Displaying similar documents to “The affineness criterion for quantum Hom-Yetter-Drinfel'd modules”

Category 𝒪 for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

Similarity:

In this paper we study the BGG-categories 𝒪 q associated to quantum groups. We prove that many properties of the ordinary BGG-category 𝒪 for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for 𝒪 and for finite dimensional U q -modules we are able...

Separable functors for the category of Doi Hom-Hopf modules

Shuangjian Guo, Xiaohui Zhang (2016)

Colloquium Mathematicae

Similarity:

Let ̃ ( k ) ( H ) A C be the category of Doi Hom-Hopf modules, ̃ ( k ) A be the category of A-Hom-modules, and F be the forgetful functor from ̃ ( k ) ( H ) A C to ̃ ( k ) A . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

Similarity:

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 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 ( ) )

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 ( ) ) . 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.

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

Similarity:

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 0 . Our main theorem shows that every U q ( 𝔮 ( n ) ) -module in the category 𝒪 int 0 has a unique crystal basis.

A cluster algebra approach to q -characters of Kirillov–Reshetikhin modules

David Hernandez, Bernard Leclerc (2016)

Journal of the European Mathematical Society

Similarity:

We describe a cluster algebra algorithm for calculating q -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra U q ( 𝔤 ^ ) . This yields a geometric q -character formula for tensor products of Kirillov–Reshetikhin modules. When 𝔤 is of type A , D , E , this formula extends Nakajima’s formula for q -characters of standard modules in terms of homology of graded quiver varieties.

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 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.

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.

Braided coproduct, antipode and adjoint action for U q ( s l 2 )

Pavle Pandžić, Petr Somberg (2024)

Archivum Mathematicum

Similarity:

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.

A construction of the Hom-Yetter-Drinfeld category

Haiying Li, Tianshui Ma (2014)

Colloquium Mathematicae

Similarity:

In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category H H via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and H H is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that ( A H , α β ) is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category...

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 ) .

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 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...

On a cubic Hecke algebra associated with the quantum group U q ( 2 )

Janusz Wysoczański (2010)

Banach Center Publications

Similarity:

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 α I n - 2 - j on ( ³ ) n with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra 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

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...

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...

Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac (2015)

Banach Center Publications

Similarity:

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 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...

Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

Similarity:

Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration M < d e g N such that t M / t + 1 M t N / t + 1 N for all t. Given a module M with square-free top and a projective cover P, she showed that d i m k H o m ( M , M ) = d i m k H o m ( P , M ) if and only if M has no proper degeneration M < d e g N where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from...

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

Similarity:

Let A be a standard Koszul standardly stratified algebra and X an A -module. The paper investigates conditions which imply that the module Ext A * ( X ) over the Yoneda extension algebra A * is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

Similarity:

Let H be a finite-dimensional bialgebra. In this paper, we prove that the category ℒℛ ( H ) of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category H H * H H * 𝒴𝒟 over the tensor product bialgebra H H * as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

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 .