Displaying similar documents to “Exponentiations over the quantum algebra U q ( s l 2 ( ) )

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000-2001)

Séminaire Équations aux dérivées partielles

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We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function u solving the eikonal equation aėȧnd a probability measure σ solving a related transport equation. We present some elementary formal identities relating certain quantum states ψ and u , σ . We show also how...

Covariantization of quantized calculi over quantum groups

Seyed Ebrahim Akrami, Shervin Farzi (2020)

Mathematica Bohemica

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We introduce a method for construction of a covariant differential calculus over a Hopf algebra A from a quantized calculus d a = [ D , a ] , a A , where D is a candidate for a Dirac operator for A . We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra A . We apply this method to the Dirac operator for the quantum SL ( 2 ) given by S. Majid. We find that the differential calculus obtained by our...

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of M N -spaces needed to represent (up to a constant C > 1 ) the M N -version of the n -dimensional operator Hilbert space O H n as a direct sum of copies of M N . We show that, when C is close to 1, this multiplicity grows...

Right coideal subalgebras of U q + ( 𝔰𝔬 2 n + 1 )

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

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We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group U q + ( 𝔰𝔬 2 n + 1 ) provided that q is not a root of 1. If q has a finite multiplicative order t > 4 ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel u q + ( 𝔰𝔬 2 n + 1 ) . In particular, the total number of right coideal subalgebras that contain the coradical equals ( 2 n ) ! ! ; the order of the Weyl group defined by the root system of type B n .

C * -basic construction between non-balanced quantum doubles

Qiaoling Xin, Tianqing Cao (2024)

Czechoslovak Mathematical Journal

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For finite groups X , G and the right G -action on X by group automorphisms, the non-balanced quantum double D ( X ; G ) is defined as the crossed product ( X op ) * G . We firstly prove that D ( X ; G ) is a finite-dimensional Hopf C * -algebra. For any subgroup H of G , D ( X ; H ) can be defined as a Hopf C * -subalgebra of D ( X ; G ) in the natural way. Then there is a conditonal expectation from D ( X ; G ) onto D ( X ; H ) and the index is [ G ; H ] . Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the...

Remarks on Sekine quantum groups

Jialei Chen, Shilin Yang (2022)

Czechoslovak Mathematical Journal

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We first describe the Sekine quantum groups 𝒜 k (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of 𝒜 k and describe their representation rings r ( 𝒜 k ) . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of r ( 𝒜 k ) .

A class of quantum doubles of pointed Hopf algebras of rank one

Hua Sun, Yueming Li (2023)

Czechoslovak Mathematical Journal

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We construct a class of quantum doubles D ( H D n ) of pointed Hopf algebras of rank one H 𝒟 . We describe the algebra structures of D ( H D n ) by generators with relations. Moreover, we give the comultiplication Δ D , counit ε D and the antipode S D , respectively.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

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We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β 1 V ( B β ) . For the interaction parameter β ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for β ( 0 , 3 / 5 ) in [28]. This allows, in the case β ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon...

The Grothendieck ring of quantum double of quaternion group

Hua Sun, Jia Pang, Yanxi Shen (2024)

Czechoslovak Mathematical Journal

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Let 𝕜 be an algebraically closed field of characteristic p 2 , and let Q 8 be the quaternion group. We describe the structures of all simple modules over the quantum double D ( 𝕜 Q 8 ) of group algebra 𝕜 Q 8 . Moreover, we investigate the tensor product decomposition rules of all simple D ( 𝕜 Q 8 ) -modules. Finally, we describe the Grothendieck ring G 0 ( D ( 𝕜 Q 8 ) ) by generators with relations.

On a cubic Hecke algebra associated with the quantum group U q ( 2 )

Janusz Wysoczański (2010)

Banach Center Publications

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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group U q ( 2 ) , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators h j : = I j α I n - 2 - j on ( ³ ) n with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra q , n ( 2 ) associated with the quantum group U q ( 2 ) . The purpose of this note is to present the construction.

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let G be a finite group and H a subgroup. Denote by D ( G ; H ) (or D ( G ) ) the crossed product of C ( G ) and H (or G ) with respect to the adjoint action of the latter on the former. Consider the algebra D ( G ) , e generated by D ( G ) and e , where we regard E as an idempotent operator e on D ( G ) for a certain conditional expectation E of D ( G ) onto D ( G ; H ) . Let us call D ( G ) , e the basic construction from the conditional expectation E : D ( G ) D ( G ; H ) . The paper constructs a crossed product algebra C ( G / H × G ) G , and proves that there is an algebra isomorphism between...

Annihilator ideals of finite dimensional simple modules of two-parameter quantized enveloping algebra U r , s ( 𝔰𝔩 2 )

Yu Wang, Xiaoming Li (2023)

Czechoslovak Mathematical Journal

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Let U be the two-parameter quantized enveloping algebra U r , s ( 𝔰𝔩 2 ) and F ( U ) the locally finite subalgebra of U under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of F ( U ) in the case when r s - 1 is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of U by generators.

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areif the Newton polygons of the corresponding partition functions coincide up to translation....