Divided Power Algebras Over an Operad

Abstract

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras.We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group, the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.• In the set Π(r; n) there is a unique element of Comp p (n) (more precisely, in the image of Comp p (n) by ι), namely r, and Π p (n) = r∈Comp p (n) Π(r; n).

Divided power algebras and distributive laws

“…This result does not come as a surprise and was hinted at in [13]. The real contribution of this article is a systematic method to compute divided power P • Q structures in terms of monomial operations coming from the divided power P and Q structures, which is compatible with the results of [14].…”

“…The main part of this section is devoted to reviewing the classical definitions of symmetric sequences, operads and their algebras. We set our notation in the same way as in [14]. For a textbook account on algebraic operads, we refer the reader to [19].…”

“…In [14], we have obtained a characterisation of divided power P-algebras in terms of monomial operations indexed by elements of P and compositions of integers. This characterisation can be refined by fixing either the operad P or the characteristic of the base field.…”

“…Outline of the paper: In Section 2, we recall the classical definitions of operads and their algebras, as well as the less classical definitions of divided power algebras over an operad. In Section 3, we build some operations on partitions and compositions of integers that are used in [14] to obtain the characterisation of divided power algebras over an operad. In Section 4, we review the definition of a distributive law on a product of monad and on a product of operad.…”

“…In Section 5, we use distributive laws to build two operads, Shift P and Der P , which encode respectively the structure of P-algebra with P-algebra endomorphism, and the structure of P-algebra with P-derivation. In Section 6, we recall the characterisation of divided P-algebras obtained in [14] in terms of monomial operations, and we give the first applications of this result to classical operads. In Section 7, we obtain our first result, Proposition 7.1, which assures that the divided power structure on a product of two operads with distributive law can itself be obtained from the divided power structure of the two factor operads through a distributive law.…”