Displaying similar documents to “Quantum 4-sphere: the infinitesimal approach”

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 ) .

Quantum detailed balance conditions with time reversal: the finite-dimensional case

Franco Fagnola, Veronica Umanità (2011)

Banach Center Publications

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We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely t r ( ρ 1 / 2 x ρ t 1 / 2 ( y ) ) = t r ( ρ 1 / 2 θ y * θ ρ t 1 / 2 ( θ x * θ ) ) for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying...

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.

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,...

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...

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.

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.

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...

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.

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])...

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...

Quantum Cohomology and Crepant Resolutions: A Conjecture

Tom Coates, Yongbin Ruan (2013)

Annales de l’institut Fourier

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We give an expository account of a conjecture, developed by Coates–Iritani–Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold 𝒳 to the quantum cohomology of a crepant resolution Y of 𝒳 . We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan–Graber. We also give a ‘quantized’ version of the conjecture, which determines...

Quantum Cohomology and Periods

Hiroshi Iritani (2011)

Annales de l’institut Fourier

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In a previous paper, the author introduced an integral structure in quantum cohomology defined by the K -theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism...

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...