The author studies categories of generalized bialgebras defined by triples \((\mathbb C,\between,\mathcal A)\), where \(\mathcal A\) is an operad, which collects operations \(\mu: \mathcal H^{\otimes r}\to \mathcal H\) governing the multiplicative structure, \(\mathbb C\) is another operad, which collects operations \(\delta: \mathcal H\to \mathcal H^{\otimes s}\) governing the comultiplicative structure, and \(\between\) is a distribution law governing the compatibility relation between the multiplicative and the comultiplicative structure of our category of bialgebras. Intituitively, the distribution law \(\between\) is a rewriting rule which changes any composite
\[ \mathcal H^{\otimes r}@>\mu>>\mathcal H@>\delta>>\mathcal H^{\otimes s} \]
of a multiplicative operation \(\mu\in\mathcal A(r)\) followed by a comultiplicative operation \(\delta\in\mathbb C(s)\) into a sum of composites
\[ \mathcal H^{\otimes r}@>{\delta_1\otimes\cdots\otimes\delta_r}>>\mathcal H^{\otimes n} @>{\sigma_*}>>\mathcal H^{\otimes n} @>{\mu_1\otimes\cdots\otimes\mu_s}>>\mathcal H^{\otimes s}, \]
where each \(\delta_i\) is an operation of the operad \(\mathbb C\), each \(\mu_j\) is an operation of \(\mathcal A\), and the map \(\sigma_*\) is a tensor permutation.
The distribution law \(\between\) is generally assumed to preserve operadic composition structures on both sides. This axiom implies that \(\between\) is determined by its action on generating operations whenever the operads \(\mathcal A\) and \(\mathbb C\) are defined by a presentation by generators and relations. In all examples, the author implicitly uses this observation and the preservation of composition structures. Nevertheless his main results hold under slightly less rigid assumptions. The reader is referred to the work of T. F. Fox and M. Markl [Contemp. Math. 202, 167–205 (1997; Zbl 0866.18008)] for a full account of the axioms of distribution laws in the context of operads.
The usual categories of biassociative bialgebras (cocommutative or not), as well as the other standard example of Lie bialgebras, fit the setting of operads in distribution. In the usual example of biassociative bialgebras, we take two copies of the associative operad \(\mathbb C = \mathcal A = \text{As}\), the operad defined by a generating operation \(m\in\text{As}(2)\) such that \(m(m,1) = m(1,m)\), and the distribution law is determined by the identity \(\delta\cdot\mu = \mu\otimes\mu\cdot(\tau_{2 3})_*\cdot\delta\otimes\delta\), where \(\mu\) (respectively, \(\delta\)) refers to the copy in \(\mathcal A\) (respectively, \(\mathbb C\)) of the generating operation \(m\in\text{As}(2)\) and the permutation \(\tau_{2 3}\) is the transposition of \((2,3)\). The author deals with a non-unital version of this structure. Many new examples of operads in distribution have been discovered by the author and his coworkers. Among others, we have an exotic distribution law on the associative operad \(\mathbb C = \mathcal A = \text{As}\) giving rise to a new category of biassociative bialgebras [J.-L. Loday and M. Ronco, J. Reine Angew. Math. 592, 123–155 (2006; Zbl 1096.16019)]. Many such examples are reviewed in the book.
The main purpose of the book under review is to prove analogues of the classical Poincaré-Birkhoff-Witt and Milnor-Moore theorems in the context of bialgebras over operads in distribution. The author gives a conceptual setting, unifying many cases addressed in the literature, in which such theorems hold.
First of all, we have to define an analogue of the primitive part of bialgebras in the context of operads in distribution. The operad \(\mathbb C\) (like \(\mathcal A\)) is supposed to vanish in arity \(0\). This convention amounts to considering non-unital (co)algebras. The author also assumes that \(\mathbb C\) (and \(\mathcal A)\) is spanned in arity \(1\) by the operadic unit, the fixed element of the operad which represents the identity operation on the associated category of algebras. Then he defines the primitive part of a \(\mathbb C\)-\(\mathcal A\)-bialgebra \(\operatorname{Prim}\mathcal H\) as the subspace of elements \(x\in\mathcal H\) such that \(\delta(x) = 0\), \(\forall\delta\in\mathbb C(n)\), \(n\geq 2\). He proves that \(\operatorname{Prim}\mathcal H\) is an algebra over an operad \(\mathcal P = \operatorname{Prim}_{\mathbb C}\mathcal A\), the primitive operad, naturally associated to the triple \((\mathbb C,\between,\mathcal A)\).
The primitive operad \(\mathcal P\) forms naturally a suboperad of \(\mathcal A\) and we have a functor \(F: \mathcal A\text{-Alg}\to\mathcal P\text{-Alg}\), from the category of \(\mathcal A\)-algebras to the category of \(\mathcal P\)-algebras, since any \(\mathcal A\)-algebra inherits an action of \(\mathcal P\) by restriction of structure. This restriction functor admits a left adjoint \(U: \mathcal P\text{-Alg}\to\mathcal A\text{-Alg}\) mapping a \(\mathcal P\)-algebra \(L\) to an associated enveloping \(\mathcal A\)-algebra \(U(L)\). The author proves that \(U(L)\) inherits a natural \(\mathbb C\)-\(\mathcal A\)-bialgebra structure.
The author assumes that the distribution law yields a \(\mathbb C\)-coalgebra structure on the free algebra \(\mathcal A(V)\) associated to any vector space \(V\). This requirement amounts to a compatibility relation between the distribution law and operadic composites which gives the minimal setting for the author’s structure results. The standard axioms of operads in distribution give a stronger compatibility relation and are not fully necessary for this purpose.
We have a natural morphism of \(\mathbb C\)-coalgebras \(\phi(V): \mathcal A(V)\to\mathbb C^c(V)\) induced by the projection \(\mathcal A(V)\to V\), where \(\mathbb C^c(V)\) is the cofree \(\mathbb C\)-coalgebra cogenerated by \(V\). In the case where the map \(\phi(V)\) is an iso, the author obtains a rigidity theorem, asserting that the primitive operad is the unit operad \(\mathcal P = I\), the operad associated to the category of vector spaces, and that any connected \(\mathcal A\)-\(\mathbb C\)-bialgebra \(\mathcal H\) is isomorphic to both the free \(\mathcal A\)-algebra \(\mathcal A(\operatorname{Prim}\mathcal H)\) and the cofree \(\mathbb C\)-coalgebra \(\mathbb C^c(\operatorname{Prim}\mathcal H)\) on its primitive part \(\operatorname{Prim}\mathcal H\). In the case where the map \(\phi(V)\) is a split epi in the category of \(\mathbb C\)-coalgebras, the author obtains a Poincaré-Birkhoff-Witt theorem, asserting that the enveloping \(\mathcal A\)-algebra of a \(\mathcal P\)-algebra \(\mathcal H = U(L)\) is a cofree \(\mathbb C\)-coalgebra over its primitive part \(\mathbb C^c(\operatorname{Prim}\mathcal H)\), and a Milnor-Moore theorem, asserting that the functors \(U: L\mapsto U(L)\) and \(\operatorname{Prim}: \mathcal H\to\operatorname{Prim}\mathcal H\) define inverse equivalences of categories between the category of \(\mathcal P\)-algebras and the category of connected \(\mathcal A\)-\(\mathbb C\)-bialgebras. In this situation, the author says that the operads \((\mathbb C,\mathcal A,\mathcal P)\) form a good triple of operads. In both cases, the arguments require to take a field of characteristic zero as ground ring.
The classical Poincaré-Birkhoff-Witt and Milnor-Moore theorems correspond to the instance of a good triple \((\text{Com,\,As,\,Lie})\) formed by the commutative operad \(\mathbb C = \text{Com}\), the associative operad \(\mathcal A = \text{As}\), and the Lie operad \(\mathcal P = \text{Lie}\). The author also observes that the classical Hopf-Borel theorem, which gives the structure of commutative cocommutative bialgebras, corresponds to the case \((\mathcal A,\between,\mathbb C) = (\text{Com},\between,\text{Com})\) of the rigidity theorem. The exotic category of biassociative bialgebras mentioned at the beginning of this review gives another instance of a category associated to operads in distribution \((\mathcal A,\between,\mathbb C) = (\text{As},\between,\text{As})\) for which the isomorphism requirement, and hence the rigidity theorem, holds.
To obtain his results, the author considers a decomposition into components of homogeneous weight \(\omega = \sum_{n\geq 1}\omega^{[n]}\) of the composite map
\[ \mathcal H\to\mathbb C^c(\mathcal H)@>{\psi(\mathcal H)}>>\mathcal A(\mathcal H)\to\mathcal H \] where \(\mathcal H\to\mathbb C^c(\mathcal H)\) is yielded by the \(\mathbb C\)-coalgebra structure of \(\mathcal H\), the linear map \(\mathcal A(\mathcal H)\to\mathcal H\) is yielded by the \(\mathcal A\)-algebra structure, and \(\psi(\mathcal H)\) is a fixed section of the morphism \(\phi(\mathcal H)\). The crux of his arguments relies on the observation that the infinite composite \[ e = (\text{id}-\omega^{[2]})(\text{id}-\omega^{[3]})\cdots(\text{id}-\omega^{[n]})\cdots \] induces a retraction onto \(\operatorname{Prim}\mathcal H\).
This proof is similar to an argument line that Frédéric Patras discovered in his thesis [Homothéties Simpliciales, Thèse de Doctorat, Université Paris 7 (1992)] in the context of classical cocommutative Hopf algebras – see notably F. Patras’s proof of the classical Poincaré-Birkhoff-Witt and Milnor-Moore theorems in [Ann. Inst. Fourier 43, No. 4, 1067–1087 (1993; Zbl 0795.16028)]. The map \(e: \mathcal H\to\mathcal H\) is a generalization, in the context of operads, of the eulerian idempotent used by Patras. The origin of this idempotent goes back to applications of K.-T. Chen’s iterated integrals to the Campbell-Baker-Hausdorff formula [Ann. Math. (2) 65, 163–178 (1957; Zbl 0077.25301)].
The first part of the book is an informal introduction to algebraic applications of the theory of operads. The generalized rigidity, Poincaré-Birkhoff-Witt, and Milnor-More theorems are proved in the second part. The third part is an outlook for applications and variants of these main results. Parts 4 and 5 give an overall review of examples occuring in the literature.
The book includes a wide bibliography.