$N$-homogeneous superalgebras

Abstract

We develop the theory of $N$-homogeneous algebras in a super-setting, with particular emphasis on the Koszul property. To any Hecke operator $\mathcal{R}$ on a vector superspace, we associate certain superalgebras $S_{\mathcal{R},N}$ and ${\wedge }_{\mathcal{R},N}$ generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are $N$-Koszul. For the special case where $\mathcal{R}$ is the ordinary supersymmetry, we derive an $N$-generalized super-version of MacMahon’s classical “master theorem”.