scholarly journals Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

2014 ◽  
Vol 47 (20) ◽  
pp. 205201 ◽  
Author(s):  
Bertfried Fauser ◽  
Peter D Jarvis ◽  
Ronald C King
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Unified Approach ◽  
Group Characters

scholarly journals Braided Frobenius algebras from certain Hopf algebras

2021 ◽  
pp. 2350012
Author(s):  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Frobenius Algebra ◽  
Compatibility Conditions ◽  
Ribbon Graphs ◽  
Frobenius Algebras ◽  
Ternary Operation ◽  
Compact Surfaces ◽  
Baxter Operator

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

scholarly journals Comodules over weak multiplier bialgebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450037 ◽  
Author(s):  
Gabriella Böhm
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Base Algebra ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Multiplier Hopf Algebras ◽  
Hopf Modules ◽  
Total Algebra ◽  
Module Structures

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

scholarly journals The Tensor Category of Linear Maps and Leibniz Algebras

1998 ◽  
Vol 5 (3) ◽  
pp. 263-276
Author(s):  
J. L. Loday ◽  
T. Pirashvili
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Type Theorem ◽  
Tensor Category ◽  
Leibniz Algebra ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Leibniz Algebras ◽  
Linear Maps

Abstract We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

scholarly journals Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
François Bergeron ◽  
Aaron Lauve
Keyword(s):  
Tensor Product ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Symmetric Polynomials ◽  
Reflection Groups ◽  
Invariant Polynomials ◽  
Product Decomposition ◽  
Combinatorial Hopf Algebras ◽  
Produit Tensoriel ◽  
International Audience

International audience We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras. Nous analysons la structure de l'algèbre $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ des polynômes symétriques en des variables commutatives. Plus précisément, au moyen de "l'action par positions'', on réalise $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ comme sous-module de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$. On découvre alors une nouvelle décomposition de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ comme produit tensoriel, obtenant ainsi un analogue des théorèmes classiques de Chevalley et Shephard-Todd. Dans le cas $|\mathbf{x}|= \infty$, nos techniques se simplifient en une forme aisément généralisables à beaucoup d'autres paires d'algèbres de Hopf familières.

On Cleft Extensions of the Tensor Product of Two Hopf Algebras

2013 ◽  
Vol 41 (11) ◽  
pp. 4299-4332
Author(s):  
Ningyuan Yao ◽  
Xiaolong Jiang
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Cleft Extensions

scholarly journals HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

2001 ◽  
Vol 44 (1) ◽  
pp. 19-26 ◽  
Author(s):  
M. D. Crossley ◽  
Sarah Whitehouse
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Group Action ◽  
Hopf Algebras ◽  
Homotopy Theory ◽  
Cohomology Ring ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Primary 16W30

AbstractLet $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

scholarly journals Representations of a non-pointed Hopf algebra

2021 ◽  
Vol 6 (10) ◽  
pp. 10523-10539
Author(s):  
Ruifang Yang ◽  
◽  
Shilin Yang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Hopf Algebras ◽  
Indecomposable Modules ◽  
Representation Ring ◽  
Pointed Hopf Algebras ◽  
Dimension One ◽  
Gk Dimension

<abstract><p>In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.</p></abstract>

REPRESENTATIONS OF SIMPLE POINTED HOPF ALGEBRAS

2004 ◽  
Vol 03 (01) ◽  
pp. 91-104 ◽  
Author(s):  
SHILIN YANG
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Hopf Algebras ◽  
Representation Type ◽  
Ground Field ◽  
Indecomposable Modules ◽  
Pointed Hopf Algebras

In this paper, the representation type of a class of pointed Hopf algebras is determined. As an application, all indecomposable modules of a simple-pointed Hopf algebra R(q,α) are classified. The Clebsch–Gordan-like formula for the decomposition of the tensor product, taken on the ground field, of two indecomposable modules of R(q,α) is obtained.

Sign in / Sign up

Export Citation Format

Share Document