Following Riley's work, for each $2$-bridge link $K(r)$ of slope $r∈mathbb{R}$ and an integer or a half-integer $n$ greater than $1$, we introduce the Heckoid orbifold $S(r;n)$ and the Heckoid group $G(r;n)=pi_1(S(r;n))$ of index $n$ for $K(r)$ . When $n$ is an integer, $S(r;n)$ is called an even Heckoid orbifold; in this case, the underlying space is the exterior of $K(r)$, and the singular set is the lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a $4$-punctured sphere $S$ in $S(r;n)$ determined by the $2$-bridge sphere of $K(r)$, when is it null-homotopic in $S(r;n)$? (2) For two distinct essential simple loops on $S$, when are they homotopic in $S(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from $2$-bridge link groups onto Heckoid groups.
{"title":"SIMPLE LOOPS ON 2-BRIDGE SPHERES IN HECKOID ORBIFOLDS FOR 2-BRIDGE LINKS","authors":"Donghi Lee, M. Sakuma","doi":"10.3934/ERA.2012.19.97","DOIUrl":"https://doi.org/10.3934/ERA.2012.19.97","url":null,"abstract":"Following Riley's work, \u0000for each $2$-bridge link $K(r)$ of slope $r∈mathbb{R}$ \u0000and an integer or a half-integer $n$ greater than $1$, \u0000we introduce the Heckoid orbifold $S(r;n)$ and the Heckoid group $G(r;n)=pi_1(S(r;n))$ of \u0000index $n$ for $K(r)$ . \u0000When $n$ is an integer, \u0000$S(r;n)$ is called an even Heckoid orbifold; \u0000in this case, the underlying space is the exterior of $K(r)$, \u0000and the singular set is the lower tunnel of $K(r)$ with index $n$. \u0000The main purpose of this note is to announce answers to \u0000the following questions for even Heckoid orbifolds. \u0000(1) For an essential simple loop on a $4$-punctured sphere $S$ \u0000in $S(r;n)$ determined by the $2$-bridge sphere of $K(r)$, \u0000when is it null-homotopic in $S(r;n)$? \u0000(2) For two distinct essential simple loops \u0000on $S$, when are they homotopic in $S(r;n)$? \u0000We also announce applications of these results to \u0000character varieties, McShane's identity, and \u0000epimorphisms from $2$-bridge link groups onto Heckoid groups.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"28 1","pages":"97-111"},"PeriodicalIF":0.0,"publicationDate":"2012-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232791","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 deals with some semi-spectral representations of logmodular algebras. More exactly, we characterize such representations by the corresponding scalar semi-spectral measures. In the case of a logmodular algebra we obtain, for $0
{"title":"Operator representations of logmodular algebras which admit $gamma-$spectral $rho-$dilations","authors":"A. Juratoni, F. Pater, O. Bundau","doi":"10.3934/ERA.2012.19.49","DOIUrl":"https://doi.org/10.3934/ERA.2012.19.49","url":null,"abstract":"This paper deals with some semi-spectral representations of \u0000logmodular algebras. More exactly, we characterize such \u0000representations by the corresponding scalar semi-spectral measures. \u0000In the case of a logmodular algebra we obtain, for $0<rho leq 1,$ \u0000several results which generalize the corresponding results of \u0000Foias-Suciu [2] in the case $rho =1.$","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"19 1","pages":"49-57"},"PeriodicalIF":0.0,"publicationDate":"2012-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232997","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 introduce a general construction for a correspondence between certain Automorphic representations in classical groups. This construction is based on the method of small representations, which we use to construct examples of CAP representations.
{"title":"Constructing automorphic representations in split classicalgroups","authors":"D. Ginzburg","doi":"10.3934/ERA.2012.19.18","DOIUrl":"https://doi.org/10.3934/ERA.2012.19.18","url":null,"abstract":"In this paper we introduce a general construction for a \u0000correspondence between certain Automorphic representations in \u0000classical groups. This construction is based on the method of small \u0000representations, which we use to construct examples of CAP \u0000representations.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"19 1","pages":"18-32"},"PeriodicalIF":0.0,"publicationDate":"2012-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232279","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 note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $Gamma$ and a boundary action $Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $rho:Gammato bf{H}$, defines a special homomorphism $W_{{Gamma}{B}}to Weyl_{bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.
{"title":"Boundaries, Weyl groups, and Superrigidity","authors":"U. Bader, A. Furman","doi":"10.3934/ERA.2012.19.41","DOIUrl":"https://doi.org/10.3934/ERA.2012.19.41","url":null,"abstract":"This note describes a unified approach to several superrigidity results, old and new, \u0000 concerning representations of lattices into simple algebraic groups over local fields. \u0000 For an arbitrary group $Gamma$ and a boundary action $Gamma$ ↷ $B$ \u0000 we associate a certain generalized Weyl group $W_{{Gamma}{B}}$ and show that any \u0000 representation with a Zariski dense unbounded image in a simple algebraic group, \u0000 $rho:Gammato bf{H}$, \u0000 defines a special homomorphism $W_{{Gamma}{B}}to Weyl_{bf H}$. \u0000 This general fact allows the deduction of the aforementioned superrigidity results.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"54 1","pages":"41-48"},"PeriodicalIF":0.0,"publicationDate":"2011-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232346","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 characterize order preserving transforms on the class of lower-semi-continuous convex functions that are defined on a convex subset of $mathbb{R}^n$ (a "window") and some of its variants. To this end, we investigate convexity preserving maps on subsets of $mathbb{R}^n$. We prove that, in general, an order isomorphism is induced by a special convexity preserving point map on the epi-graph of the function. In the case of non-negative convex functions on $K$, where $0in K$ and $f(0) = 0$, one may naturally partition the set of order isomorphisms into two classes; we explain the main ideas behind these results.
我们刻画了在$mathbb{R}^n$(一个“窗口”)的凸子集及其变体上定义的下半连续凸函数类上的保序变换。为此，我们研究了$mathbb{R}^n$子集上的保凸映射。在一般情况下，我们证明了一个序同构是由一个特殊的保凸点映射在函数的外延图上引起的。对于K$上的非负凸函数，其中$0 In K$且$f(0) = 0$，可以很自然地将序同构集划分为两类;我们将解释这些结果背后的主要思想。
{"title":"Order isomorphisms in windows","authors":"S. Artstein-Avidan, D. Florentin, V. Milman","doi":"10.3934/ERA.2011.18.112","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.112","url":null,"abstract":"We characterize order preserving transforms on the class of \u0000lower-semi-continuous convex functions that are defined on a convex \u0000subset of $mathbb{R}^n$ (a \"window\") and some of its variants. To this \u0000end, we investigate convexity preserving maps on subsets of $mathbb{R}^n$. \u0000We prove that, in general, an order isomorphism is induced by a \u0000special convexity preserving point map on the epi-graph of the \u0000function. In the case of non-negative convex functions on $K$, where \u0000$0in K$ and $f(0) = 0$, one may naturally partition the set of \u0000order isomorphisms into two classes; we explain the main ideas \u0000behind these results.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"112-118"},"PeriodicalIF":0.0,"publicationDate":"2011-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232551","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 $T:C^1(RR)to C(RR)$ be an operator satisfying the derivation equation $T(fcdot g)=(Tf)cdot g + f cdot (Tg),$ where $f,gin C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form $(Tf)(x) = c(x) , f'(x) + d(x) , f(x) , ln |f(x)|$ for $f in C^1(RR), x in RR$, where $c, d in C(RR)$ are suitable continuous functions, with the convention $0 ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f ln |f|$. We can also determine the solutions of the generalized derivation equation $T(fcdot g)=(Tf)cdot (A_1g) + (A_2f) cdot (Tg), $ where $f,gin C^1(RR)$, for operators $T:C^1(RR)to C(RR)$ and $A_1, A_2:C(RR)to C(RR)$ fulfilling some weak additional properties.
设$T:C^1(RR)到C(RR)$是一个满足微分方程$T(fcdot g)=(Tf)cdot g + fcdot (Tg)的算子，$ where $f,gin C^1(RR)$，以及一些弱附加假设。那么$T$必须是$(Tf)(x) = c(x) ， f'(x) + d(x) ， f(x) ， ln |f(x)|$对于$f in c ^1(RR)， x in RR$，其中$c, d in c(RR)$是合适的连续函数，约定$0 ln 0 = 0$。如果假设$T$的定义域为$C(RR)$，则$C =0$，而$T$本质上由熵函数$f ln |f|$给出。我们还可以确定广义导数方程$T(fcdot g)=(Tf)cdot (A_1g) + (A_2f) cdot (Tg)的解，$其中$f,gin C^1(RR)$，对于算子$T:C^1(RR)to C(RR)$和$A_1, A_2:C(RR)to C(RR)$满足一些弱附加性质。
{"title":"Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$","authors":"Hermann Köenig, V. Milman","doi":"10.3934/ERA.2011.18.54","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.54","url":null,"abstract":"Let $T:C^1(RR)to C(RR)$ be an operator satisfying the derivation equation $T(fcdot g)=(Tf)cdot g + f cdot (Tg),$ where $f,gin C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form $(Tf)(x) = c(x) , f'(x) + d(x) , f(x) , ln |f(x)|$ for $f in C^1(RR), x in RR$, where $c, d in C(RR)$ are suitable continuous functions, with the convention $0 ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f ln |f|$. We can also determine the solutions of the generalized derivation equation $T(fcdot g)=(Tf)cdot (A_1g) + (A_2f) cdot (Tg), $ where $f,gin C^1(RR)$, for operators $T:C^1(RR)to C(RR)$ and $A_1, A_2:C(RR)to C(RR)$ fulfilling some weak additional properties.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"54-60"},"PeriodicalIF":0.0,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232231","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}
An element of a free Leibniz algebra is called Jordan if it belongs to a free Leibniz-Jordan subalgebra. Elements of the Jordan commutant of a free Leibniz algebra are called weak Jordan. We prove that an element of a free Leibniz algebra over a field of characteristic 0 is weak Jordan if and only if it is left-central. We show that free Leibniz algebra is an extension of a free Lie algebra by left-center. We find the dimensions of the homogeneous components of the Jordan commutant and the base of its multilinear part. We find criterion for an element of free Leibniz algebra to be Jordan.
{"title":"Jordan elements and Left-Center of a Free Leibniz algebra","authors":"A. Dzhumadil'daev","doi":"10.3934/ERA.2011.18.31","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.31","url":null,"abstract":"An element of a free Leibniz algebra is called Jordan if it belongs to a free Leibniz-Jordan subalgebra. Elements of the Jordan commutant of a free Leibniz algebra are called weak Jordan. We prove that an element of a free Leibniz algebra over a field of characteristic 0 is weak Jordan if and only if it is left-central. We show that free Leibniz algebra is an extension of a free Lie algebra by left-center. We find the dimensions of the homogeneous components of the Jordan commutant and the base of its multilinear part. We find criterion for an element of free Leibniz algebra to be Jordan.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"31-49"},"PeriodicalIF":0.0,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232221","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}
Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.
{"title":"Realization of joint spectral radius via Ergodic theory","authors":"Xiongping Dai, Yu Huang, Mingqing Xiao","doi":"10.3934/ERA.2011.18.22","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.22","url":null,"abstract":"Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that \u0000the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"8 1","pages":"22-30"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3934/ERA.2011.18.22","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232175","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 degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate. In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.
{"title":"Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes","authors":"Benjamin P. Mirabelli, Maksim Maydanskiy","doi":"10.3934/era.2011.18.131","DOIUrl":"https://doi.org/10.3934/era.2011.18.131","url":null,"abstract":"The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate. \u0000In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"131-143"},"PeriodicalIF":0.0,"publicationDate":"2011-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232162","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 describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key role in our approach is played by a transitive action of the automorphism group of the free algebra $ k $ on the Calogero-Moser varieties $ CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for $ A_1(k) $ to the class of Morita equivalent algebras.
我们描述了与第一Weyl代数等价的代数Morita自同构群的结构。特别地，我们利用群作用于图的Bass-Serre理论，用合并积的形式给出了这些群的几何表示。自由代数$ k $的自同构群对在[5]中定义的Calogero-Moser变元$ CC_n $的传递作用在我们的方法中发挥了关键作用。最后，我们提出了$ A_1(k) $的Dixmier猜想到Morita等价代数类的一个自然推广。
{"title":"On subgroups of the Dixmier group and Calogero-Moser spaces","authors":"Y. Berest, A. Eshmatov, F. Eshmatov","doi":"10.3934/ERA.2011.18.12","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.12","url":null,"abstract":"We describe the structure of the automorphism groups of algebras \u0000Morita equivalent to the first Weyl algebra $ A_1(k) $. \u0000In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key role in our approach is played by a transitive action of the automorphism group of the free algebra $ k $ on the Calogero-Moser varieties $ CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture \u0000for $ A_1(k) $ to the class of Morita equivalent algebras.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"12-21"},"PeriodicalIF":0.0,"publicationDate":"2011-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232612","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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