Displaying similar documents to “A class of quantum doubles of pointed Hopf algebras of rank one”

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 .

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 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 . 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 bicrossed products of H 4 and H 8

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let H 4 and H 8 be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through H 8 and H 4 (equivalently, any bicrossed product between the Hopf algebras H 8 and H 4 ) must be isomorphic to one of the following four Hopf algebras: H 8 H 4 , H 32 , 1 , H 32 , 2 , H 32 , 3 . The set of all matched pairs ( H 8 , H 4 , , ) is explicitly described, and then the associated bicrossed product is given by generators and relations.

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 ( X op ) * 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...

The structures of Hopf * -algebra on Radford algebras

Hassan Suleman Esmael Mohammed, Hui-Xiang Chen (2019)

Czechoslovak Mathematical Journal

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We investigate the structures of Hopf * -algebra on the Radford algebras over . All the * -structures on H are explicitly given. Moreover, these Hopf * -algebra structures are classified up to equivalence.

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.

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...

On the real X -ranks of points of n ( ) with respect to a real variety X n

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let  X n be an integral and non-degenerate m -dimensional variety defined over . For any P n ( ) the real X -rank r X , ( P ) is the minimal cardinality of S X ( ) such that P S . Here we extend to the real case an upper bound for the X -rank due to Landsberg and Teitler.

Truncation and Duality Results for Hopf Image Algebras

Teodor Banica (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Associated to an Hadamard matrix H M N ( ) is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with G S N . We study a certain family of discrete measures μ r [ 0 , N ] , coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type 0 N ( x / N ) p d μ r ( x ) = 0 N ( x / N ) r d ν p ( x ) , where μ r , ν r are the truncations of the spectral measures μ,ν associated to H , H t . We also prove, using these truncations μ r , ν r , that for any deformed Fourier matrix H = F M Q F N we have μ = ν.

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 H (or 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 ) D ( G ; H ) . The paper constructs a crossed product algebra C ( G / H × G ) G , and proves that there is an algebra isomorphism between...

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...

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.

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng Xuan, Wei-Xue Shi (2016)

Mathematica Bohemica

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A topological space X has a rank 2-diagonal if there exists a diagonal sequence on X of rank 2 , that is, there is a countable family { 𝒰 n : n ω } of open covers of X such that for each x X , { x } = { St 2 ( x , 𝒰 n ) : n ω } . We say that a space X satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. We mainly prove that if X is a DCCC normal space with a rank 2-diagonal, then the cardinality of X is at most 𝔠 . Moreover, we prove that if X is a first...

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- n commutative algebras equipped with an ordered basis is an affine scheme 𝔅 n of finite type over , with geometrically connected fibers. It is smooth if and only if n 3 . It is reducible if n 8 (and the converse holds, at least if we remove the fibers above 2 and 3 ). The relative dimension of 𝔅 n is 2 27 n 3 + O ( n 8 / 3 ) . The subscheme parameterizing étale algebras is isomorphic to GL n / S n , which is of dimension only n 2 . For n 8 , there exist algebras that are not limits of étale algebras. The dimension...