Displaying similar documents to “Quantum detailed balance conditions with time reversal: the finite-dimensional case”

On two quantum versions of the detailed balance condition

Franco Fagnola, Veronica Umanità (2010)

Banach Center Publications

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Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form x , y s : = t r ( ρ 1 - s x * ρ s y ) (s ∈ [0,1])...

Quantum 4-sphere: the infinitesimal approach

F. Bonechi, M. Tarlini, N. Ciccoli (2003)

Banach Center Publications

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We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to q ( S U ( 2 ) ) .

Free dynamical quantum groups and the dynamical quantum group S U Q d y n ( 2 )

Thomas Timmermann (2012)

Banach Center Publications

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We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical S U q ( 2 ) studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

An idempotent for a Jordanian quantum complex sphere

Bartosz Zieliński (2003)

Banach Center Publications

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A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group S L h ( 2 ) .

The geometric reductivity of the quantum group S L q ( 2 )

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

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We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants A G is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple,...

Quantum SU(2) and the Baum-Connes conjecture

Christian Voigt (2012)

Banach Center Publications

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We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group S U q ( 2 ) of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.

Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac (2015)

Banach Center Publications

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Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if δ : A A m i n H is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra A δ H . For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated...

Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors

Sungjin Ra, Hakho Hong (2024)

Applications of Mathematics

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This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in 3 , we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions...

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Shuangjian Guo, Shengxiang Wang (2016)

Colloquium Mathematicae

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Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and B = A c o H . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map β : A B A A H , a B b S - 1 ( b [ 1 ] ) α ( b [ 0 ] [ - 1 ] ) β - 1 ( a ) β ( b [ 0 ] [ 0 ] ) , is surjective, we prove that the induction functor A B - : ̃ ( k ) B A H is an equivalence of categories.

Bernstein’s analyticity theorem for quantum differences

Tord Sjödin (2007)

Czechoslovak Mathematical Journal

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We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f ’s quantum differences are nonnegative then f has a power series representation on I . Further, if the quantum differences have fixed sign on I then f is analytic on I .

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

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We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if Γ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space k Γ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra U ( Γ ) is a bialgebra in the category of A-comodules.

Exponentiations over the quantum algebra U q ( s l 2 ( ) )

Sonia L’Innocente, Françoise Point, Carlo Toffalori (2013)

Confluentes Mathematici

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We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra U q ( s l 2 ( ) ) . We discuss two cases, according to whether the parameter q is a root of unity. We show that the universal enveloping algebra of s l 2 ( ) embeds in a non-principal ultraproduct of U q ( s l 2 ( ) ) , where q varies over the primitive roots of unity.

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim (2015)

Journal of the European Mathematical Society

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In this paper, we develop the crystal basis theory for the quantum queer superalgebra U q ( 𝔮 ( n ) ) . We define the notion of crystal bases and prove the tensor product rule for U q ( 𝔮 ( n ) ) -modules in the category 𝒪 int 0 . Our main theorem shows that every U q ( 𝔮 ( n ) ) -module in the category 𝒪 int 0 has a unique crystal basis.

On the Q -deformed Heisenberg uncertainty relations and discrete time

Hrubý, Jaroslav

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The opportunity for verifying the basic principles of quantum theory and possible q -deformation appears in quantum cryptography (QC) – a new discipline of physics and information theory.The author, member of the group of cryptology of Praha, presents in this paper the possibility to verify the q -deformation of Heisenberg uncertainty relation q -deformed QM and possible discretization on the base of a model presented in the fourth section.In the seven sections, the author discusses these...

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.

Q-adapted quantum stochastic integrals and differentials in Fock scale

Viacheslav Belavkin, Matthew Brown (2011)

Banach Center Publications

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In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic (QS) integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures, on a space-time σ-field , of the QS integration. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative D is proved. Finally, Q-adapted dynamics is discussed, including Bosonic (Q = I), Fermionic (Q = -I), and monotone...

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...