The Schur-Weyl duality and Invariants for classical Lie superalgebras

TL;DR

This paper characterizes invariants of classical Lie superalgebras using a super-analog of Schur-Weyl duality, identifying generators of their centers and exploring algebraic structures related to supersymmetrization.

Contribution

It provides a unified framework for invariants of classical Lie superalgebras via super Schur-Weyl duality, including new proofs and descriptions of centers.

Findings

Determined generators of the center for most classical Lie superalgebras.

Established surjectivity of the projection from tensor algebra to universal enveloping algebra.

Provided an algebraic proof of the triviality of the center for rak{p}_n.

Abstract

In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{g}$-invariants of the tensor algebra $T(\mathfrak{g})$, the supersymmetric algebra $S(\mathfrak{g})$, and the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of a classical Lie superalgebra $\mathfrak{g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak{g})^{\mathfrak{g}}$ of the projection from $T(\mathfrak{g})$ to $\mathrm{U}(\mathfrak{g})$ is surjective, which enables us to determine the generators of the center $\mathcal{Z}(\mathfrak{g})$ except for $\mathfrak{g}=\mathfrak{osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal{Z}(\mathfrak{p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams.