Dual Hopf Algebra Research Articles

Quantum geometry of Boolean algebras and de Morgan duality

We take a fresh look at the geometrization of logic using the recently developed tools of “quantum Riemannian geometry” applied in the digital case over the field \mathbb{F}_2=\{0,1\} , extending de Morgan duality to this context of differential forms and connections. The 1 -forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph 0-1-2 has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of \mathbb{Z}_3 . For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an n -gon with n>4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over \mathbb{F}_2 .

The Exponential Map for Hopf Algebras

We give an analogue of the classical exponential map on Lie groups for Hopf $*$-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert $C^{*} $-bimodule of $\frac{1}{2}$ densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups $S_{3}$ and $\mathbb{Z}$, Woronowicz's matrix quantum group $\mathbb{C}_{q}[SU_2] $ and the Sweedler-Taft algebra.

On holomorphic reflexivity conditions for complex Lie groups

AbstractWe consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$.

Primitive elements of the Hopf algebras of tableaux

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.

BMS algebras in 4 and 3 dimensions, their quantum deformations and duals

BMS symmetry is a symmetry of asymptotically flat spacetimes in vicinity of the null boundary of spacetime and it is expected to play a fundamental role in physics. It is interesting therefore to investigate the structures and properties of quantum deformations of these symmetries, which are expected to shed some light on symmetries of quantum spacetime. In this paper we discuss the structure of the algebra of extended BMS symmetries in 3 and 4 spacetime dimensions, realizing that these algebras contain an infinite number of distinct Poincaré subalgebras, a fact that has previously been noted in the 3 dimensional case only. Then we use these subalgebras to construct an infinite number of different Hopf algebras being quantum deformations of the BMS algebras. We also discuss different types of twist-deformations and the dual Hopf algebras, which could be interpreted as noncommutative, extended quantum spacetimes.

On Isotypic Decompositions for Non-Semisimple Hopf Algebras

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

A pair of dual Hopf algebras on permutations

<abstract> Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra. </abstract>

The Steenrod algebra from the group theoretical viewpoint

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A⁎, whose group structure is given by the composition of formal power series ([4] Appendix). The aim of this paper is to show some group theoretic properties of Gp by making use of this presentation of Gp(A⁎). We give a decreasing filtration of subgroup schemes of Gp which we use for estimating the length of the lower central series of finite subgroup schemes of Gp. We also give a successive quotient maps Gp→ρ0Gp〈1〉→ρ1Gp〈2〉→ρ2⋯→ρk−1Gp〈k〉→ρkGp〈k+1〉→ρk+1⋯ of affine group schemes over a prime field Fp such that the kernel of ρk is a maximal abelian subgroup.

Hopf-Frobenius Algebras and a Simpler Drinfeld Double

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.

Miraculous cancellations for quantum SL 2

In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter $q$ is a root of unity. The current paper is devoted to giving a more representation theoretic interpretation of this phenomenon, in terms of the quantum group $U_q(sl_2)$ and its dual Hopf algebra $SL_2^q$.

Generalised noncommutative geometry on finite groups and Hopf quivers

We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of S_3 .We show how quiver geometries arise naturally in the context of quantum principal bundles.We provide a formulation of bimodule Riemannian geometry for quantum metrics on a quiver, with a fully worked example on 2 points; in the quiver case, metric data assigns matrices not real numbers to the edges of a graph. The paper builds on the general theory in our previous work [19].

Planar algebras, cabling and the Drinfeld double

We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra H into the two-cabling of the planar algebra of the dual Hopf algebra H^* .

From Quadratic Gaussian to Quantum Groups: Exploiting Duality in Modelling IR-FX Hybrids (Presentation Slides)

Mathematical finance explores the consistency relationships between the prices of securities imposed by elementary economic principles. Commonplace among these are replicability and the absence of arbitrage, both essentially algebraic constraints on the valuation map from a security to its price. The discussion is framed in terms of observables, the securities, and states, the linear and positive maps from security to price. Founded on the principles of replicability and the absence of arbitrage, mathematical finance then equates to the theory of positive linear maps and their scale invariances. This recognises that the defining principles are in essence algebraic, and may be satisfied in circumstances more general than the classical probabilistic setting. These slides consider the dual Hopf algebraic structures of observables and states, and how they may be utilised to create efficient models for pricing. This naturally leads to the study of models based on restrictions of the dual Hopf algebras, such as the Quadratic Gauss model, that retain much of the phenomenology of their parents within a more tractable domain, and extensions of the dual Hopf algebras, such as the Linear Dirac model, with novel features unattainable in the classical case.

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.

The Pieri Rule for Dual Immaculate Quasi-Symmetric Functions

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third author to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.

Duality for generalised differentials on quantum groups

We study generalised differential structures (Ω1,d) on an algebra A, where A⊗A→Ω1 given by a⊗b→adb need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs (Λ1,ω) where Λ1 is a right module and ω a right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω:A+→Λ1 in the category of right crossed (or Drinfeld–Radford–Yetter) modules over A. When A=U(g) the generalised left covariant differential structures are classified by cocycles ω∈Z1(g,Λ1). We then introduce and study the dual notion of a codifferential structure (Ω1,i) on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra (Ω,d) augmented by a codifferential i of degree −1. Here Ω is a graded super-Hopf algebra extending the Hopf algebra Ω0=A and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. Accordingly, group 1-cocycles correspond precisely to codifferential structures on algebraic groups and function algebras. Among general constructions, we show that first order data (Λ1,ω) on a Hopf algebra A extends canonically to a strongly bicovariant differential graded algebra via the braided super-shuffle algebra. The theory is also applied to quantum groups with Ω1(Cq(G)) dually paired to Ω1(Uq(g)).

A pre-Lie algebra associated to a linear endomorphism and related algebraic structures

We construct a functor $$T$$ from the category of endomorphisms of vector spaces to the category of Com-PreLie algebras. For any endomorphism $$f$$ of a vector space $$V$$ , we describe the enveloping algebra of the pre-Lie algebra $$T(V,f)$$ , the dual Hopf algebra and the associated group of characters. For $$f=\mathrm{Id}_V$$ , we find the algebra of formal diffeomorphisms, seen as a subalgebra of the Connes–Kreimer Hopf algebra of rooted trees in the context of QFT; for other well-chosen nilpotent $$f$$ , we obtain the groups of Fliess operators in Control Theory. An algebraic study of these Com-PreLie Hopf algebras is carried out: gradations, generation, subobject generated by $$V$$ , etc.

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].

On a symmetry of Müger’s centralizer for the Drinfeld double of a semisimple Hopf algebra

In this paper we prove a formula that relates Muger’s centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for Muger’s centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.