Remarks on Sekine quantum groups

Jialei Chen; Shilin Yang

Displaying similar documents to “Remarks on Sekine quantum groups”

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.

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...

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.

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...

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...

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 ${(ℂ X^{op})}^{*} ⋊ ℂ G$ . 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...

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 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.

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...

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...

Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk, Dariusz Klein (2004)

Colloquium Mathematicae

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Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{λ} G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $I n d_{m} (S^{λ} G)$ the set of isomorphism classes of indecomposable $S^{λ} G$ -modules of S-rank m. We exhibit rings $S^{λ} G$ for which there exists a function $f_{λ} : ℕ \to ℕ$ such that $f_{λ} (n) \geq n$ and $I n d_{f_{λ} (n)} (S^{λ} G)$ is an infinite set for every natural n > 1. In special cases $f_{λ} (ℕ)$ contains every natural...

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 $ℂ H$ (or $ℂ G$ ) 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 $E : D (G) \to D (G; H)$ . The paper constructs a crossed product algebra $C (G / H \times G) ⋊ ℂ G$ , 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.