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.

Braided coproduct, antipode and adjoint action for $U_{q} (s l_{2})$

Pavle Pandžić, Petr Somberg (2024)

Archivum Mathematicum

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Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_{q} ({𝔰𝔩}_{n})$ , we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_{q} ({𝔰𝔩}_{2})$ . The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.

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.

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 $r (𝒜_{k})$ .

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...

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.

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.

Quantised ${𝔰𝔩}_{2}$ -differential algebras

Andrey Krutov, Pavle Pandžić (2024)

Archivum Mathematicum

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We propose a definition of a quantised ${𝔰𝔩}_{2}$ -differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${𝔰𝔩}_{2}$ are natural examples of such algebras.

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 $d a = [D, a]$ , $a \in A$ , 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 $A^{\circ}$ . 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...

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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Quantized semisimple Lie groups

Rita Fioresi, Robert Yuncken (2024)

Archivum Mathematicum

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The goal of this expository paper is to give a quick introduction to $q$ -deformations of semisimple Lie groups. We discuss principally the rank one examples of $𝒰_{q} ({𝔰𝔩}_{2})$ , $𝒪 ({SU}_{q} (2))$ , $𝒟 ({SL}_{q} (2, ℂ))$ and related algebras. We treat quantized enveloping algebras, representations of $𝒰_{q} ({𝔰𝔩}_{2})$ , generalities on Hopf algebras and quantum groups, $*$ -structures, quantized algebras of functions on $q$ -deformed compact semisimple groups, the Peter-Weyl theorem, $*$ -Hopf algebras associated to complex semisimple Lie groups and the Drinfeld...

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 $p \neq 2$ , 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.

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”