Hopf Monoids Research Articles

Hopf monoids in varieties

Commutative varieties provide a natural setting for generalizing Hopf algebra theory over commutative rings, since they satisfy the various conditions identified in the category theoretical analysis of this theory to guarantee for example the existence of all naturally occurring forgetful functors in this context, the existence of universal measuring comonoids and the existence of generalized finite duals. It will be shown in addition, that crucial properties of the latter, known from the case of Hopf algebra theory over commutative rings, can be generalized Hopf algebra theory over a commutative variety. The attempt to generalize its construction leads to a couple of questions concerning properties of the monoidal structure of a commutative variety, which seem to be of a more general interest.

Hopf measuring comonoids and enrichment

We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable $P(A,B)$-comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved --- in an appropriate, multiplier-valued sense --- which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved --- which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.

Hopf comonads on naturally Frobenius map-monoidales

We study monoidal comonads on a naturally Frobenius map-monoidale M in a monoidal bicategory M. We regard them as bimonoids in the duoidal hom-category M(M,M), and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in M, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf monoids in braided monoidal categories, of small groupoids, of Hopf algebroids over commutative base algebras, of weak Hopf algebras, and of Hopf monads in the sense of Bruguières and Virelizier.

Strong forms of self-duality for Hopf monoids in species

A vector species is a functor from the category of finite sets with bijections to vector spaces (over a fixed field); informally, one can view this as a sequence of S n S_n -modules, one for each natural number n n . A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. A vector species has a basis if and only if it is given by a sequence of S n S_n -modules which are permutation representations. We say that a Hopf monoid is freely self-dual if it is connected and finite-dimensional, and if it has a basis in which the structure constants of its product and coproduct coincide. Such Hopf monoids are self-dual in the usual sense, and we show that they are furthermore both commutative and cocommutative. We prove more specific classification theorems for freely self-dual Hopf monoids whose products (respectively, coproducts) are linearized in the sense that they preserve the basis; we call such Hopf monoids strongly self-dual (respectively, linearly self-dual). In particular, we show that every strongly self-dual Hopf monoid has a basis isomorphic to some species of block-labeled set partitions, on which the product acts as the disjoint union. In turn, every linearly self-dual Hopf monoid has a basis isomorphic to the species of maps to a fixed set, on which the coproduct acts as restriction. It follows that every linearly self-dual Hopf monoid is strongly self-dual. Our final results concern connected Hopf monoids which are finite- dimensional, commutative, and cocommutative. We prove that such a Hopf monoid has a basis in which its product and coproduct are both linearized if and only if it is strongly self-dual with respect to a basis equipped with a certain partial order, generalizing the refinement partial order on set partitions.

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

The category Hopf ℂ of Hopf monoids in a symmetric monoidal category ℂ, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ℂ preserve directed colimits one has the following results: (1) If, in ℂ, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ℂ and comonadic over the category of monoids in C. (2) If, in ℂ, extremal monomorphisms are stable under tensor squaring, then Hopf ℂ is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ℂ.

The characteristic polynomial of the Adams operators on graded connected Hopf algebras

The Adams operators $\Psi_n$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. We study the Adams operators when $H$ is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator $\Psi_n$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$ (the case $n=-1$). We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and $q$-Hopf algebras.

Strong forms of linearization for Hopf monoids in species

A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. We say that a Hopf monoid is strongly linearized if it has a "basis" preserved by its product and coproduct in a certain sense. We prove several equivalent characterizations of this property, and show that any strongly linearized Hopf monoid which is commutative and cocommutative possesses four bases which one can view as analogues of the classical bases of the algebra of symmetric functions. There are natural functors which turn Hopf monoids into graded Hopf algebras, and applying these functors to strongly linearized Hopf monoids produces several notable families of Hopf algebras. For example, in this way we give a simple unified construction of the Hopf algebras of superclass functions attached to the maximal unipotent subgroups of three families of classical Chevalley groups.

The Hopf monoid on nonnesting supercharacters of pattern groups

We construct supercharacter theories for a collection of unipotent matrix groups and produce a Hopf monoid from the supercharacters. These supercharacter theories are coarser than those defined by Diaconis--Isaacs for algebra groups and have supercharacters and superclasses indexed by nonnesting labeled set partitions. We compute the supercharacter tables and describe the product and coproduct of the Hopf monoid combinatorially. We also show that this Hopf monoid is free.

On the Hadamard Product of Hopf Monoids

AbstractCombinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with theHadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

The antipode and primitive elements in the Hopf monoid of supercharacters

From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum functions, giving new cancelation free formulae for the standard Hopf algebra of supercharacters and symmetric functions in noncommuting variables. We also give partial results for the antipode on the supercharacter basis, and explicitly describe the primitives of this Hopf monoid.

Yetter-Drinfeld modules over weak braided Hopf monoids and center categories

In this paper, we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter-Drinfeld modules over a weak braided Hopf monoid. We apply the general result to the calculus of the center in module categories.

Hopf monoids from class functions on unitriangular matrices

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.

Lagrange's Theorem for Hopf Monoids in Species

AbstractFollowing Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange‘s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies k of a Hopf monoid h to be a Hopf submonoid: the quotient of any one of the generating series of h by the corresponding generating series of k must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopfmonoid in the formof certain polynomial inequalities and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.

Lagrange's Theorem for Hopf Monoids in Species

We prove Lagrange's theorem for Hopf monoids in the category of connected species. We deduce necessary conditions for a given subspecies $\textrm{k}$ of a Hopf monoid $\textrm{h}$ to be a Hopf submonoid: each of the generating series of $\textrm{k}$ must divide the corresponding generating series of $\textrm{k}$ in ℕ〚x〛. Among other corollaries we obtain necessary inequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In the set-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence. Nous prouvons le théorème de Lagrange pour les monoïdes de Hopf dans la catégorie des espèces connexes. Nous déduisons des conditions nécessaires pour qu'une sous-espèce $\textrm{k}$ d'un monoïde de Hopf $\textrm{h}$ soit un sous-monoïde de Hopf: chacune des séries génératrices de $\textrm{k}$ doit diviser la série génératrice correspondante de $\textrm{h}$ dans ℕ〚x〛. Parmi d'autres corollaires nous trouvons des inégalités nécessaires pour qu'une suite d'entiers soit la suite des dimensions d'un monoïde de Hopf. Dans le cas ensembliste les inégalités sont linéaires et exigent que la transformée binomiale de la suite soit non négative.

Weak Hopf monoids in braided monoidal categories

We develop the theory of weak bimonoids in braided monoidal categories and show that they are in one-to-one correspondence with quantum categories with a separable Frobenius object-of-objects. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R⊗R.

Universal constructions for Hopf algebras

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are the various subcategory-embeddings arsing naturally in this context reflexive or coreflexive? (c) When does a category of Hopf monoids have all limits or colimits? These problems are also shown to be intimately related. Particular emphasis is given to the case of Hopf algebras, i.e., when the chosen symmetric monoidal category is the category of modules over a commutative unital ring.