Page 1 Next

Displaying 1 – 20 of 61

Showing per page

A construction of the Hom-Yetter-Drinfeld category

Haiying Li, Tianshui Ma (2014)

Colloquium Mathematicae

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 H H . Finally,...

A Maschke type theorem for relative Hom-Hopf modules

Shuangjian Guo, Xiu-Li Chen (2014)

Czechoslovak Mathematical Journal

Let ( H , α ) be a monoidal Hom-Hopf algebra and ( A , β ) a right ( H , α ) -Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor F from the category of relative Hom-Hopf modules to the category of right ( A , β ) -Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the ( H , α ) -coaction to be separable. This leads to a generalized...

A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

Iván Ezequiel Angiono (2015)

Journal of the European Mathematical Society

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Additive deformations of braided Hopf algebras

Malte Gerhold, Stefan Kietzmann, Stephanie Lachs (2011)

Banach Center Publications

Additive deformations of bialgebras in the sense of J. Wirth [PhD thesis, Université Paris VI, 2002], i.e. deformations of the multiplication map fulfilling a certain compatibility condition with respect to the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.

Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras

Shuangjian Guo, Xiaohui Zhang, Shengxiang Wang (2016)

Colloquium Mathematicae

We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of Doi Hom-Hopf...

Categorification of Hopf algebras of rooted trees

Joachim Kock (2013)

Open Mathematics

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring...

Classifying bicrossed products of two Sweedler's Hopf algebras

Costel-Gabriel Bontea (2014)

Czechoslovak Mathematical Journal

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras E that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products H 4 H 4 . There are three steps in our approach. First, we explicitly describe the set of all matched pairs ( H 4 , H 4 , , ) by proving that, with the exception of the trivial pair, this set is parameterized by the ground field k . Then, for...

Cobraided smash product Hom-Hopf algebras

Tianshui Ma, Haiying Li, Tao Yang (2014)

Colloquium Mathematicae

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products ( A R B , α β ) . Moreover, necessary and sufficient conditions for ( A R B , α β ) to be a cobraided Hom-Hopf algebra are given.

Deformed commutators on comodule algebras over coquasitriangular Hopf algebras

Zhongwei Wang, Guoyin Zhang, Liangyun Zhang (2015)

Colloquium Mathematicae

We construct quantum commutators on comodule algebras over coquasitriangular Hopf algebras, so that they are quantum group coinvariant and have the generalized antisymmetry and Leibniz properties. If the coquasitriangular Hopf algebra is additionally cotriangular, then the quantum commutators satisfy a generalized Jacobi identity, and turn the comodule algebra into a quantum Lie algebra. Moreover, we investigate the projective and injective dimensions of some Doi-Hopf modules over a quantum commutative...

Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two

Tomasz Brzeziński (2015)

Colloquium Mathematicae

Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

Eulerian idempotent and Kashiwara-Vergne conjecture

Emily Burgunder (2008)

Annales de l’institut Fourier

By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution ( F , G ) of the first equation of the Kashiwara-Vergne conjecture x + y - log ( e y e x ) = ( 1 - e - ad x ) F ( x , y ) + ( e ad y - 1 ) G ( x , y ) . Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.

Free dynamical quantum groups and the dynamical quantum group S U Q d y n ( 2 )

Thomas Timmermann (2012)

Banach Center Publications

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.

From left modules to algebras over an operad: application to combinatorial Hopf algebras

Muriel Livernet (2010)

Annales mathématiques Blaise Pascal

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from 𝕊 -modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.Let 𝒪 denote the forgetful functor from 𝕊 -modules to graded vector spaces. Left modules over an operad 𝒫 are treated as 𝒫 -algebras in the category of 𝕊 -modules. We generalize the results obtained...

Galois H-objects with a normal basis in closed categories. A cohomological interpretation.

José N. Alonso Alvarez, José Manuel Fernández Vilaboa (1993)

Publicacions Matemàtiques

In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a...

Grothendieck ring of quantum double of finite groups

Jingcheng Dong (2010)

Czechoslovak Mathematical Journal

Let k G be a group algebra, and D ( k G ) its quantum double. We first prove that the structure of the Grothendieck ring of D ( k G ) can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of G . As a special case, we then give an application to the group algebra k D n , where k is a field of characteristic 2 and D n is a dihedral group of order 2 n .

Currently displaying 1 – 20 of 61

Page 1 Next