vallarte, G. Salgado, O. A. Sánchez-Valenzuela
Extensions of Lie algebras equipped with Sasakian or Frobenius-K"ahler geometrical structures are studied. Conditions are given so that a double extension of a Sasakian Lie algebra be Sasakian again. Conditions are also given for obtaining either a Sasakian or a Frobernius-K"ahler Lie algebra upon respectively extending a Frobernius-K"ahler or a Sasakian Lie algebra by adjoining a derivation of the source algebra. Low-dimensional examples are included.
{"title":"On extensions of Frobenius-Kähler and Sasakian Lie algebras","authors":"M. C. ROdríguez-Vallarte, G. Salgado, O. A. Sánchez-Valenzuela","doi":"arxiv-2408.11236","DOIUrl":"https://doi.org/arxiv-2408.11236","url":null,"abstract":"Extensions of Lie algebras equipped with Sasakian or Frobenius-K\"ahler\u0000geometrical structures are studied. Conditions are given so that a double\u0000extension of a Sasakian Lie algebra be Sasakian again. Conditions are also\u0000given for obtaining either a Sasakian or a Frobernius-K\"ahler Lie algebra upon\u0000respectively extending a Frobernius-K\"ahler or a Sasakian Lie algebra by\u0000adjoining a derivation of the source algebra. Low-dimensional examples are\u0000included.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192744","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 define and explore the bounded skew power series ring $R^+[[x;sigma,delta]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(sigma,delta)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $delta$ is an inner $sigma$-derivation and no positive power of $sigma$ is inner as an automorphism of $Q$, then $Q^+[[x;sigma,delta]]$ is often prime, and even simple under certain mild restrictions on $delta$. It follows from this result that $R^+[[x;sigma,delta]]$ is itself prime.
{"title":"Bounded skew power series rings for inner $σ$-derivations","authors":"Adam Jones, William Woods","doi":"arxiv-2408.10545","DOIUrl":"https://doi.org/arxiv-2408.10545","url":null,"abstract":"We define and explore the bounded skew power series ring\u0000$R^+[[x;sigma,delta]]$ defined over a complete, filtered, Noetherian prime\u0000ring $R$ with a commuting skew derivation $(sigma,delta)$. We establish\u0000precise criteria for when this ring is well-defined, and for an appropriate\u0000completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $delta$\u0000is an inner $sigma$-derivation and no positive power of $sigma$ is inner as\u0000an automorphism of $Q$, then $Q^+[[x;sigma,delta]]$ is often prime, and even\u0000simple under certain mild restrictions on $delta$. It follows from this result\u0000that $R^+[[x;sigma,delta]]$ is itself prime.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"143 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192745","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}
This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $mathfrak R$-contramodules. In other words, a complex of flat $mathfrak R$-contramodules is contraacyclic in the sense of Becker if and only if it is an acyclic complex with flat $mathfrak R$-contramodules of cocycles.
{"title":"A contramodule generalization of Neeman's flat and projective module theorem","authors":"Leonid Positselski","doi":"arxiv-2408.10928","DOIUrl":"https://doi.org/arxiv-2408.10928","url":null,"abstract":"This paper builds on top of arXiv:2306.02734. We consider a complete,\u0000separated topological ring $mathfrak R$ with a countable base of neighborhoods\u0000of zero consisting of open two-sided ideals. The main result is that the\u0000homotopy category of projective left $mathfrak R$-contramodules is equivalent\u0000to the derived category of the exact category of flat left $mathfrak\u0000R$-contramodules. In other words, a complex of flat $mathfrak R$-contramodules\u0000is contraacyclic in the sense of Becker if and only if it is an acyclic complex\u0000with flat $mathfrak R$-contramodules of cocycles.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192743","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}
Habib Ammari, Silvio Barandun, Ping Liu, Alexander Uhlmann
Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the $k$-Toeplitz matrix cases.
{"title":"Generalised Brillouin Zone for Non-Reciprocal Systems","authors":"Habib Ammari, Silvio Barandun, Ping Liu, Alexander Uhlmann","doi":"arxiv-2408.05073","DOIUrl":"https://doi.org/arxiv-2408.05073","url":null,"abstract":"Recently, it has been observed that the Floquet-Bloch transform with real\u0000quasiperiodicities fails to capture the spectral properties of non-reciprocal\u0000systems. The aim of this paper is to introduce the notion of a generalised\u0000Brillouin zone by allowing the quasiperiodicities to be complex in order to\u0000rectify this. It is proved that this shift of the Brillouin zone into the\u0000complex plane accounts for the unidirectional spatial decay of the eigenmodes\u0000and leads to correct spectral convergence properties. The results in this paper\u0000clarify and prove rigorously how the spectral properties of a finite structure\u0000are associated with those of the corresponding semi-infinitely or infinitely\u0000periodic lattices and give explicit characterisations of how to extend the\u0000Hermitian theory to non-reciprocal settings. Based on our theory, we\u0000characterise the generalised Brillouin zone for both open boundary conditions\u0000and periodic boundary conditions. Our results are consistent with the physical\u0000literature and give explicit generalisations to the $k$-Toeplitz matrix cases.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935657","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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