Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-$\mathcal {O}$ Classification

Jürg Fröhlich; Emmanuel Thiran

Displaying similar documents to “Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE- $𝒪$ Classification”

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 $k \oplus_{Γ}$ 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 $U_{q} (s l_{2} (ℂ))$

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 $U_{q} (s l_{2} (ℂ))$ . We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s l_{2} (ℂ)$ embeds in a non-principal ultraproduct of $U_{q} (s l_{2} (ℂ))$ , where $q$ varies over the primitive roots of unity.

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 $δ : A \to A ⨂_{m i n} H$ 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...

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,...

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 $β : A \otimes_{B} A \to A \otimes H$ , $a \otimes_{B} b \mapsto S^{- 1} (b_{[1]}) α (b_{[0] [- 1]}) \otimes β^{- 1} (a) β (b_{[0] [0]})$ , is surjective, we prove that the induction functor $A \otimes_{B} - : ̃ {(_{k})}_{B} \to_{A}^{H}$ is an equivalence of categories.

Instanton Singularities in Euclidean $Φ^{4}$ Quantum Field Theories

J. Feldman (1986)

Recherche Coopérative sur Programme n°25

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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 $h_{j} : = I_{j} \otimes α \otimes I_{n - 2 - j}$ on ${(ℂ ³)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra $ℋ_{q, n} (2)$ associated with the quantum group $U_{q} (2)$ . The purpose of this note is to present the construction.

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...

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 $𝒪_{int}^{\geq 0}$ . Our main theorem shows that every $U_{q} (𝔮 (n))$ -module in the category $𝒪_{int}^{\geq 0}$ has a unique crystal basis.

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.

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...

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 $_{q} (S U (2))$ .

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 $S L_{h} (2)$ .

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 $β \in (0, 2 / 3)$ , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N \to \infty$ 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 $β \in (0, 3 / 5)$ in [28]. This allows, in the case $β \in (0, 3 / 5)$ , for the application of the Klainerman–Machedon...

Generalized $P_{0}$ -lattices of order ω+

T. Traczyk, W. Zarębski (1976)

Colloquium Mathematicae

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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 $B_{n}$ .

Fractional integral operators on $B^{p, λ}$ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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We introduce function spaces $B^{p, λ}$ with Morrey-Campanato norms, which unify $B^{p, λ}$ , $C M O^{p, λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.

Generalized $P_{1}$ - and $P_{2}$ -lattices of order ω+

W. Zarębski (1977)

Colloquium Mathematicae

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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...

Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2016)

Journal of the European Mathematical Society

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We consider the wave and Schrödinger equations on a bounded open connected subset $Ω$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $ω$ of $Ω$ during a time interval $[0, T]$ with $T > 0$ . It is well known that, if the pair $(ω, T)$ satisfies the Geometric Control Condition ( $ω$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions...

Configurations of rank- $40 r$ extremal even unimodular lattices ( $r = 1, 2, 3$ )

Scott Duke Kominers, Zachary Abel (2008)

Journal de Théorie des Nombres de Bordeaux

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We show that if $L$ is an extremal even unimodular lattice of rank $40 r$ with $r = 1, 2, 3$ , then $L$ is generated by its vectors of norms $4 r$ and $4 r + 2$ . Our result is an extension of Ozeki’s result for the case $r = 1$ .

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

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This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $Ω \subset ℝ^{n}$ , with Dirichlet boundary conditions. The observation is done on a subset $ω$ of Lebesgue measure $| ω | = L | Ω |$ , where $L \in (0, 1)$ is fixed. We denote...

Displaying similar documents to “Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE- 𝒪 Classification”

Displaying similar documents to “Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE- $𝒪$ Classification”