On a cubic Hecke algebra associated with the quantum group $U_q(2)$

Janusz Wysoczański

Displaying similar documents to “On a cubic Hecke algebra associated with the quantum group $U_{q} (2)$ ”

Exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$

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

Confluentes Mathematici

Similarity:

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

Similarity:

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...

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

Similarity:

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...

Right coideal subalgebras of $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

Similarity:

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

Similarity:

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...

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000-2001)

Séminaire Équations aux dérivées partielles

Similarity:

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...

Remarks on Sekine quantum groups

Jialei Chen, Shilin Yang (2022)

Czechoslovak Mathematical Journal

Similarity:

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 $r (𝒜_{k})$ .

A class of quantum doubles of pointed Hopf algebras of rank one

Hua Sun, Yueming Li (2023)

Czechoslovak Mathematical Journal

Similarity:

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 the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

Similarity:

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...

The Grothendieck ring of quantum double of quaternion group

Hua Sun, Jia Pang, Yanxi Shen (2024)

Czechoslovak Mathematical Journal

Similarity:

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.

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

Similarity:

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...

Geometric theta-lifting for the dual pair ${𝕊𝕆}_{2 m}, 𝕊 p_{2 n}$

Sergey Lysenko (2011)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $> 2$ . Consider the dual pair $H = {SO}_{2 m}, G = {Sp}_{2 n}$ over $X$ with $H$ split. Write ${Bun}_{G}$ and ${Bun}_{H}$ for the stacks of $G$ -torsors and $H$ -torsors on $X$ . The theta-kernel ${Aut}_{G, H}$ on ${Bun}_{G} \times {Bun}_{H}$ yields theta-lifting functors $F_{G} : D ({Bun}_{H}) \to D ({Bun}_{G})$ and $F_{H} : D ({Bun}_{G}) \to D ({Bun}_{H})$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Displaying similar documents to “On a cubic Hecke algebra associated with the quantum group U q ( 2 ) ”

Displaying similar documents to “On a cubic Hecke algebra associated with the quantum group $U_{q} (2)$ ”