The Experts below are selected from a list of 143799 Experts worldwide ranked by ideXlab platform
Pavel Pyatov - One of the best experts on this subject based on the ideXlab platform.
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spectral extension of the Quantum Group cotangent bundle
Communications in Mathematical Physics, 2009Co-Authors: A P Isaev, Pavel PyatovAbstract:The structure of a cotangent bundle is investigated for Quantum linear Groups GL q (n) and SL q (n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SL q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.
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spectral extension of the Quantum Group cotangent bundle
arXiv: Quantum Algebra, 2008Co-Authors: A P Isaev, Pavel PyatovAbstract:The structure of a cotangent bundle is investigated for Quantum linear Groups GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SLq(n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators -- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SLq(n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.
Roland Vergnioux - One of the best experts on this subject based on the ideXlab platform.
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orthogonal free Quantum Group factors are strongly 1 bounded
Advances in Mathematics, 2018Co-Authors: Michael Brannan, Roland VergniouxAbstract:Abstract We prove that the orthogonal free Quantum Group factors L ( F O N ) are strongly 1-bounded in the sense of Jung. In particular, they are not isomorphic to free Group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the Quantum Cayley tree associated to F O N , and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.
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the connes embedding property for Quantum Group von neumann algebras
Transactions of the American Mathematical Society, 2016Co-Authors: Benoit Collins, Michael Brannan, Roland VergniouxAbstract:For a compact Quantum Group G of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^∞(G) into an ultrapower of the hyperfinite II_1-factor (the Connes embedding property for L^∞(G)). We establish a connection between the Connes embedding property for L^∞(G) and the structure of certain Quantum subGroups of G, and use this to prove that the II_1-factors L^∞(O_N^+) and L^∞(U_N^+) associated to the free orthogonal and free unitary Quantum Groups have the Connes embedding property for all N >= 4. As an application, we deduce that the free entropy dimension of the standard generators of L^∞(O_N^+) equals 1 for all N >= 4. We also mention an application of our work to the problem of classifying the Quantum subGroups of O_N^+.
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the connes embedding property for Quantum Group von neumann algebras
arXiv: Operator Algebras, 2014Co-Authors: Benoit Collins, Michael Brannan, Roland VergniouxAbstract:For a compact Quantum Group $\mathbb G$ of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra $L^\infty(\mathbb G)$ into an ultrapower of the hyperfinite II$_1$-factor (the Connes embedding property for $L^\infty(\mathbb G)$). We establish a connection between the Connes embedding property for $L^\infty(\mathbb G)$ and the structure of certain Quantum subGroups of $\mathbb G$, and use this to prove that the II$_1$-factors $L^\infty(O_N^+)$ and $L^\infty(U_N^+)$ associated to the free orthogonal and free unitary Quantum Groups have the Connes embedding property for all $N \ge 4$. As an application, we deduce that the free entropy dimension of the standard generators of $L^\infty(O_N^+)$ equals $1$ for all $N \ge 4$. We also mention an application of our work to the problem of classifying the Quantum subGroups of $O_N^+$.
Ke-jia Zhang - One of the best experts on this subject based on the ideXlab platform.
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A novel Quantum Group signature scheme without using entangled states
Quantum Information Processing, 2015Co-Authors: Guang-bao Xu, Ke-jia ZhangAbstract:In this paper, we propose a novel Quantum Group signature scheme. It can make the signer sign a message on behalf of the Group without the help of Group manager (the arbitrator), which is different from the previous schemes. In addition, a signature can be verified again when its signer disavows she has ever generated it. We analyze the validity and the security of the proposed signature scheme. Moreover, we discuss the advantages and the disadvantages of the new scheme and the existing ones. The results show that our scheme satisfies all the characteristics of a Group signature and has more advantages than the previous ones. Like its classic counterpart, our scheme can be used in many application scenarios, such as e-government and e-business.
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a secure Quantum Group signature scheme based on bell states
Physica Scripta, 2013Co-Authors: Hui-juan Zuo, Ting-ting Song, Ke-jia Zhang, Weiwei ZhangAbstract:In this paper, we propose a new secure Quantum Group signature with Bell states, which may have applications in e-payment system, e-government, e-business, etc. Compared with the recent Quantum Group signature protocols, our scheme is focused on the most general situation in practice, i.e. only the arbitrator is trusted and no intermediate information needs to be stored in the signing phase to ensure the security. Furthermore, our scheme has achieved all the characteristics of Group signature—anonymity, verifiability, traceability, unforgetability and undeniability, by using some current developed Quantum and classical technologies. Finally, a feasible security analysis model for Quantum Group signature is presented.
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Cryptanalysis of the Quantum Group Signature Protocols
International Journal of Theoretical Physics, 2013Co-Authors: Ke-jia Zhang, Ting-ting Song, Ying Sun, Hui-juan ZuoAbstract:Recently, the researches of Quantum Group signature (QGS) have attracted a lot of attentions and some typical protocols have been designed for e-payment system, e-government, e-business, etc. In this paper, we analyze the security of the Quantum Group signature with the example of two novel protocols. It can be seen that both of them cannot be implemented securely since the arbitrator cannot solve the disputes fairly. In order to show that, some possible attack strategies, which can be used by the malicious participants, are proposed. Moreover, the further discussions of QGS are presented finally, including some insecurity factors and improved ideas.
Michael Brannan - One of the best experts on this subject based on the ideXlab platform.
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orthogonal free Quantum Group factors are strongly 1 bounded
Advances in Mathematics, 2018Co-Authors: Michael Brannan, Roland VergniouxAbstract:Abstract We prove that the orthogonal free Quantum Group factors L ( F O N ) are strongly 1-bounded in the sense of Jung. In particular, they are not isomorphic to free Group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the Quantum Cayley tree associated to F O N , and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.
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the connes embedding property for Quantum Group von neumann algebras
Transactions of the American Mathematical Society, 2016Co-Authors: Benoit Collins, Michael Brannan, Roland VergniouxAbstract:For a compact Quantum Group G of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^∞(G) into an ultrapower of the hyperfinite II_1-factor (the Connes embedding property for L^∞(G)). We establish a connection between the Connes embedding property for L^∞(G) and the structure of certain Quantum subGroups of G, and use this to prove that the II_1-factors L^∞(O_N^+) and L^∞(U_N^+) associated to the free orthogonal and free unitary Quantum Groups have the Connes embedding property for all N >= 4. As an application, we deduce that the free entropy dimension of the standard generators of L^∞(O_N^+) equals 1 for all N >= 4. We also mention an application of our work to the problem of classifying the Quantum subGroups of O_N^+.
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the connes embedding property for Quantum Group von neumann algebras
arXiv: Operator Algebras, 2014Co-Authors: Benoit Collins, Michael Brannan, Roland VergniouxAbstract:For a compact Quantum Group $\mathbb G$ of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra $L^\infty(\mathbb G)$ into an ultrapower of the hyperfinite II$_1$-factor (the Connes embedding property for $L^\infty(\mathbb G)$). We establish a connection between the Connes embedding property for $L^\infty(\mathbb G)$ and the structure of certain Quantum subGroups of $\mathbb G$, and use this to prove that the II$_1$-factors $L^\infty(O_N^+)$ and $L^\infty(U_N^+)$ associated to the free orthogonal and free unitary Quantum Groups have the Connes embedding property for all $N \ge 4$. As an application, we deduce that the free entropy dimension of the standard generators of $L^\infty(O_N^+)$ equals $1$ for all $N \ge 4$. We also mention an application of our work to the problem of classifying the Quantum subGroups of $O_N^+$.
Stefaan Vaes - One of the best experts on this subject based on the ideXlab platform.
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the unitary implementation of a locally compact Quantum Group action
Journal of Functional Analysis, 2001Co-Authors: Stefaan VaesAbstract:Abstract In this paper we study actions of locally compact Quantum Groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact Group action. This result is an important tool in the study of Quantum Groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact Quantum Group ( M , Δ ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N α ⊂ N ↪ M α ⋉ N is a basic construction. Here N α denotes the fixed point algebra and M α ⋉ N is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion N α ⊂ N is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal. 137 , 466–543; 1998, J. Funct. Anal. 154 , 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand. 84 , 297–319): every integrable outer action with infinite fixed point algebra is a dual action.
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the unitary implementation of a locally compact Quantum Group action
arXiv: Operator Algebras, 2000Co-Authors: Stefaan VaesAbstract:In this paper we study actions of locally compact Quantum Groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact Group action. This result is an important tool in the study of Quantum Groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When alpha is an action of a locally compact Quantum Group on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion of the fixed point algebra in the algebra N in the crossed product, is a basic construction. When alpha is an outer and integrable action on a factor N we prove that the inclusion of the fixed point algebra in the algebra N is irreducible, of depth 2 and regular, giving a converse to the results of Enock and Nest. Finally we prove the equivalence of minimal and outer actions and we generalize a theorem of Yamanouchi: every integrable outer action with infinite fixed point algebra is a dual action.
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locally compact Quantum Groups in the von neumann algebraic setting
arXiv: Operator Algebras, 2000Co-Authors: Johan Kustermans, Stefaan VaesAbstract:In this paper we complete in several aspects the picture of locally compact Quantum Groups. First of all we give a definition of a locally compact Quantum Group in the von Neumann algebraic setting and show how to deduce from it a C*-algebraic Quantum Group. Further we prove several results about locally compact Quantum Groups which are important for applications, but were not yet settled in our paper "Locally compact Quantum Groups". We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact Quantum Group with its dual. Loosely speaking we show how the antipode of the locally compact Quantum Group determines the modular Group and modular conjugation of the dual locally compact Quantum Group.
