{"title":"Acyclic Complexes and Gorenstein Rings","authors":"S. Estrada, A. Iacob, Holly Zolt","doi":"10.1142/s1005386720000474","DOIUrl":"https://doi.org/10.1142/s1005386720000474","url":null,"abstract":"For a given class of modules $mathcal{A}$, we denote by $widetilde{mathcal{A}}$ the class of exact complexes $X$ having all cycles in $mathcal{A}$, and by $dw(mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $mathcal{A}$. We use the notations $mathcal{GI}$ $(mathcal{GF}, mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $mathcal{DI}$ for Ding injective modules, and $mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are equivalent over any ring $R$: (1) Every exact complex of injective modules is totally acyclic. (2) Every exact complex of Gorenstein injective modules is in $widetilde{mathcal{GI}}$. (3) Every complex in $dw(mathcal{GI})$ is dg-Gorenstein injective. We show that the analogue result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. if moreover, the ring is $n$-perfect for some integer $n ge 0$, then the three equivalent statements for flat and Gorenstein flat modules are also equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings: Let $R$ be a commutative coherent ring. The following statements are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules. (2) every exact complex of injectives has all its cycles Ding injective modules and every $R$-module M such that $M^+$ is Gorenstein flat is Ding injective. If moreover the ring $R$ has finite Krull dimension then statements (1), (2) above are also equivalent to (3) $R$ is a Gorenstein ring (in the sense of Iwanaga).","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117082784","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}
{"title":"A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras","authors":"D. Burde, W. Moens","doi":"10.1016/j.jalgebra.2020.05.006","DOIUrl":"https://doi.org/10.1016/j.jalgebra.2020.05.006","url":null,"abstract":"","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117740478","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}
{"title":"Drinfel’d construction for Hom–Hopf T-coalgebras","authors":"Dongdong Yan, Shuanhong Wang","doi":"10.1142/s0129167x20500585","DOIUrl":"https://doi.org/10.1142/s0129167x20500585","url":null,"abstract":"Let $H$ be a Hom-Hopf T-coalgebra over a group $pi $ (i.e., crossed Hom-Hopf $pi $-coalgebra). First, we introduce and study the left-right $alpha $-Yetter-Drinfel'd category $mathcal{YD}(H)_{alpha}$ over $H$ with $alpha in G$, and construct a class of new braided T-categories. Then, we prove that Yetter-Drinfel'd module category $mathcal{YD}(H)$ is a full subcategory of the center $mathcal{Z}(Rep(H))$ of the category of representations of $H$. Next, we define the quasi-triangular structure of $H$, and obtain its representation crossed category $Rep(H)$ is quasi-braided. Finally, the Drinfel'd construction $D(H)$ of $H$ is constructed, and an equivalent relation between $mathcal{YD}(H)$ and the representation of $D(H)$ is given.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129711548","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}
{"title":"On the structure of hyperfields obtained as quotients of fields","authors":"M. Baker, T. Jin","doi":"10.1090/PROC/15207","DOIUrl":"https://doi.org/10.1090/PROC/15207","url":null,"abstract":"We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients of fields. The main ingredients in the proof are the Weil bounds from number theory and a result from Ramsey theory.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128867786","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}
{"title":"Degree-One Rational Cherednik Algebras for the Symmetric Group","authors":"B. Foster-Greenwood, C. Kriloff","doi":"10.3842/SIGMA.2021.039","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.039","url":null,"abstract":"Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of gl_n-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the sl_n-type rational Cherednik algebras H_{0,c}.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117349581","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}
Doston Jumaniyozov, I. Kaygorodov, A. Khudoyberdiyev
{"title":"The algebraic and geometric classification of nilpotent noncommutative Jordan algebras","authors":"Doston Jumaniyozov, I. Kaygorodov, A. Khudoyberdiyev","doi":"10.1142/S0219498821502029","DOIUrl":"https://doi.org/10.1142/S0219498821502029","url":null,"abstract":"We give algebraic and geometric classifications of complex $4$-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of $3$ rigid algebras and $2$ one-parametric families of algebras.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133309824","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}
{"title":"Unipotent factorization of vector bundle automorphisms","authors":"Jakob Hultgren, E. F. Wold","doi":"10.1142/s0129167x21500130","DOIUrl":"https://doi.org/10.1142/s0129167x21500130","url":null,"abstract":"We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over smooth manifolds. This generalises work of Thurston-Wasserstein and Wasserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"27 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131751020","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}
{"title":"Codimension growth of Lie algebras with a generalized action","authors":"Geoffrey Janssens","doi":"10.1090/proc/15868","DOIUrl":"https://doi.org/10.1090/proc/15868","url":null,"abstract":"Let $L$ be a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct the first example of an $S$-graded Lie algebra having a non-integer, even irrational, exponential growth rate $lim_{nrightarrow infty} sqrt[n]{c_n^{S}(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand, returning to general $H$, if $L$ is semisimple and also semisimple for the $H$-action we prove the analog of Amitsur's conjecture (i.e. $lim_{nrightarrow infty} sqrt[n]{c_n^{H}(L)} in mathbb{Z}$). Moreover if $H=FS$ is a semigroup algebra the semisimplicity on $L$ can be dropped which is in strong contract to the associative setting.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115115155","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}
{"title":"Generalizing the notion of rank to noncommutative quadratic forms","authors":"M. Vancliff, Padmini Veerapen","doi":"10.1090/conm/592/11863","DOIUrl":"https://doi.org/10.1090/conm/592/11863","url":null,"abstract":"In 2010, Cassidy and Vancliff extended the notion of a quadratic form on n generators to the noncommutative setting. In this article, we suggest a notion of rank for such noncommutative quadratic forms, where n = 2 or 3. Since writing an arbitrary quadratic form as a sum of squares fails in this context, our methods entail rewriting an arbitrary quadratic form as a sum of products. In so doing, we find analogs for 2 x 2 minors and determinant of a 3 x 3 matrix in this noncommutative setting.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129024903","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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