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De Finetti Theorems for the Unitary Dual Group 酉对偶群的De Finetti定理
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-11 DOI: 10.3842/SIGMA.2022.067
Isabelle Baraquin, Guillaume C'ebron, U. Franz, Laura Maassen, Moritz Weber
{"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":null,"pages":null},"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}
引用次数: 1
The Exponential Map for Hopf Algebras Hopf代数的指数映射
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-09 DOI: 10.3842/SIGMA.2022.017
Ghaliah Alhamzi, E. Beggs
{"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":null,"pages":null},"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}
引用次数: 0
The Double Fock Space of Type B B型双垛空间
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-28 DOI: 10.3842/SIGMA.2023.040
Marek Bo.zejko, W. Ejsmont
{"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":null,"pages":null},"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}
引用次数: 1
Node Polynomials for Curves on Surfaces 曲面上曲线的节点多项式
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-23 DOI: 10.3842/SIGMA.2022.059
S. Kleiman, R. Piene
{"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":null,"pages":null},"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}
引用次数: 0
The Weighted Ambient Metric 加权环境度量
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-22 DOI: 10.3842/SIGMA.2022.086
Jeffrey S. Case, Ayush Khaitan
{"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":null,"pages":null},"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}
引用次数: 3
Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds 不可约量子旗流形上相对线模的双模连接
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-20 DOI: 10.3842/SIGMA.2022.070
A. Carotenuto, R. O. Buachalla
{"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":null,"pages":null},"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}
引用次数: 2
Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Cyclide Coordinates 拉普拉斯方程的一个基本解在平环坐标系中的展开式
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-17 DOI: 10.3842/SIGMA.2022.041
L. Bi, H. Cohl, H. Volkmer
{"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":null,"pages":null},"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}
引用次数: 2
Mapping Class Group Representations Derived from Stated Skein Algebras 从有状态Skein代数导出的映射类群表示
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-15 DOI: 10.3842/SIGMA.2022.064
J. Korinman
{"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":null,"pages":null},"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}
引用次数: 2
Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians 三角Gaudin算子和动力学Hamiltonian算子的差分算子和对偶性
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-13 DOI: 10.3842/SIGMA.2022.081
F. Uvarov
{"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":null,"pages":null},"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}
引用次数: 1
The Generalized Fibonacci Oscillator as an Open Quantum System 开放量子系统的广义斐波那契振子
IF 0.9 3区 物理与天体物理
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-02-04 DOI: 10.3842/SIGMA.2022.035
F. Fagnola, C. Ko, H. Yoo
{"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":null,"pages":null},"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}
引用次数: 1
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