Hopf Algebra Research Articles

Multiplier Hopf Coquasigroup: Motivation and Biduality

Inspired by the multiplier Hopf algebra theory introduced by A. Van Daele, this paper introduces a new algebraic structure, a multiplier Hopf coquasigroup, by constructing the integral dual of an infinite-dimensional Hopf quasigroup with faithful integrals. Then, it shows that the biduality theorem also holds for Hopf quasigroups and multiplier Hopf coquasigroups of the discrete type.

A group theoretic description of the κ-Poincaré Hopf algebra

It is well known in the literature that the momentum space associated to the κ-Poincaré algebra is described by the Lie group AN(3). In this letter we show that the full κ-Poincaré Hopf algebra structure can be obtained from rather straightforward group-theoretic manipulations starting from the Iwasawa decomposition of the SO(1,4) group.

Cyclic cohomology and multiplier Hopf algebras

In this paper, we study cyclic cohomology theory on a regular multiplier Hopf algebra. Given a modular pair consisting of a group-like element, a character and in involution for a regular multiplier Hopf algebra, we associate a cyclic module. The approach is based on the Connes-Moscovici cyclic module of Hopf algebras and we obtain the same cyclic modules when our algebra has an identity.

Monotone Cumulant-Moment Formula and Schröder Trees

We prove a formula to express multivariate monotone cumulants of random variables in terms of their moments by using a Hopf algebra of decorated Schröder trees.

Centers of braided tensor categories

Let C be a finite braided multitensor category. Then the end B=∫X∈CX⊗X⁎ is natural a Hopf algebra in C. We show that the Drinfeld center of C is isomorphic to the category of left B-comodules in C, and the decomposition of B into a direct sum of indecomposable C-subcoalgebras leads to a decomposition of B-ComodC into a direct sum of indecomposable C-module subcategories.As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple and cosemisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.

On Hopf algebraic structures of quantum toroidal algebras

ABSTRACT We define an algebra using a simplified set of generators for the quantum toroidal algebra and show that there exists an epimorphism from to . We derive a closed formula of the comultiplication on the generators of that extends that of the quantum affine algebra . As a consequence, we show that is a Hopf algebra for n = 1, 2 and give conjectural formulas in the general case. We further show that is isomorphic to a double algebra.

FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory $${\varPhi }$$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $$H^{\text {cut}}_{\text {FG}}({\varPhi })$$ and lattices of intermediate structures. Feynman diagrams in $$H_{\text {FG}}({\varPhi })$$ are applied to describe states in $${\varPhi }$$ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space $$\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$$ . We associate Hopf subalgebras of $$H_{\text {FG}}({\varPhi })$$ generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of $${\varPhi }$$ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.

Hopf algebras of parking functions and decorated planar trees

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that it is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon.We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements is given by intervals in each of these orders.Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

The "Young" and "Reverse" Dichotomy of Polynomials

A ``flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young" and ``reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of q-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.

Antipodal symmetry of two-loop MHV amplitudes

I present a conjecture that all two-loop MHV amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory possess an antipodal symmetry when evaluated on parity-even kinematics. The symmetry acts as a change of basis on the symbol letters, followed by the antipode operation associated with the Hopf algebra structure of multiple polylogarithms. At the symbol level, I provide the symmetry map explicitly for amplitudes with up to eight external particles, and also provide evidence at all multiplicities. Intriguingly, the map acts as an isomorphism on the normal fans of the Newton polytopes of the symbol letters. The conjectured symmetry is one of the rare known cases where the antipode map shows up in physically important examples.

Classification of finite-dimensional Hopf algebras over dual Radford algebras

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.

Multiparameter quantum groups at roots of unity

We address the study of multiparameter quantum groups (MpQGs) at roots of unity, namely quantum universal enveloping algebras $U\_\mathbf{q}(\mathfrak{g})$ depending on a matrix of parameters $\mathbf{q}=(q\_{ij}){i,j\in I}$. This is performed via the construction of quantum root vectors and suitable “integral forms” of $U\mathbf{q}(\mathfrak{g})$, a restricted one—generated by quantum divided powers and quantum binomial coefficients—and an unrestricted one—where quantum root vectors are suitably renormalized. The specializations at roots of unity of either form are the “MpQGs at roots of unity” we look for. In particular, we study special subalgebras and quotients of our MpQGs at roots of unity—namely, the multiparameter version of small quantum groups—and suitable associated quantum Frobenius morphisms, that link the MpQGs at roots of 1 with MpQGs at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion, often at the core of our strategy, is that every MpQG is actually a $2$-cocycle deformation of the algebra structure of (a lift of) the “canonical” one-parameter quantum group by Jimbo–Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $\mathbf{q}$, our quantum groups yield (through the choice of integral forms and their specializations) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.

Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field \Bbbk of positive characteristic p are studied. We show that every IHOE over \Bbbk satisfies a polynomial identity (PI), with PI-degree a power of p , and that it is a filtered deformation of a commutative polynomial ring. We classify all 2 -step IHOEs over \Bbbk , thus generalising the classification of 2 -dimensional connected unipotent algebraic groups over \Bbbk . Further properties of 2 -step IHOEs are described: for example their simple modules are classified, and every 2 -step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie \Bbbk -algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over \Bbbk .

Group-Graded By-Product Construction and Group Double Centralizer Properties

For a group π with unit e, we introduce and study the notion of a π-graded Hopf algebra. Then we introduce and construct a new braided monoidal category HHeYDπ over a π-graded Hopf algebra H. We introduce the notion of a π-double centralizer property and investigate this property by studying a braided π-graded Hopf algebra U(gln(V))⋉πH, where V is an n-dimensional vector space in HHeYDπ and U(gln(V)) is the braided universal enveloping algebra of gln(V) which is not the usual Hopf algebra. Finally, some examples and special cases are given.

Some interactions between Hopf Galois extensions and noncommutative rings

In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

A shuffle algebra point of view on operator-valued probability theory

We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to the operator-valued case. Given an operator-valued probability space with an algebra B acting on it (on the left and on the right), we associate operators in the operad of multilinear maps on B to the operator-valued distribution and free cumulants of a random variable. These operators define a representation of a PRO of non-crossing partitions. Using concepts from higher category theory, specifically 2-monoidal categories, we define a notion of unshuffle Hopf algebra on an underlying PRO. We introduce a PRO of words insertions and show that both the latter and the PRO of non-crossing partitions are unshuffle Hopf algebras. The two relate by mean of a map of unshuffle bialgebra (in a 2-monoidal sense) which we call the splitting map. Ultimately, we obtain a left half-shuffle fixed point equation corresponding to free moment-cumulant relations in a shuffle algebra of bicollection homomorphisms on the PRO of words insertions. Right half-shuffle and shuffle laws are interpreted in the framework of boolean and monotone non-commutative probability theory, respectively.

Convolution bialgebra of a Lie groupoid and transversal distributions

For a Lie groupoid G over a smooth manifold M we construct the adjoint action of the étale Lie groupoid G# of germs of local bisections of G on the Lie algebroid g of G. With this action, we form the associated convolution Cc∞(M)/R-bialgebra Cc∞(G#,g). We represent this Cc∞(M)/R-bialgebra in the algebra of transversal distributions on G. This construction extends the Cartier-Gabriel decomposition of the Hopf algebra of distributions with finite support on a Lie group.

Gauge Symmetries and Renormalization

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.

The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

Cohomological obstructions and weak crossed products over weak Hopf algebras

Let H be a cocommutative weak Hopf algebra and let (B,φB) a weak left H-module algebra. In this paper, for a twisted convolution invertible morphism σ:H2→B we define its obstruction θσ as a Sweedler 3-cocycle with values in the center of B. We obtain that the class of this obstruction vanish in third Sweedler cohomology group HφZ(B)3(H,Z(B)) if, and only if, there exists a twisted convolution invertible 2-cocycle α:H2→B such that H⊗B can be endowed with a weak crossed product structure with α keeping a cohomological-like relation with σ. Then, as a consequence, the class of the obstruction of σ vanish if, and only if, there exists a cleft extension of B by H.