arXiv subjects
Publications and source records attributed to Alexander Sherman.
We study the Duflo-Serganova functor $\operatorname{DS}_x$ for the queer Lie superalgebra $\mathfrak{q}_n$ and for all odd $x$ with $[x,x]$ semisimple. For the case when the rank of $x$ is $1$ we give a formula for multiplicities in terms of the arc diagram attached to $λ$. Further, we prove that $\operatorname{DS}_x(L)$ is semisimple if $L$ is a simple finite-dimensional module and $x$ is of rank $1$ satisfying $x^2=0$.
Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $(\mathfrak{g},\mathfrak{k})$, or more generally supersymmetric spaces $G/K$. We define certain invariant distributions on $G/K$, which we call ghost distributions, and which in some sense are induced from invariant distributions on $G_0/K_0$. Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from $G$ to a symmetric subgroup $K'$ which is related (and sometimes conjugate) to $K$. We discuss the case of $G\times G/G$ for an arbitrary quasireductive supergroup $G$, where our results prove the existence of a polynomial which determines projectivity of irreducible $G$-modules. Finally, a generalization of Gorelik's ghost centre is defined called the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation.
We present two novel proofs of the known classification of connected affine algebraic supergroups $G$ such that $\operatorname{Rep}G$ is semisimple. The proofs are geometrically motivated, although both rely on an algebraic lemma that characterizes $\mathfrak{osp}(1|2n)$ amongst simple Lie superalgebras.
We give a definition of the notion of spherical varieties in the world of complex supervarieties with actions of algebraic supergroups. A characterization of affine spherical supervarieties is given which generalizes a characterization in the classical case. We also explain some general properties of the monoid of highest weights. Several examples are given that are interesting in their own right and highlight differences with the classical case, including the regular representation, symmetric supervarieties, and actions of split supergroups.
We present a classification of all spherical indecomposable representations of classical and exceptional Lie superalgebras. We also include information about stabilizers, symmetric algebras, and Borels for which sphericity is achieved. In one such computation, the symmetric algebra of the standard module of $\mathfrak{osp}(m|2n)$ is computed, which in particular gives the representation-theoretic structure of polynomials on the complex supersphere.
Civil Asset Forfeiture (CAF) is a longstanding and controversial legal process viewed on the one hand as a powerful tool for combating drug crimes and on the other hand as a violation of the rights of US citizens. Data used to support both sides of the controversy to date has come from government sources representing records of the events at the time of occurrence. Court dockets represent litigation events initiated following the forfeiture, however, and can thus provide a new perspective on the CAF legal process. This paper will show new evidence supporting existing claims about the growth of the practice and bias in its application based on the quantitative analysis of data derived from these court cases.
We give model-theoretic characterizations of UHF algebras and of AF algebras as C*-algebras that omit certain sets of types.