Given a principal ideal domain $R$ of characteristic zero, containing 1/2, and a two-cone $X$ of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra $FH(\Omega X;R)$ to be isomorphic with the universal enveloping algebra of some $R$-free graded Lie algebra; as usual, $F$ stands for free part, $H$ for homology, and $\Omega$ for the Moore loop space functor.

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space $V$. The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on $V$ (resp. graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras and observe that the ${\text{Lod}}_{\infty }$ category contains the ${\text{L}}_{\infty }$ category as...

We compute the unique nonzero cohomology group of a generic $G^0$- linearized locally free $𝒪$-module, where $G^0$ is the identity component of a complex classical Lie supergroup $G$ and $P\hookrightarrow G^0$ is an arbitrary parabolic subsupergroup. In particular we prove that for $G\ne \mathbb{P}\left(m\right),S\mathbb{P}\left(m\right)$ this cohomology group is an irreducible $G^0$-module. As an application we generalize the character formula of typical irreducible $G^0$-modules to a natural class of atypical modules arising in this way.

Natural graded Lie brackets on the space of cochains of n-Leibniz and n-Lie algebras are introduced. It turns out that these brackets agree under the natural embedding introduced by Gautheron. Moreover, n-Leibniz and n-Lie algebras turn to be canonical structures for these brackets in a similar way in which associative algebras (respectively, Lie algebras) are canonical structures for the Gerstenhaber bracket (respectively, Nijenhuis-Richardson bracket).