Hochschild polytopes

Abstract The ( m , n )-multiplihedron is a polytope whose faces correspond to m -painted n -trees, and whose oriented skeleton is the Hasse diagram of the rotation lattice on binary m -painted n -trees. Deleting certain inequalities from the facet description of the ( m , n )-multiplihedron, we construct the ( m , n )-Hochschild polytope whose faces correspond to m -lighted n -shades, and whose oriented skeleton is the Hasse diagram of the rotation lattice on unary m -lighted n -shades. Moreover, there is a natural shadow map from m -painted n -trees to m -lighted n -shades, which turns out to define a meet semilattice morphism of rotation lattices. In particular, when $$m=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , our Hochschild polytope is a deformed permutahedron whose oriented skeleton is the Hasse diagram of the Hochschild lattice.