Let k be a field, let G be a smooth affine k-group and V a finite-dimensional G-module. We say V is emph{rigid} if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is emph{geometrically rigid} (resp.~emph{absolutely rigid}) if V is rigid after base change of G and V to bar k (resp.~any field extension of k). We show that all simple G-modules are geometrically rigid, though not in general absolutely rigid. More precisley, we show that if V is a simple G-module, then there is a finite purely inseparable extension k_V/k naturally attached to V such that V_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G turns on an investigation of algebras of the form Kotimes_k E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension k_V/k through an analogous version for artinian algebras. In the second half of the paper we apply recent results on the structure and representation theory of pseudo-reductive groups to gives a concrete description of k_V. Namely, we combine the main structure theorem of the Conrad--Prasad classification of pseudo-reductive G together with our previous high weight theory. For V a simple G-module, we calculate the minimal field of definition of the geometric Jacobson radical of End_G(V) in terms of the high weight of G and the Conrad--Prasad classification data; this gives a concrete construction of the field k_V as a subextension of the minimal field of definition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone the dimension formula for G we had previously established; we also use it to give a description of End_G(V) which includes a dimension formula.
设 k 是一个域，G 是一个光滑仿射 k 群，V 是一个有限维 G 模块。如果 G 对 V 的每个不可分解和子的作用的索序列和根序列一致，我们就说 V 是 emph{刚性的（rigid）；如果在把 G 和 V 改为 bar k（respect.~任何 k 的域扩展）之后，V 仍然是刚性的，我们就说 V 是 emph{几何刚性的（geometrically rigid）（respect.~emph{绝对刚性的（absolutely rigid））。我们证明了所有简单 G 模块都是几何刚性的，尽管一般来说不是绝对刚性的。更确切地说，我们证明了如果 V 是一个简单 G 模块，那么有一个无限的纯不可分的扩展 k_V/k 自然地连接到 V，使得 V_{k_V} 作为 G_{k_V} 模块是绝对刚性的。对连通 G 的证明依赖于对 K/otimes_k E 形式的代数的研究，其中 K 和 E 都是 k 的域扩展；我们给出了这样一个代数的例子，它作为自身的模块是不刚性的。我们通过对artinian代数的类似版本，建立了纯不可分场扩展k_V/k的存在性。在论文的后半部分，我们应用伪还原群的结构和表示理论的最新成果，给出了 k_V 的具体描述。也就是说，我们将伪还原 G 的康拉德--普拉萨德分类的主要结构定理与之前的高权重理论结合起来。对于简单的 G 模块 V，我们根据 G 的高权重和康拉德--普拉萨德分类数据计算了 End_G(V) 的几何雅各布森根的最小定义域；这给出了 k_V 作为 G 的几何单能根的最小定义域的子扩展的具体构造。我们还观察到康拉德--普拉萨德分类法可以用来兑现我们之前建立的 G 的维度公式；我们还用它给出了包含维度公式的 End_G(V) 的描述。
{"title":"Geometric rigidity of simple modules for algebraic groups","authors":"Michael Bate, David I. Stewart","doi":"arxiv-2409.05221","DOIUrl":"https://doi.org/arxiv-2409.05221","url":null,"abstract":"Let k be a field, let G be a smooth affine k-group and V a finite-dimensional\u0000G-module. We say V is emph{rigid} if the socle series and radical series\u0000coincide for the action of G on each indecomposable summand of V; say V is\u0000emph{geometrically rigid} (resp.~emph{absolutely rigid}) if V is rigid after\u0000base change of G and V to bar k (resp.~any field extension of k). We show that\u0000all simple G-modules are geometrically rigid, though not in general absolutely\u0000rigid. More precisley, we show that if V is a simple G-module, then there is a\u0000finite purely inseparable extension k_V/k naturally attached to V such that\u0000V_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G\u0000turns on an investigation of algebras of the form Kotimes_k E where K and E\u0000are field extensions of k; we give an example of such an algebra which is not\u0000rigid as a module over itself. We establish the existence of the purely\u0000inseparable field extension k_V/k through an analogous version for artinian\u0000algebras. In the second half of the paper we apply recent results on the structure and\u0000representation theory of pseudo-reductive groups to gives a concrete\u0000description of k_V. Namely, we combine the main structure theorem of the\u0000Conrad--Prasad classification of pseudo-reductive G together with our previous\u0000high weight theory. For V a simple G-module, we calculate the minimal field of\u0000definition of the geometric Jacobson radical of End_G(V) in terms of the high\u0000weight of G and the Conrad--Prasad classification data; this gives a concrete\u0000construction of the field k_V as a subextension of the minimal field of\u0000definition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone\u0000the dimension formula for G we had previously established; we also use it to\u0000give a description of End_G(V) which includes a dimension formula.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192574","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 develop methods for determining key properties (simplicity and the dimension of radical) of flip subalgebras in Matsuo algebras. These are interesting classes of commutative non-associative algebras that were introduced within the broader paradigm of axial algebras.
{"title":"Radicals in flip subalgebras","authors":"Bernardo G. Rodrigues, Sergey Shpectorov","doi":"arxiv-2409.05236","DOIUrl":"https://doi.org/arxiv-2409.05236","url":null,"abstract":"We develop methods for determining key properties (simplicity and the\u0000dimension of radical) of flip subalgebras in Matsuo algebras. These are\u0000interesting classes of commutative non-associative algebras that were\u0000introduced within the broader paradigm of axial algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192678","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 $H$ be a Hopf group coalgebra with a bijective antipode and $A$ an $H$-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of Hopf group coalgebras.
{"title":"Possion Hopf module Fundamental theorem for Hopf group coalgebras","authors":"Daowei Lu, Dingguo Wang","doi":"arxiv-2409.04687","DOIUrl":"https://doi.org/arxiv-2409.04687","url":null,"abstract":"Let $H$ be a Hopf group coalgebra with a bijective antipode and $A$ an\u0000$H$-comodule Poisson algebra. In this paper, we mainly generalize the\u0000fundamental theorem of Poisson Hopf modules to the case of Hopf group\u0000coalgebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192681","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 work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras that are not Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type must be nilpotent and the Hom-Lie algebra has trivial center. We also prove that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras -- similar to the double extension procedure for Lie algebras -- and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension.
{"title":"Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid","authors":"R. García-Delgado","doi":"arxiv-2409.04546","DOIUrl":"https://doi.org/arxiv-2409.04546","url":null,"abstract":"In this work we give an inductive way to construct quadratic Hom-Lie algebras\u0000with twist maps in the centroid. We focus on those Hom-Lie algebras that are\u0000not Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type\u0000must be nilpotent and the Hom-Lie algebra has trivial center. We also prove\u0000that there exists a maximal ideal containing the kernel and the image of the\u0000twist map. Then we state an inductive way to construct this type of Hom-Lie\u0000algebras -- similar to the double extension procedure for Lie algebras -- and\u0000prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist\u0000map in the centroid, which is not a Lie algebra, can be constructed using this\u0000type of double extension.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192682","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 study the modules $M$ every simple subfactors of which is a homomorphic image of $M$ and call them co-Kasch modules. These modules are dual to Kasch modules $M$ every simple subfactors of which can be embedded in $M$. We show that a module is co-Kasch if and only if every simple module in $sigma[M]$ is a homomorphic image of $M$. In particular, a projective right module $P$ is co-Kasch if and only if $P$ is a generator for $sigma[P]$. If $R$ is right max and right $H$-ring, then every right $R$-module is co-Kasch; and the converse is true for the rings whose simple right modules have locally artinian injective hulls. For a right artinian ring $R$, we prove that: (1) every finitely generated right $R$-module is co-Kasch if and only if every right $R$-module is a co-Kasch module if and only if $R$ is a right $H$-ring; and (2) every finitely generated projective right $R$-module is co-Kasch if and only if the Cartan matrix of $R$ is a diagonal matrix. For a Pr"ufer domain $R$, we prove that, every nonzero ideal of $R$ is co-Kasch if and only if $R$ is Dedekind. The structure of $mathbb{Z}$-modules that are co-Kasch is completely characterized.
{"title":"Co-Kasch Modules","authors":"Rafail Alizade, Engin Büyükaşık","doi":"arxiv-2409.04059","DOIUrl":"https://doi.org/arxiv-2409.04059","url":null,"abstract":"In this paper we study the modules $M$ every simple subfactors of which is a\u0000homomorphic image of $M$ and call them co-Kasch modules. These modules are dual\u0000to Kasch modules $M$ every simple subfactors of which can be embedded in $M$.\u0000We show that a module is co-Kasch if and only if every simple module in\u0000$sigma[M]$ is a homomorphic image of $M$. In particular, a projective right\u0000module $P$ is co-Kasch if and only if $P$ is a generator for $sigma[P]$. If\u0000$R$ is right max and right $H$-ring, then every right $R$-module is co-Kasch;\u0000and the converse is true for the rings whose simple right modules have locally\u0000artinian injective hulls. For a right artinian ring $R$, we prove that: (1)\u0000every finitely generated right $R$-module is co-Kasch if and only if every\u0000right $R$-module is a co-Kasch module if and only if $R$ is a right $H$-ring;\u0000and (2) every finitely generated projective right $R$-module is co-Kasch if and\u0000only if the Cartan matrix of $R$ is a diagonal matrix. For a Pr\"ufer domain\u0000$R$, we prove that, every nonzero ideal of $R$ is co-Kasch if and only if $R$\u0000is Dedekind. The structure of $mathbb{Z}$-modules that are co-Kasch is\u0000completely characterized.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192702","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 $(K,M)$ be a pair satisfying some mild condition, where $K$ is a class of $R$-modules and $M$ is a class of $R$-homomorphisms. We show that if $f:Arightarrow B$ and $g:Brightarrow A$ are $M$-embeddings and $A,B$ are $K_M$-injective, then $A$ is isomorphic to $B$, positively answering an question proposed by Marcos and Jiri [6].
{"title":"The Schröder-Bernstein problem for relative injective modules","authors":"Xiaolei Zhang","doi":"arxiv-2409.03972","DOIUrl":"https://doi.org/arxiv-2409.03972","url":null,"abstract":"Let $(K,M)$ be a pair satisfying some mild condition, where $K$ is a class\u0000of $R$-modules and $M$ is a class of $R$-homomorphisms. We show that if\u0000$f:Arightarrow B$ and $g:Brightarrow A$ are $M$-embeddings and $A,B$ are\u0000$K_M$-injective, then $A$ is isomorphic to $B$, positively answering an\u0000question proposed by Marcos and Jiri [6].","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192690","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}
Kevin Aguyar Brix, Adam Dor-On, Roozbeh Hazrat, Efren Ruiz
We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Va{v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Corti~nas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.
{"title":"Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebra","authors":"Kevin Aguyar Brix, Adam Dor-On, Roozbeh Hazrat, Efren Ruiz","doi":"arxiv-2409.03950","DOIUrl":"https://doi.org/arxiv-2409.03950","url":null,"abstract":"We prove that a unital shift equivalence induces a graded isomorphism of\u0000Leavitt path algebras when the shift equivalence satisfies an alignment\u0000condition. This yields another step towards confirming the Graded\u0000Classification Conjecture. Our proof uses the bridging bimodule developed by\u0000Abrams, the fourth-named author and Tomforde, as well as a general lifting\u0000result for graded rings that we establish here. This general result also allows\u0000us to provide simplified proofs of two important recent results: one\u0000independently proven by Arnone and Va{v s} through other means that the graded\u0000$K$-theory functor is full, and the other proven by Arnone and Corti~nas that\u0000there is no unital graded homomorphism between a Leavitt algebra and the path\u0000algebra of a Cuntz splice.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192684","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 work, we propose a systematic derivation of normal forms for dispersive equations using decorated trees introduced in arXiv:2005.01649. The key tool is the arborification map which is a morphism from the Butcher-Connes-Kreimer Hopf algebra to the Shuffle Hopf algebra. It originates from Ecalle's approach to dynamical systems with singularities. This natural map has been used in many applications ranging from algebra, numerical analysis and rough paths. This connection shows that Hopf algebras also appear naturally in the context of dispersive equations and provide insights into some crucial decomposition.
{"title":"Derivation of normal forms for dispersive PDEs via arborification","authors":"Yvain Bruned","doi":"arxiv-2409.03642","DOIUrl":"https://doi.org/arxiv-2409.03642","url":null,"abstract":"In this work, we propose a systematic derivation of normal forms for\u0000dispersive equations using decorated trees introduced in arXiv:2005.01649. The\u0000key tool is the arborification map which is a morphism from the\u0000Butcher-Connes-Kreimer Hopf algebra to the Shuffle Hopf algebra. It originates\u0000from Ecalle's approach to dynamical systems with singularities. This natural\u0000map has been used in many applications ranging from algebra, numerical analysis\u0000and rough paths. This connection shows that Hopf algebras also appear naturally\u0000in the context of dispersive equations and provide insights into some crucial\u0000decomposition.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192688","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 conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras.
{"title":"Fully noncentral Lie ideals and invariant additive subgroups in rings","authors":"Eusebio Gardella, Tsiu-Kwen Lee, Hannes Thiel","doi":"arxiv-2409.03362","DOIUrl":"https://doi.org/arxiv-2409.03362","url":null,"abstract":"We prove conditions ensuring that a Lie ideal or an invariant additive\u0000subgroup in a ring contains all additive commutators. A crucial assumption is\u0000that the subgroup is fully noncentral, that is, its image in every quotient is\u0000noncentral. For a unital algebra over a field of characteristic $neq 2$ where every\u0000additive commutator is a sum of square-zero elements, we show that a fully\u0000noncentral subspace is a Lie ideal if and only if it is invariant under all\u0000inner automorphisms. This applies in particular to zero-product balanced\u0000algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192687","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. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez
This paper focuses on defining an analog of differential-graded triangular matrix algebra in the context of differential-graded categories. Given two dg-categories $mathcal{U}$ and $mathcal{T}$ and $M in text{DgMod}(mathcal{U} otimes mathcal{T}^{text{op}})$, we construct the differential graded triangular matrix category $Lambda := left( begin{smallmatrix} mathcal{T} & 0 M & mathcal{U} end{smallmatrix} right)$. Our main result is that there is an equivalence of dg-categories between the dg-comma category $(text{DgMod}(mathcal{T}),text{GDgMod}(mathcal{U}))$ and the category $text{DgMod}left( left( begin{smallmatrix} mathcal{T} & 0 M & mathcal{U} end{smallmatrix} right)right)$. This result is an extension of a well-known result for Artin algebras (see, for example, [2,III.2].
{"title":"Diferential graded triangular matrix categories","authors":"M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez","doi":"arxiv-2409.03910","DOIUrl":"https://doi.org/arxiv-2409.03910","url":null,"abstract":"This paper focuses on defining an analog of differential-graded triangular\u0000matrix algebra in the context of differential-graded categories. Given two\u0000dg-categories $mathcal{U}$ and $mathcal{T}$ and $M in\u0000text{DgMod}(mathcal{U} otimes mathcal{T}^{text{op}})$, we construct the\u0000differential graded triangular matrix category $Lambda := left(\u0000begin{smallmatrix} mathcal{T} & 0 M & mathcal{U} end{smallmatrix}\u0000right)$. Our main result is that there is an equivalence of dg-categories\u0000between the dg-comma category\u0000$(text{DgMod}(mathcal{T}),text{GDgMod}(mathcal{U}))$ and the category\u0000$text{DgMod}left( left( begin{smallmatrix} mathcal{T} & 0 M &\u0000mathcal{U} end{smallmatrix} right)right)$. This result is an extension of a\u0000well-known result for Artin algebras (see, for example, [2,III.2].","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192685","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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