TY - JOUR

T1 - DONKIN–KOPPINEN FILTRATION FOR GL(m|n) AND GENERALIZED SCHUR SUPERALGEBRAS

AU - Marko, F.

AU - Zubkov, A. N.

N1 - Funding Information:
A. N. Zubkov is the work was supported by the UAEU grant G00003324.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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U2 - 10.1007/s00031-022-09714-y

DO - 10.1007/s00031-022-09714-y

M3 - Article

AN - SCOPUS:85127434483

SN - 1083-4362

JO - Transformation Groups

JF - Transformation Groups

ER -