Abstract: The paper contains results that characterize the Donkin–Koppinen filtration of the coordinate superalgebra K[G] of the general linear supergroup G = GL(m|n) by its subsupermodules CΓ = OΓ(K[G]). Here, the supermodule CΓ is the largest subsupermodule of K[G] whose composition factors are irreducible supermodules of highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ of the poset X(T)+ of dominant weights of G. A decomposition of G as a product of subsuperschemes U–×Gev×U+ induces a superalgebra isomorphism ϕ*K[U–]⊗K[Gev]⊗K[U+]≃K[G]. We show that CΓ=ϕ*(K[U–]⊗MΓK[U+]), where MΓ=OΓ(K[Gev]). Using the basis of the module MΓ, given by generalized bideterminants, we describe a basis of CΓ. Since each CΓ is a subsupercoalgebra of K[G], its dual CΓ∗=SΓ is a (pseudocompact) superalgebra called the generalized Schur superalgebra. There is a natural superalgebra morphism πΓ : Dist(G) → SΓ such that the image of the distribution algebra Dist(G) is dense in SΓ. For the ideal X(T)l+, of all weights of fixed length l, the generators of the kernel of πX(T)l+ are described.

Cite: Marko F. , Zubkov A.N.
Donkin–Koppinen filtration for gl(m|n) and generalized Schur superalgebras
Transformation Groups. 2023. V.28. N2. P.911–949. DOI: 10.1007/s00031-022-09714-y WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Aug 15, 2020
Accepted:	Apr 18, 2021
Published online:	Mar 31, 2022
Published print:	Jun 15, 2023

Identifiers:

Web of science:	WOS:000777220100002
Scopus:	2-s2.0-85127434483
Elibrary:	48424162
OpenAlex:	W3049042986

Citing:

DB	Citing
Scopus	1
OpenAlex	3

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