Isabelle Baraquin, Guillaume C'ebron, U. Franz, Laura Maassen, Moritz Weber
We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.
我们证明了酉对偶群(也称为布朗代数)的几个de Finetti定理。首先，我们给出了具有相同分布的$R$对角线元素的有限de Finetti定理。这是令人惊讶的，因为它适用于有限序列，而不是经典群和量子群的de Finetti定理;而且，它不涉及任何已知的独立性概念。其次，考虑到$W^*$-概率空间中的无限序列，我们的表征可以归结为算子值的自由中心圆元素，就像一元量子群$U_n^+$的情况一样。第三，上述de Finetti定理建立在对偶群作用的基础上，对偶群作用是将布朗代数视为对偶群时的自然作用。然而，我们也可以给布朗代数配备一个双代数作用，它在某种程度上更接近量子群设置。然后，我们得到了一个no-go de Finetti定理:在Brown代数的双代数作用下，在$W^*$-概率空间中，不变性产生零序列。另一方面，如果我们在$W^*$-概率空间中放弃忠实状态的假设，我们得到了一个类似对偶群作用的非平凡半de Finetti定理。
{"title":"De Finetti Theorems for the Unitary Dual Group","authors":"Isabelle Baraquin, Guillaume C'ebron, U. Franz, Laura Maassen, Moritz Weber","doi":"10.3842/SIGMA.2022.067","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.067","url":null,"abstract":"We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45516917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We give an analogue of the classical exponential map on Lie groups for Hopf ∗ -algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert C ∗ -bimodule of 12 densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups S 3 and Z , Woronowicz’s matrix quantum group C q [ SU 2 ] and the Sweedler–Taft algebra.
{"title":"The Exponential Map for Hopf Algebras","authors":"Ghaliah Alhamzi, E. Beggs","doi":"10.3842/SIGMA.2022.017","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.017","url":null,"abstract":". We give an analogue of the classical exponential map on Lie groups for Hopf ∗ -algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert C ∗ -bimodule of 12 densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups S 3 and Z , Woronowicz’s matrix quantum group C q [ SU 2 ] and the Sweedler–Taft algebra.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42010765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce the notion of a double Fock space of type B. We will show that this new construction is compatible with combinatorics of counting positive and negative inversions on a hyperoctahedral group.
{"title":"The Double Fock Space of Type B","authors":"Marek Bo.zejko, W. Ejsmont","doi":"10.3842/SIGMA.2023.040","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.040","url":null,"abstract":"In this article, we introduce the notion of a double Fock space of type B. We will show that this new construction is compatible with combinatorics of counting positive and negative inversions on a hyperoctahedral group.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"31 13","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $rle 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.
我们完成了一个定理的证明，我们在[数学]中宣布并部分证明了它。数学学报，2004,69-90，数学. ag /0111299。这个定理涉及曲面族上的曲线族。它有两部分。第一个在那篇论文中得到了证明。它描述了一个自然循环，枚举族中恰好有$r$普通节点的曲线。第二部分在此得到证明。证明了对于rle 8$，这个循环的类是由族的陈氏类的积的参数空间的压下中的一个可计算的普适多项式给出的。
{"title":"Node Polynomials for Curves on Surfaces","authors":"S. Kleiman, R. Piene","doi":"10.3842/SIGMA.2022.059","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.059","url":null,"abstract":"We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $rle 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45033892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence and uniqueness of weighted ambient metrics and weighted Poincaré metrics for smooth metric measure spaces.
我们证明了光滑度量空间的加权环境度量和加权Poincaré度量的存在性和唯一性。
{"title":"The Weighted Ambient Metric","authors":"Jeffrey S. Case, Ayush Khaitan","doi":"10.3842/SIGMA.2022.086","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.086","url":null,"abstract":"We prove the existence and uniqueness of weighted ambient metrics and weighted Poincaré metrics for smooth metric measure spaces.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49337964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It was recently shown (by the second author and D'{i}az Garc'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $mathcal{O}_q(G/L_S)$ admits a unique $mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and D'{i}az Garc'{i}a. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.
{"title":"Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds","authors":"A. Carotenuto, R. O. Buachalla","doi":"10.3842/SIGMA.2022.070","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.070","url":null,"abstract":"It was recently shown (by the second author and D'{i}az Garc'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $mathcal{O}_q(G/L_S)$ admits a unique $mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and D'{i}az Garc'{i}a. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43181557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive an expansion for the fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of ''flat rings''. These internal and external flat-ring harmonic functions are expressed in terms of simply-periodic Lamé functions. In a limiting case we obtain the expansion of the fundamental solution in toroidal coordinates.
{"title":"Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Cyclide Coordinates","authors":"L. Bi, H. Cohl, H. Volkmer","doi":"10.3842/SIGMA.2022.041","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.041","url":null,"abstract":"We derive an expansion for the fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of ''flat rings''. These internal and external flat-ring harmonic functions are expressed in terms of simply-periodic Lamé functions. In a limiting case we obtain the expansion of the fundamental solution in toroidal coordinates.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45839786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.
我们使用所陈述的skein代数构造了具有一个边界分量的紧连通定向曲面的映射子群的有限维投影表示。
{"title":"Mapping Class Group Representations Derived from Stated Skein Algebras","authors":"J. Korinman","doi":"10.3842/SIGMA.2022.064","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.064","url":null,"abstract":"We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49106489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=langle alpha_{i}^{x}p_{ij}(x),, i=1,dots, n,, j=1,dots, n_{i}rangle$, where $alpha_{i}in{mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $check{S}^{dagger}_{W}$ of the quotient difference operator $check{S}_{W}$ satisfying $widehat{S} =check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $dim W$ annihilating $W$, and $widehat{S}$ is a linear difference operator with constant coefficients depending on $alpha_{i}$ and $deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $operatorname{ord} check{S}^{dagger}_{W}$, which is annihilated by $check{S}^{dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $zinmathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(mathfrak{gl}_{k},mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.
{"title":"Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians","authors":"F. Uvarov","doi":"10.3842/SIGMA.2022.081","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.081","url":null,"abstract":"We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=langle alpha_{i}^{x}p_{ij}(x),, i=1,dots, n,, j=1,dots, n_{i}rangle$, where $alpha_{i}in{mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $check{S}^{dagger}_{W}$ of the quotient difference operator $check{S}_{W}$ satisfying $widehat{S} =check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $dim W$ annihilating $W$, and $widehat{S}$ is a linear difference operator with constant coefficients depending on $alpha_{i}$ and $deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $operatorname{ord} check{S}^{dagger}_{W}$, which is annihilated by $check{S}^{dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $zinmathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(mathfrak{gl}_{k},mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49637092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an open quantum system with Hamiltonian $H_S$ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature $beta$. We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, that turns out to be unique and faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of $H_S$.
{"title":"The Generalized Fibonacci Oscillator as an Open Quantum System","authors":"F. Fagnola, C. Ko, H. Yoo","doi":"10.3842/SIGMA.2022.035","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.035","url":null,"abstract":"We consider an open quantum system with Hamiltonian $H_S$ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature $beta$. We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, that turns out to be unique and faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of $H_S$.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49264802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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