Infinitesimal or cocommutative dipterous bialgebras and good triples of operads
Leroux Philippe
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22 pages, 2 figures. Version 3: New results added: The triple of operads (2As,Dipt,Vect) is good and an injective coding of trees into m-ary trees. Version 4: The name Grove has been changed to the name Mag^\infty (known in the literature) + minor modifications
The works of Poincare, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com, As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on themselves and look for good triples were $As$ is replaced by the dipterous operad Dipt. Since the work of Loday and Ronco, the triple of operads (As, Dipt, B_\infty) endowed with the semi-Hopf compatibility relations is known to be good. In this paper, we prove that the triple of operads (As, Dipt, Grove) endowed with the so-called (nonunital) semi-infinitesimal compatibility relations is good. For that, explicit constructions of the free dipterous algebra and the free grove-algebra over a K-vector space V are given. These constructions turn out to be related to rooted planar trees and the little an large Schroeder numbers. Many examples of dipterous algebras are given, notably the free L-dipterous algebras. As a corollary of our results, we also recover that the triple of operads (2As, Dipt, Vect) endowed both with the unital semi-Hopf and with the unital semi-infinitesimal compatibility relations is good, where 2As denotes the operad of 2-associative algebras. We also open this paper on a good triple, related to the Connes-Kreimer Hopf algebra in quantum field theory, (Com, Dipt, Prim_{Com} Dipt) endowed with the Hopf compatibility relations and also present a general theorem giving good triples from entangled dipterous like operads named associative molecules.
We construct a bialgebra object in the category of linear maps LM from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgebras, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.
In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra $V$, it is proved that the set $G(V)$ of group-like elements is naturally an abelian semigroup, whereas the set $P(V)$ of primitive elements is a vertex Lie algebra. For $g\in G(V)$, denote by $V_g$ the connected component containing $g$. Among the main results, it is proved that if $V$ is a cocommutative vertex bialgebra, then $V=\oplus_{g\in G(V)}V_g$, where $V_{\bf 1}$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra ${\mathcal{V}}_{P(V)}$ associated to the ver…
Let ${\Bbb F}_{n}$ be the free group of rank $n$ and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C$^{*}$-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n}$ $(1\leq n<\infty$). We introduce a new comultiplication $Δ_φ$ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,Δ_φ)$ is a non-cocommutative C$^{*}$-bialgebra. With respect to $Δ_φ$, the tensor product $π\otimes_φπ'$ of any two representations $π$ and $π'$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of sev…
We show that the inductive limit of a certain inductive system of quasi-cocommutative C$^{*}$-bialgebras is not quasi-cocommutative. This implies that the category of quasi-cocommutative C$^{*}$-bialgebras is not closed with respect to the inductive limit.
Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $Δ_φ$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ)$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ,R)$ is triangular.
The paper is devoted to prove a version of Milnor-Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from 2, we prove that, for a given connected braided bialgebra $A$ having a $λ$-cocommutative infinitesimal braiding for some regular element $λ\neq 0$ in the base field, then the infinitesimal braiding of $A$ is of Hecke-type of mark $λ$ and $A$ is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements.
We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.
Realizing the possibility suggested by Hardouin [6], we show that her own Picard-Vessiot Theory for iterative $q$-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [2], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative $q$-difference modules over an iterative $q$-difference ring $R$, by modules over a certain cocommutative $\times_R$-bialgebra. Recall that the notion of $\times_R$-bialgebras was defined by Sweedler [17], as a generalization of bialgebras.
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution spa…
We introduce the notion of Gamma-Lie bialgebra, where Gamma is a group. These objects give rise to cocommutative co-Poisson algebras, for which we construct quantization functors. This enlarges the class of co-Poisson algebras for which a quantization is known. Our result relies on our earlier work, where we showed that twists of Lie bialgebras can be quantized; we complement this work by studying the behavior of this quantization under compositions of twists.
We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincaré-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid for cocommutative bialgebras, to a large class of generalized bialgebras. Technically we work in the theory of operads which permits us to give a conceptual proof of our main theorem. It unifies several results, generalizing…
Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the first into Drinfeld twistings of the second. They possess a composition operation extending composition of bialgebra homomorphisms. Gauge transformations of twists, compatible with adjacent homomorphisms, give rise to gauge transformation of twisted homomorphisms, which behave nicely with respect to compositions. Here we study (gauge classes of) twisted automorphisms of cocommutative Hopf algebras. After revising well-known relations between twists, twisted forms of bialgebras and $R$-matrices (for commutative bialgebras)…
Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well known fact that the representation theory of the quantized group is completely analogous to its classical counterpart. The subalgebras of cocommutative elements in the corresponding FRT-bialgebras (defined by Faddeev, Reshetikhin, and Takhtadzhyan) are explicitly determined, using a bialgebra embedding of the FRT-bialgebra into the tensor product of the qua…
Let ${\sf CK}_{*}$ denote the C$^{*}$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication $Δ_ϕ$ and a counit $ε$ on ${\sf CK}_{*}$ such that $Δ_ϕ$ is a nondegenerate $*$-homomorphism from ${\sf CK}_{*}$ to ${\sf CK}_{*}\otimes {\sf CK}_{*}$ and $ε$ is a $*$-homomorphism from ${\sf CK}_{*}$ to ${\bf C}$. From this, ${\sf CK}_{*}$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra. Furthermore, C$^{*}$-bialgebra automorphisms, a tensor product of representations and C$^{*}$-subbialgebras of ${\sf CK}_{*}$ are investigated.
The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $Δ_φ$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},Δ_φ)$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},Δ_φ)$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=Δ_φ(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.
Let ${\cal A}_{0}(*)$ denote the direct sum of a certain set of UHF algebras and let ${\cal A}(*)\equiv {\bf C}\oplus {\cal A}_{0}(*)$. We introduce a non-cocommutative comultiplication $Δ_ϕ$ on ${\cal A}(*)$, and give an example of comodule-C$^{*}$-algebra of the C$^{*}$-bialgebra $({\cal A}(*),Δ_ϕ)$. With respect to $Δ_ϕ$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.
An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommut…
We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal pro…
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same cod…
Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of all Cuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication $Δ_ϕ$ and a counit $ε$. Let ${\rm BI}({\cal O}_{*})$ denote the set of all closed biideals of the C$^{*}$-bialgebra $({\cal O}_{*},Δ_ϕ,ε)$ and let ${\cal P}({\bf P})$ denote the power set of the set of all prime numbers. We show a one-to-one correspondence between ${\rm BI}({\cal O}_{*})$ and ${\cal P}({\bf P})$. Furthermore, we show that for any ${\cal I}$ in ${\rm BI}({\cal O}_{*})$, there exists a C$^{*}$-subbialgebra ${\cal B}_{\cal I}$ of …
We introduce a certain differential graded bialgebra, neither commutative nor cocommutative, that governs perturbations of a differential on complexes supplied with an abstract Hodge decomposition. This leads to a conceptual treatment of the Homological Perturbation Lemma and its multiplicative version. As an application we give an explicit form of the decomposition theorem for $A_\infty$ algebras and $A_\infty$ modules and, more generally, for twisted objects in differential graded categories.
A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the subalgebra of coinvariants) is obtained for such coactions of a cosemisimple Hopf algebra. This is applied to two adjoint coactions of quantized function algebras of classical groups on the associated FRT bialgebra. Provided that the Hopf algebra is cosemisimple and coquasitriangular, the algebras of coinvariants form two finitely generated, commutative, grad…
For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah--Bott--Goldman Poisson structures on moduli spaces of representations of surface groups.
Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*_R a certain deformation of the multiplication of H^* via R. We prove that H^*_R is a quantum commutative left H\otimes H^{op cop}-module algebra. If H is the Drinfel'd double of A then H^*_R is the Heisenberg double of A. We study the relation between H^*_R and Majid's "covariantised product". We give a formula for the canonical element of the Heisenberg double of A, solution to the pentagon equation, in terms of the R-matrix of the Drinfel'd double of A. We generalize a theorem of Jiang-Hua Lu on …
The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, given any principal ideal domain R, for each prime h in R we establish sort of an "inner" Galois' correspondence on the category HA of torsionless Hopf algebras over R, via the definition of two functors (from HA to itself) such that the image of the first, resp. of the second, is the full …
