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We introduce the notion of species relative to a fixed hyperplane arrangement. Roughly speaking, a species is a family of vector spaces, one for each face of the arrangement, along with linear isomorphisms between vector spaces indexed by faces of the same support. Next, we introduce the notion of a monoid in species. It consists of a species equipped with "product'' maps from a vector space indexed by a face to a vector space indexed by a smaller face. These are subject to naturality, associativity, unitality axioms. There is also a dual notion of a comonoid in species defined using `"coproduct'' maps, and a mixed self-dual notion of a bimonoid in species. We also define commutativity for a monoid and dually cocommutativity for a comonoid. A bimonoid could be commutative, cocommutative, both or neither. Commutative monoids, cocommutative comonoids, bicommutative bimonoids are convenient to formulate using flats rather than faces. In addition to the above, we discuss related objects such as q-bimonoids (which include bimonoids, signed bimonoids, 0-bimonoids), signed commutative monoids, and partially commutative monoids. The latter interpolate between monoids and commutative monoids. The above notion of species when specialized to the braid arrangements relates to the classical notion of Joyal species.
This chapter reviews important notions and results on hyperplane arrangements required in the book. The discussion includes: 1) geometric objects such as faces, flats, bifaces, partial-flats, nested faces, lunes, bilunes, cones; 2) algebraic objects such as the Tits monoid, Birkhoff monoid, Janus monoid and their linearized algebras; Lie and Zie elements; 3) combinatorial objects such as distance functions and Varchenko matrices; descent, lune, Witt identities; incidence algebras, zeta and Möbius functions along with their noncommutative and two-sided analogues, the Zaslavsky formula and its noncommutative analogue.
We introduce the Hadamard product on the category of species (relative to a fixed hyperplane arrangement). A key property of this product is that it preserves monoids, comonoids, bimonoids. In fact, for any scalars p and q, the Hadamard product of a p-bimonoid and a q-bimonoid is a pq-bimonoid. Similarly, the Hadamard product of (co)commutative (co)monoids is again (co)commutative. These facts can be seen as formal consequences of the bilax property of the Hadamard functor. We construct the internal hom for the Hadamard product of species, and discuss its bilax property and the related constructions of the convolution monoid, coconvolution comonoid, biconvolution bimonoid. Moreover, we also construct the internal hom for the Hadamard product of monoids, comonoids and bimonoids, making critical use of the fact that these are functor categories just like the category of species. The internal hom for (co, bi)commutative bimonoids is intimately connected to the internal hom for the tensor product of modules over the Birkhoff algebra, Tits algebra, Janus algebra. We construct the universal measuring comonoid from one monoid to another monoid. It allows us to enrich the category of monoids over the category of comonoids. This enriched category possesses powers and copowers which we describe explicitly. The power is in fact the convolution monoid. The copower is a certain quotient of the free monoid on the Hadamard product of the given comonoid and monoid. We introduce the bimonoid of star families. It is constructed out of a cocommutative comonoid and a bimonoid. It builds on the internal hom for the Hadamard product of comonoids. Moreover, it has a commutative counterpart which we call the bicommutative bimonoid of star families. This one builds on the internal hom for cocommutative comonoids. There is also an analogous construction starting with a bimonoid and a commutative monoid which builds on the universal measuring comonoid. These bimonoids play an important role in the study of exp-log correspondences. We introduce the signature functor on species. It is defined by taking Hadamard product with the signed exponential species. The latter carries the structure of a signed bimonoid. This sets up an equivalence between the categories of bimonoids and signed bimonoids.
We introduce the notion of a Lie monoid in species (relative to a fixed hyperplane arrangement). This goes hand-in-hand with the notions of (co)monoid, (co)commutative (co)monoid, bimonoid. In contrast to monoids and commutative monoids, Lie monoids are considerably harder to formulate and study. Recall that left modules over the commutative and associative operads are commutative monoids and monoids, respectively. In a similar vein, left modules over the Lie operad are defined to be Lie monoids. One can also formulate Lie monoids in terms of a Lie bracket subject to antisymmetry and Jacobi identity. This arises from the presentation of the Lie operad. We work throughout with the definition of Lie monoids as left modules over the Lie operad; however, for additional clarity, we also illustrate many constructions and results using the Lie bracket formulation. Every monoid carries the structure of a Lie monoid via the commutator bracket. For a bimonoid, the commutator bracket restricts to its primitive part. Thus, the primitive part of any bimonoid carries the structure of a Lie monoid. In the other direction, to every Lie monoid, one can associate its universal enveloping monoid. These constructions can be expressed as adjunctions. The free Lie monoid on a species can be described using the Lie operad of the arrangement. There is an adjunction between the categories of species and bimonoids involving the primitive part functor. The resulting monad on species coincides with the monad induced by the Lie operad. In particular, this explains why the primitive part of a bimonoid is a Lie monoid. It also shows that the primitive part of the free bimonoid on a species is the free Lie monoid on that species. As a special case, the Zie species carries the structure of a Lie monoid and moreover, it is isomorphic to the free Lie monoid on the exponential species. All the above considerations carry over to the signed setting. Thus, we have the notion of a signed Lie monoid which is a left module over the signed Lie operad. A signed Lie monoid can also be formulated using signed antisymmetry and signed Jacobi identity which involve the signed distance function on faces of the arrangement. We also briefly discuss the dual notion of Lie comonoids. They are left comodules over the Lie cooperad. They can also be formulated in terms of a Lie cobracket. Related notions are the cocommutator cobracket, cofree Lie comonoid, universal coenveloping comonoid.
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. In each case, the result provides explicit inverse isomorphisms between two universally constructed bimonoids. We call these the Hoffman-Newman-Radford (HNR) isomorphisms. For a cocommutative comonoid, the free bimonoid on that comonoid is isomorphic to the free bimonoid on the same comonoid but with the trivial coproduct. The product is concatenation in both, but the coproducts differ, it is dequasishuffle in the former and deshuffle in the latter. An explicit isomorphism can be constructed in either direction, one direction involves a noncommutative zeta function, while the other direction involves a noncommutative Möbius function.These are the HNR isomorphisms. There is a dual result starting with a commutative monoid.In this case, the coproduct is deconcatenation in both, but the products differ, it is quasishuffle in the former and shuffle in the latter. Interestingly, these ideas can be used to prove that noncommutative zeta functions and noncommutative Möbius functions are inverse to each other in the lune-incidence algebra. There is a commutative analogue of the above results in which the universally constructed bimonoids are bicommutative. Now the HNR isomorphisms are constructed using the zeta function and Möbius function of the poset of flats. As an application, we explain how they can be used to diagonalize the mixed distributive law for bicommutative bimonoids. There is also a q-analogue, for q not a root of unity. In this case, the HNR isomorphisms involve the two-sided q-zeta and q-Möbius functions. As an application, we explain how they can be used to study the nondegeneracy of the mixed distributive law for q-bimonoids.
We discuss the free monoid and the cofree comonoid on a species (relative to a fixed hyperplane arrangement). In addition, we discuss the free bimonoid on a comonoid, and dually the cofree bimonoid on a monoid. More generally, for any scalar q, we have the free q-bimonoid on a comonoid and the cofree q-bimonoid on a monoid. An important special case is when the starting (co)monoid has trivial (co)product. We employ the terms concatenation and q-(quasi)shuffle for the products, and deconcatenation and q-de(quasi)shuffle for the coproducts. For q = 1, the q-(quasi)shuffle product is commutative, while the q-de(quasi)shuffle coproduct is cocommutative. The concatenation product and deconcatenation coproduct do not depend on q, and do not satisfy any commutativity property. In addition, we also discuss the free commutative monoid and the cofree cocommutative comonoid on a species and related constructions. These have signed analogues. We discuss the q-norm map between free and cofree q-bimonoids. It is an isomorphism when q is not a root of unity. Invertibility of the Varchenko matrix associated to the q-distance function plays a critical role here. We also discuss the (co)free graded (co)monoid on a graded species. Every species can be viewed as a graded species concentrated in degree 1. The free graded monoid on a species has a unique coproduct which turns it into a graded q-bimonoid. This is precisely the q-deshuffle coproduct. Dually, the q-shuffle product is the unique product which turns into a graded q-bimonoid.
This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal category. However, that is not the case here; the relevant concept is that of monads and algebras over monads. We construct a monad on the category of species, and observe that algebras over it are the same as monoids in species. Dually, we construct a comonad whose coalgebras are the same as comonoids in species. In addition, we construct a mixed distributive law between this monad and comonad such that bialgebras over the resulting bimonad are the same as bimonoids in species. Moreover, the mixed distributive law can be deformed by a parameter q such that the resulting bialgebras are the same as q-bimonoids. The above monad, comonad, bimonad have commutative counterparts which relate to commutative monoids, cocommutative comonoids, bicommutative bimonoids in species. We briefly discuss the Mesablishvili-Wisbauer rigidity theorem. As a consequence, the category of species is equivalent to the category of 0-bimonoids, as well as to the category of bicommutative bimonoids. These ideas are developed in more detail later. We extend the notion of species from a hyperplane arrangement to the more general setting of a left regular band (LRB).
For any q-bimonoid (relative to a fixed hyperplane arrangement), we define its antipode as a map from the bimonoid to itself whose face-components are defined by taking an alternating sum with each summand being the composite of a coproduct component followed by a product component. We refer to this formula as the Takeuchi formula. Up to signs, it equals the 0-logarithm of the identity map on the bimonoid. There is also a commutative analogue of this formula for bicommutative bimonoids. We study interactions of the antipode with morphisms of bimonoids, the duality functor, bimonoid filtrations, the signature functor. The antipode is intimately related to the antipodal map on the arrangement; this is brought forth by its interaction with op and cop constructions. We compute logarithm of the antipode map using the noncommutative Zaslavsky formula, and moreover relate it to logarithm of the identity map. Understanding the cancelations in the Takeuchi formula for a given bimonoid is often a challenging combinatorial problem. We solve this problem for the exponential bimonoid, the bimonoid of chambers, the bimonoid of faces, the bimonoid of flats, and so on. More generally, we provide cancelation-free formulas for the antipode of any bimonoid which arises from a universal construction. In a similar vein, for any set-bimonoid, this problem can be interpreted as the calculation of the Euler characteristic of a cell complex. The descent and lune identities provide motivating examples of such a calculation. The antipode map is closely related to the Takeuchi elements associated to arrangements. More precisely, the face-component of the antipode map is the characteristic operation by the Takeuchi element of the arrangement over the support of that face. This connection makes it possible to study the antipode using properties of the Takeuchi elements.
We discuss some important rigidity theorems related to universal constructions. They usually take the form of an adjoint equivalence between suitable categories. The Loday-Ronco theorem says that the category of 0-bimonoids is equivalent to the category of species. In particular, 0-bimonoids are both free and cofree. This theorem is a special case of a more general result in which 0-bimonoids are replaced by q-bimonoids with q not a root of unity. We refer to this result as the rigidity of q-bimonoids. Invertibility of the Varchenko matrix associated to the q-distance function on faces plays a critical role here. The Leray-Samelson theorem says that the category of bicommutative bimonoids is equivalent to the category of species. In particular, bicommutative bimonoids are both free commutative and cofree cocommutative. There is also a signed analogue of Lerayâ€“Samelson which applies to signed bicommutative signed bimonoids. The Borelâ€“Hopf theorem says that any cocommutative bimonoid is cofree on its primitive part, and dually, any commutative bimonoid is free on its indecomposable part. This result also has a signed analogue which applies to signed (co)commutative signed bimonoids. We present three broad approaches to these rigidity theorems. The first approach is elementary and proceeds by an induction on the primitive filtration of the bimonoid. Here a key role is played by how the bimonoid axiom works on the primitive part. The second approach is more direct and proceeds by constructing an explicit inverse to the appropriate universal map. The universal map is defined using a zeta function and the inverse using a Möbius function. These maps have connections to the exponential and logarithm operators. The third approach is also constructive and employs (commutative, usual or two-sided) characteristic operations by suitable families of idempotents in the (Birkhoff, Tits or q-Janus) algebra, respectively, to decompose the given bimonoid. All results in this chapter are independent of the characteristic of the base field.
We introduce the notion of dispecies relative to a fixed hyperplane arrangement. The category of dispecies carries a monoidal structure which we call the substitution product. Operads are monoids in this monoidal category. We describe the free operad on a dispecies, and then proceed to operad presentations with an emphasis on binary quadratic operads. Apart from the substitution product, the category of dispecies also carries the Hadamard product which turns it into a 2-monoidal category. Hopf operads are bimonoids in this 2-monoidal category. We use these ideas to construct the black and white circle products on binary quadratic operads. We discuss three main examples of operads, namely, commutative, associative, Lie. These are all binary quadratic. Further, under a suitable notion of quadratic duality, the commutative and Lie operads are duals of each other, while the associative operad is self-dual. These can be viewed as extensions of well-known facts from the classical theory of May operads. The category of species is a left module category over the monoidal category of dispecies (under the substitution product). Hence, to each operad, one can associate the category of its left modules. A left module over the associative operad is the same as a monoid in species, over the commutative operad is the same as a commutative monoid in species, over the Lie operad is the same as a Lie monoid in species. To every operad, one can attach an (associative) algebra called its incidence algebra. The incidence algebra of the commutative operad is the flat-incidence algebra, of the associative operad is the lune-incidence algebra, and of the Lie operad is the Tits algebra. The incidence algebra of any connected quadratic operad is elementary and its quiver can be explicitly described. Operads can also be defined in the more general setting of left regular bands. Interestingly, the commutative, associative, Lie operads extend to this setting.