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Level algebras and $\boldsymbol{s}$-lecture hall polytopes

Given a family of lattice polytopes, a common endeavor in Ehrhart theory is the classification of those polytopes in the family that are Gorenstein, or more generally level. In this article, we consider these questions for $\boldsymbol{s}$-lecture hall polytopes, which are a family of simplices arising from $\boldsymbol{s}$-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of $\boldsymbol{s}$-inversion sequences. Moreover, for a large subfamily of $\boldsymbol{s}$-lecture hall polytopes, we provide a more geometric classification of the Gorenstein property in terms of its tangent cones. We then show how one can use the classification of level $\boldsymbol{s}$-lecture hall polytopes to construct infinite families of level $\boldsymbol{s}$-lecture hall polytopes, and to describe level $\boldsymbol{s}$-lecture hall polytopes in small dimensions.