Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad

Abstract

We compute the dimensions of the components for the operad of two compatible brackets and for the bi-Hamiltonian operad. We also obtain character formulas for the representations of symmetric groups and SL 2 in these spaces.

Compatible associative products and trees

“…Thus, the third statement implies the other two, so we shall restrict ourselves to proving only the former. From the results from [Strohmayer 2008] (combined with results from [Dotsenko and Khoroshkin 2007] on SL 2 -modules), it follows that as an S n × SL 2 -module,…”

“…One can compare the methods and structure of this paragraph to the same in [Dotsenko and Khoroshkin 2007] in the case of the operad of two compatible Lie brackets. In this section, we prefer to think of components of our operad in terms of the multilinear elements in free algebras.…”

Prop profile of bi-Hamiltonian structures

“…We call such structures Lie 2 1-bialgebra and denote the corresponding prop by Lie 1 2 Bi. Using results from [8], [6], and [23], we show that its dioperadic part is Koszul, which makes it possible to compute its minimal resolution Lie 1 2 Bi 1 ( § 5.4) and leads us to the following conclusion. In fact we prove a stronger result.…”

“…A pair of Lie algebras are called compatible if the sum of their Lie brackets is again a Lie bracket. We denote the operad of compatible Lie algebras, introduced in [6], by Lie 2 . As a byproduct of the resolution of Lie 1 2 Bi we obtain a minimal resolution of Lie 2 ( § 5.5).…”

Applications of quotient posets

“…We will consider the weighted partition poset, Π w n introduced in [4]. The elements of Π w n are set partitions of {1, 2, .…”