Geodesic
Borel Subalgebras of Schur Superalgebras
Algebra i logika, Tome 44 (2005) no. 3, pp. 305-334.

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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 $S(1|1, r)$.
Mots-clés : Borel subalgebra, simple module
Keywords: Schur superalgebra. 
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A. N. Zubkov. Borel Subalgebras of Schur Superalgebras. Algebra i logika, Tome 44 (2005) no. 3, pp. 305-334. http://geodesic.mathdoc.fr/item/AL_2005_44_3_a2/

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