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A REALIZATION OF THE ENVELOPING SUPERALGEBRA $ {\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$

Part of: Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 September 2021

JIE DU Open the ORCID record for JIE DU [Opens in a new window] ,
QIANG FU Open the ORCID record for QIANG FU [Opens in a new window]  and
YANAN LIN Open the ORCID record for YANAN LIN [Opens in a new window]
JIE DU
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia [email protected]
QIANG FU*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai, 200092, China [email protected], [email protected]
YANAN LIN
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China [email protected]
*
†Corresponding author.
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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.

MSC classification

Primary: 20G43: Schur and $q$-Schur algebras
Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Type
Article
Information
Nagoya Mathematical Journal , Volume 247 , September 2022 , pp. 516 - 551
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

Supported by the National Natural Science Foundation of China (11671297, 11871404)

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