scholarly journals Hopf Algebra of Sashes

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Shirley Law
Keyword(s):  
Hopf Algebra ◽  
Partial Order ◽  
Natural Partial Order ◽  
Algebra Structure ◽  
Natural Basis ◽  
General Lattice ◽  
Combinatorial Description ◽  
Combinatorial Objects ◽  
Dual Hopf Algebra ◽  
International Audience

International audience A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes. Une construction générale dans la théorie des treillis dû à Reading construit des sous-algèbres de Hopf de l’algèbre de Hopf de permutations de Malvenuto et Reutenauer (MR). Les produits et coproduits de ces sous-algèbres de Hopf sont définis extrinsèquement en termes du plongement dans MR. Le but de cette communication est de trouver une description combinatoire intrinsèque d’une de ces sous-algèbres de Hopf en particulier. Cette algèbre Hopf a une base naturelle donnée par des permutations que nous appelons permutations Pell. Les permutations Pell sont en bijection avec des objets combinatoires que nous appelons écharpes, c’est-à-dire des pavages d’un rectangle 1-par-n avec trois espèces de tuiles : des carrés noirs 1-par-1, des carrés blancs 1-par-1, et des rectangles blancs 1-par-2. La bijection induit une structure d’algèbre de Hopf sur les écharpes. On décrit le produit et le coproduit en termes d’écharpes, et l’ordre partiel naturel sur les écharpes. On décrit également le coproduit dual et le produit dualde l’algèbre de Hopf dual des écharpes.

Bicovariant differential calculus on the quantum superspace ℝq(1|2)

2016 ◽  
Vol 15 (09) ◽  
pp. 1650172 ◽  
Author(s):  
Salih Celik
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Function Algebra ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Bicovariant Differential Calculus ◽  
Quantum Supergroup ◽  
Dual Hopf Algebra

Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

scholarly journals Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Nguyen Hoang-Nghia ◽  
Adrian Tanasa ◽  
Christophe Tollu
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebra Structure ◽  
Isomorphism Classes ◽  
Hopf Algebra Structure ◽  
International Audience

International audience We endow the set of isomorphism classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.

scholarly journals Combinatorics of Positroids

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Suho Oh
Keyword(s):  
Special Class ◽  
Combinatorial Description ◽  
Combinatorial Objects ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets

International audience Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.

scholarly journals Combinatorial Hopf algebra of supercharacters of type D

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Carolina Benedetti
Keyword(s):  
Hopf Algebra ◽  
Finite Field ◽  
Type A ◽  
The Other ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Type D ◽  
International Audience ◽  
Supercharacter Theory ◽  
Triangular Group

International audience We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types. Dotamos con una estructura de álgebra de Hopf la teoría de supercaracteres del grupo de matrices unipotentes triangulares superiores de tipo{D} sobre un cuerpo finito. Ademas, discutimos brevemente los tipos {B} y {C}. El tipo A fue explorado por M. Aguiar et al (2010), por lo tanto este resumen extendido es una contribución para entender combinatoriamente la teoría de supercaracteres de los otros tipos de Lie clásicos. Nous construisons une structure d'algèbre de Hopf sur la thérie des supercharactères du groupe de matrices unipotentes triangulaires supéieures de type {D}. Nous donnons aussi quelques commentaires à l'égard des types {B} et {C} . Le type {A} a été explorée par M. Aguiar et al. (2010), donc ce résumé étendu est une contribution à la théorie combinatoire des supercharactères pour les autres types de Lie classiques. \par

scholarly journals Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Natural Basis ◽  
Dual Algebra ◽  
Power Sum ◽  
Hopf Algebra Structure

We introduce a coloured generalization  $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions  described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables.  We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

scholarly journals A Hopf algebra of subword complexes (Extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Nantel Bergeron ◽  
Cesar Ceballos
Keyword(s):  
Hopf Algebra ◽  
Cluster Algebras ◽  
Algebra Structure ◽  
Acyclic Orientations ◽  
Hopf Algebra Structure ◽  
International Audience

International audience We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

scholarly journals DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

1999 ◽  
Vol 11 (05) ◽  
pp. 553-629 ◽  
Author(s):  
FRANK HAUSSER ◽  
FLORIAN NILL
Keyword(s):  
Hopf Algebra ◽  
Associative Algebra ◽  
Quantum Groups ◽  
Product Formula ◽  
Crossed Product ◽  
Group Symmetry ◽  
Algebra Structure ◽  
Quantum Double ◽  
Field Algebra ◽  
Dual Hopf Algebra

A two-sided coaction [Formula: see text] of a Hopf algebra [Formula: see text] on an associative algebra ℳ is an algebra map of the form [Formula: see text] , where (λ,ρ) is a commuting pair of left and right [Formula: see text] -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra [Formula: see text] on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product[Formula: see text] to be the algebra generated by ℳ and [Formula: see text] with relations given by [Formula: see text] We give a natural generalization of this construction to the case where [Formula: see text] is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on [Formula: see text] extending [Formula: see text], even though the analogue of an ordinary crossed product [Formula: see text] in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case [Formula: see text] and λ=ρ=Δ we obtain an explicit definition of the quantum double [Formula: see text] for quasi-Hopf algebras [Formula: see text] , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that [Formula: see text] is itself a (weak) quasi-bialgebra and that any diagonal crossed product [Formula: see text] naturally admits a two-sided [Formula: see text] -coaction. In particular, the above-mentioned lattice models always admit the quantum double [Formula: see text] as a localized cosymmetry, generalizing results of Nill and Szlachányi [42]. A complete proof that [Formula: see text] is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

scholarly journals Homomorphisms and congruences on ωα-bisimple semigroups

1973 ◽  
Vol 15 (4) ◽  
pp. 441-460 ◽  
Author(s):  
J. W. Hogan
Keyword(s):  
Partial Order ◽  
Ordered Set ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

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