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The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the $E_n$-bar construction on the cochains of $X$ and the homotopy groups of $X$. This pairing gives us a set of invariants of homotopy classes of maps from $S^m$ to a simplicial set $X$, this pairing can detect more homotopy classes of maps than the classical Hopf invariant. The second part of the paper is devoted to combining the $E_n$-Hopf invariants with the Koszul duality theory for $E_n$-operads to get a relation between the $E_n$-Hopf invariants of a space $X$ and the $E_{n+1}$-Hopf invariants of the suspension of $X$. This is done by studying the suspension morphism for the $E_\infty$-operad, which is a morphism from the $E_{\infty}$-operad to the desuspension of the $E_\infty$-operad. We show that it induces a functor from $E_\infty$-algebras to $E_\infty$-algebras, which has the property that it sends an $E_\infty$-model for a simplicial set $X$ to an $E_\infty$-model for the suspension of $X$. We use this result to give a relation between the $E_n$-Hopf invariants of maps from $S^m$ into $X$ and the $E_{n+1}$-Hopf invariants of maps from $S^{m+1}$ into the suspension of $X$. One of the main results we show here, is that this relation can be used to define invariants of stable homotopy classes of maps., Comment: 26 pages

Realizations of $\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\kappa$-deformed $\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\kappa$-Poincar\'e-Weyl and $\kappa$-Poincar\'e-Hopf algebras are discussed. Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\kappa$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed., Comment: 35 pages, improved version accepted for publication in EPJC