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Lie theory for Hopf operads. (English) Zbl 1149.18005

The authors study Hopf algebras in the category of \(\mathbf S\)-modules, where an \(\mathbf S\)-module refers to a collection \(M = \{M(n)\}_{n\in\mathbb N}\) such that \(M(n)\) is a representation of the symmetric group \(\mathbf S_n\) for \(n\in\mathbb N\).
For any \(\mathbf k\)-module \(E\), we can form a module of symmetric tensors \(S(M,E) = \bigoplus_{n=0}^{\infty} (M(n)\otimes E^{\otimes n})_{\mathbf S_n}\) with coefficients in a given \(\mathbf S\)-module \(M\). The map \(S(M): E\mapsto S(M,E)\) defines a functor on \(\mathbf k\)-modules and the category of \(\mathbf S\)-modules is equipped with structures that reflect pointwise operations on functors of this form \(S(M): E\mapsto S(M,E)\). In particular, the category of \(\mathbf S\)-modules is equipped with a tensor product, denoted by \(\cdot\) in the paper under review, such that \(S(M\cdot N,E) = S(M,E)\otimes S(N,E)\). The notion of a Hopf algebra in \(\mathbf S\)-modules is defined with respect to this tensor structure.
The authors observe that a Hopf operad \(\mathcal P\) such that \(\mathcal P(0) = \mathbf k\) defines a Hopf algebra in \(\mathbf S\)-modules. Recall that a Hopf operad is an operad in coalgebras. They prove that the primitive part of the Hopf algebra in \(\mathbf S\)-modules defined by a Hopf operad inherits an operad structure.
The authors study the example of the associative operad \(\mathcal A s\) and of the Poisson operad \(\mathcal P ois\). They observe that the primitive parts of \(\mathcal A s\) and \(\mathcal P ois\) are both isomorphic to the Lie operad \(\mathcal L ie\).
The last-mentioned result is a consequence of generalizations, in the context of \(\mathbf S\)-modules, of the usual Milnor-Moore and Poincaré-Birkhoff-Witt theorems [C. R. Stover, J. Pure Appl. Algebra 86, No. 3, 289–326 (1993; Zbl 0793.16016)]: any connected cocommutative Hopf algebra in \(\mathbf S\)-modules \(H\) is the enveloping algebra of a Lie algebra \(H = U(\mathcal G)\), where \(\mathcal G = \text{Prim} H\), and comes equipped with a natural filtration such that \(\text{gr} H = S(\mathcal G)\), the abelian Hopf algebra generated by \(\mathcal G\). For this abelian Hopf algebra \(S(\mathcal G)\), we also have \(\text{Prim} S(\mathcal G) = \mathcal G\). Besides, the Lie bracket of \(\mathcal G\) extends to a biderivation on \(S(\mathcal G)\) so that the symmetric algebra \(S(\mathcal G)\) inherits a natural Poisson structure.
For the associative operad \(\mathcal A s\), an immediate adjunction argument gives \(\mathcal A s = U(\mathcal L ie)\), where the Lie operad \(\mathcal L ie\) is considered as a Lie algebra in \(\mathbf S\)-modules. Hence the Milnor-Moore theorem implies \(\text{Prim}\mathcal A s = \text{Prim} U(\mathcal L ie) = \mathcal L ie\). Let \(\mathcal C om\) denote the commutative operad. The symmetric algebra in \(\mathbf S\)-modules \(\text{gr}\mathcal A s = \text{gr} U(\mathcal L ie) = S(\mathcal L ie)\) can be identified with a composite \(\mathbf S\)-module \(\mathcal C om\circ\mathcal L ie\), which is a representation of the Poisson operad \(\mathcal P ois\). Therefore we also have \(\text{Prim}\mathcal P ois = \text{Prim} S(\mathcal L ie) = \mathcal L ie\).
The identity \(\text{Prim}\mathcal A s = \mathcal L ie\) had already been noticed and used by the reviewer [B. Fresse, Trans. Am. Math. Soc. 352, No. 9, 4113–4141 (2000; Zbl 0958.18005)].

MSC:

18D50 Operads (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S30 Universal enveloping algebras of Lie algebras
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References:

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