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- W3199137316 abstract "Abstract In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ of the loop algebra $widehat {mathfrak {gl}}_{m|n}$ of ${mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${widehat {{mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ is to determine the action of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $widehat {mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $;{mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ and ${widehat {{mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $mathbb Z$ -form for ${mathcal U}_{mathbb Q}(widehat {mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics." @default.
- W3199137316 created "2021-09-27" @default.
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- W3199137316 date "2021-09-09" @default.
- W3199137316 modified "2023-10-18" @default.
- W3199137316 title "A REALIZATION OF THE ENVELOPING SUPERALGEBRA" @default.
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- W3199137316 doi "https://doi.org/10.1017/nmj.2021.11" @default.
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