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On a super-analog of the Schur-Weyl Duality

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arxiv 2208.07434 v1 pith:AIIGDUW3 submitted 2022-08-15 math.RT

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keywords dualityalgebrasgroupactionsenvelopingisomorphismmathbbotimes
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abstract

Two super-analogs of the Schur-Weyl duality are considered: the duality of actions in $(\mathbb{C}^{m|n})^{\otimes N}$ of the Lie superalgebra $\mathfrak{gl}(m,n)$ and the symmetric group $S_N$, and the duality of actions of the Lie superalgebra $Q(n)$ and a certain finite group $Se(N)$ in $(\mathbb{C}^{n|n})^{\otimes N}$. We construct an isomorphism of symmetric and universal enveloping algebras of these Lie superalgebras called special symmetrization. Using this isomorphism of vector spaces we describe explicitly the duality between the centers of the corresponding universal enveloping algebras and the group algebras.

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  1. The Schur-Weyl duality and Invariants for classical Lie superalgebras

    math.RT 2024-11 accept novelty 7.0 of 10

    Schur-Weyl duality yields a surjective map from tensor-algebra invariants to the center of U(g) for gl_{m|n}, q_n, osp_{2m+1|2n}, and p_n, and a new proof that Z(p_n)=0.

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