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 Algebra Logika, 2005, Volume 44, Number 3, Pages 305–334 (Mi al114)

Borel Subalgebras of Schur Superalgebras

A. N. Zubkov

Omsk State Pedagogical University

Abstract: It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w. r. t. its natural anti-isomorphism (Bruhat – Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin – Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras $S(1|1, r)$.

Keywords: Borel subalgebra, simple module, Schur superalgebra

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English version:
Algebra and Logic, 2005, 44:3, 168–184

Bibliographic databases:

UDC: 512.552.22

Citation: A. N. Zubkov, “Borel Subalgebras of Schur Superalgebras”, Algebra Logika, 44:3 (2005), 305–334; Algebra and Logic, 44:3 (2005), 168–184

Citation in format AMSBIB
\Bibitem{Zub05} \by A.~N.~Zubkov \paper Borel Subalgebras of Schur Superalgebras \jour Algebra Logika \yr 2005 \vol 44 \issue 3 \pages 305--334 \mathnet{http://mi.mathnet.ru/al114} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2170689} \zmath{https://zbmath.org/?q=an:1150.16028} \transl \jour Algebra and Logic \yr 2005 \vol 44 \issue 3 \pages 168--184 \crossref{https://doi.org/10.1007/s10469-005-0018-8} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-22344453835} 

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• http://mi.mathnet.ru/eng/al/v44/i3/p305

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Zubkov, “Some Properties of General Linear Supergroups and of Schur Superalgebras”, Algebra and Logic, 45:3 (2006), 147–171
2. Marko F., Zubkov A.N., “Schur superalgebras in characteristic $p$. II”, Bull. London Math. Soc., 38:1 (2006), 99–112
3. La Scala R., Zubkov A., “Costandard modules over Schur superalgebras in characteristic $p$”, J. Algebra Appl., 7:2 (2008), 147–166
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