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

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.

On coideal subalgebras of abelian cocentral extensions and a generalization of Wall's conjecture

It is shown that any coideal subalgebra of a finite-dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [R. Guralnick and F. Xu, On a subfactor generalization of Wall's conjecture, J. Algebra 332 (2011) 457–468].

Hopf Algebra of Sashes

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.

Duality on the quantum space(3) with two parameters

In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

Flag modules and the hit problem for the Steenrod algebra

The ‘hit problem’ of F. P. Peterson in algebraic topology asks for a minimal generating set for the polynomial algebraP(n) =2[x1,. . .,xn] as a module over the Steenrod algebra2. An equivalent problem is to find an2-basis for the subringK(n) of elementsfin the dual Hopf algebraD(n), a divided power algebra, such thatSqk(f)=0 for allk> 0. The Steenrod kernelK(n) is a2GL(n,2)-module dual to the quotientQ(n) ofP(n) by the hit elements+2P(n). A submoduleS(n) ofK(n) is obtained as the image of a family of maps from the permutation moduleFl(n) ofGL(n,2) on complete flags in ann-dimensional vector spaceVover2. We use the Schubert cell decomposition of the flags to calculateS(n) in degrees$d =\sum_{i=1}^n (2^{\lambda_i}-1)$, where λ1> λ2> ⋅⋅⋅ > λn≥ 0. When λn= 0, we define a2GL(n,2)-module map δ:Qd(n) →Q2d+n−1(n) analogous to the well-known isomorphismQd(n) →Q2d+n(n) of M. Kameko. When λn−1≥ 2, we show that δ is surjective and δ*:S2d+n−1(n)→Sd(n) is an isomorphism.

On Hopf DeMeyer-Kanzaki Galois extensions

LetHbe a finite-dimensional Hopf algebra over a fieldk,Ba leftH-module algebra, andH∗the dual Hopf algebra ofH. For anH∗-Azumaya Galois extensionBwith centerC, it is shown thatBis anH∗-DeMeyer-Kanzaki Galois extension if and only ifCis a maximal commutative separable subalgebra of the smash productB#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

On Hopf Galois Hirata extensions

LetHbe a finite-dimensional Hopf algebra over a fieldK,H*the dual Hopf algebra ofH, andBa rightH*-Galois and Hirata separable extension ofBH. ThenBis characterized in terms of the commutator subringVB(BH)ofBHinBand the smash productVB(BH)#H. A sufficient condition is also given forBto be anH*-Galois Azumaya extension ofBH.

DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

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].