Dual Hopf Algebra Research Articles

Hopf Algebra of Sashes

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.

Conjugation invariants in the Leibniz–Hopf algebra

The Leibniz–Hopf algebra is the Hopf algebra whose underlying algebra is the free associative algebra over Z on generators S1,S2,S3,… and whose diagonal is given by Δ(Sn)=∑i=0nSi⊗Sn−i (where S0 is to be understood as 1). We calculate a basis for the free submodule formed by the fixed points of this algebra under the Hopf algebra conjugation operation, χ. We also give bases for the submodules of fixed points in the dual Hopf algebra and in the mod p reductions of both the Leibniz–Hopf algebra and its dual.

Hopf subalgebras and Hopf ideals of certain semisimple Hopf algebras

In the paper, for semisimple Hopf algebras that have only one non-one-dimensional irreducible representation, all Hopf ideals are described and, under some restriction concerning the number of group elements in the dual Hopf algebra, some series of Hopf subalgebras are found. Moreover, the quotient Hopf algebras of these semisimpleHopf algebras are described.

The Hopf algebra structure of the character rings of classical groups

The character ring of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra of symmetric functions. Here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that and also admit natural Hopf algebra structures that are isomorphic to that of , and hence to . The isomorphisms are determined explicitly, along with the specification of standard bases for and analogous to those used for . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur–Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras and are not self-dual. The dual Hopf algebras and are identified. Finally, the Hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

Duality on the quantum space(3) with two parameters

Abstract 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.

On finite-dimensional semisimple Hopf algebras of dimension n(n+ 1)

We study finite-dimensional semisimple Hopf algebras over an algebraically closed field which have only one summand of dimension greater than 1 in their semisimple decompositions and assume that the group of group elements in the dual Hopf algebra is cyclic and has minimal order. Under given constraints, we obtain a detailed description of the comultiplication and the antipode.

Combinatorial Hopf Algebras and Towers of Algebras—Dimension, Quantization and Functorality

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$. In the case $r=1$ we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of the construction. This paper is a full version of the summary arXiv: 0710.3744.

Coset Decomposition for Semisimple Hopf Algebras

The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.

Schubert polynomials for the affine Grassmannian of the symplectic group

We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.

Combinatorial Hopf Algebras and Towers of Algebras

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$. Bergeron et Li ont donné un ensemble d'axiomes qui garanti que les groupes de Grothendieck d'une tour d'algèbres $\bigoplus_{n \geq 0}A_n$ peuvent être dotés d'une structure d'algèbres de Hopf graduées duales. Hivert et Nzeutzhap, et indépen\-damment Lam et Shimozono, ont construit des graphes gradués duals à partir d'éléments primitifs dans des algèbres de Hopf. Dans cet article, nous appliquons la composition de ces constructions aux tours des algèbres. Nous prouvons que si une tour $\bigoplus_{n \geq 0}A_n$ donne des algèbres de Hopf graduées duales, alors nous devons avoir $\dim (A_n)=r^nn!$ où $r = \dim (A_1)$.

Differential geometry of the quantum 3-dimensional space

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra. Differential geometry of the quantum Lie algebra of the extended quantum plane and its Hopf algebra structure is obtained. Its dual Hopf algebra is also given.

When is a Cleft Extension H-Azumaya?

We show which H op-cleft extensions \( k\# _{\sigma } H^{{op}} \) of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map \( \theta _{\sigma } :H \to H* \) defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).

Structure of the Loday–Ronco Hopf algebra of trees

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto–Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. In the dual basis for the graded dual Hopf algebra, our formula for the coproduct gives an explicit isomorphism with a free associative algebra. We also obtain a transparent proof of its isomorphism with the non-commutative Connes–Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes–Kreimer Hopf algebra is related to symmetric functions.

Krull relations in Hopf Galois extensions: Lifting and twisting

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂ A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t -Lying Over (t -LO), and Going Up (GU), analogous to the classical Krull relations for prime ideals; we also defined three new “dual” Krull relations. We say that H itself is said to have one of the Krull relations if the relation holds for all faithfully flat H -Galois extensions. We showed in [MS] that H has one of the three “dual” Krull relations if and only if the dual Hopf algebra H ∗ of H has the original relation (hence the name). An important example of Hopf Galois extensions is given by Hopf crossed products A = Rσ # H . Moreover, Galois extensions can be useful in studying crossed products, since they satisfy a “transitivity” property which crossed products lack. That is, if K is a normal Hopf subalgebra of H with Hopf quotient H , then, in general, one cannot write A = Rσ # H = (Rσ K)#τ H , by an example of [S2]. Another basic example of a Hopf Galois extension is given by a Hopf algebra A with a normal Hopf subalgebra R of finite index such that A is faithfully flat over R: for then R ⊂ A is faithfully flat H -Galois.

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.

The Drinfel'd double versus the Heisenberg double for an algebraic quantum group

Let A be a regular multiplier Hopf algebra with integrals. The dual of A, denoted by Â, is a multiplier Hopf algebra so that 〈 A ̂ ,A〉 is a pairing of multiplier Hopf algebras. We consider the Drinfel'd double, D= A ̂ ⋈A cop , associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair 〈 A ̂ ,A〉 has a “canonical multiplier” W∈M( A ̂ ⊗A) . The image of W in M( D⊗ D) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra D̂ via the right action of D on D̂ which defines the pair 〈 D ̂ ,D〉 . As expected from the finite-dimensional case, we find that the deformation of the product in D̂ is related to the Heisenberg double A# A ̂ .

Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

We extend the Larson–Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra AT of the underlying weak Hopf algebra A.

Bialgebroid actions on depth two extensions and duality

A general notion of depth two for ring homomorphism N→ M is introduced. The step two centralizers A=End N M N and B=(M⊗ NM) N in the Jones tower above N→ M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R= C M ( N) and C=End N– M ( M⊗ N M). We show A and B to possess dual left and right R-bialgebroid structures which generalize Lu's fundamental bialgebroids over an algebra. There are actions of A and B on M and E′= End NM with Galois properties. If M | N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.

On Hopf DeMeyer‐Kanzaki Galois extensions

Let H be a finite‐dimensional Hopf algebra over a field k, B a left H‐module algebra, and H∗ the dual Hopf algebra of H. For an H∗‐Azumaya Galois extension B with center C, it is shown that B is an H∗‐DeMeyer‐Kanzaki Galois extension if and only if C is a maximal commutative separable subalgebra of the smash product B#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

On Hopf Galois Hirata extensions

Let H be a finite‐dimensional Hopf algebra over a field K, H* the dual Hopf algebra of H, and B a right H*‐Galois and Hirata separable extension of BH. Then B is characterized in terms of the commutator subring VB(BH) of BH in B and the smash product VB(BH)#H. A sufficient condition is also given for B to be an H*‐Galois Azumaya extension of BH.