Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex

Ge Feng; Naihong Hu; Marc Rosso

doi:10.1016/j.jalgebra.2025.09.009

Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex

Ge Feng
, Naihong Hu^*
, Marc Rosso

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce the quantum Manin (m|n)-superspace A_q^m|n equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra Ω_q(m|n) equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra Ω_q and its Manin dual Ω_q^! are made into U_q(gl(m|n))-module superalgebras, either for q generic, or for q root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple U_q(gl(m|n))-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.

Original language	English
Pages (from-to)	492-531
Number of pages	40
Journal	Journal of Algebra
Volume	687
DOIs	https://doi.org/10.1016/j.jalgebra.2025.09.009
State	Published - 1 Feb 2026

Keywords

Quantum (dual) Grassmann superalgebra with quantum divided power superstructure
Quantum Manin superspace
Quantum differential operator, module (super-)algebra, quantum Deligne-Morgan-Manin de Rham super complex

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10.1016/j.jalgebra.2025.09.009

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@article{5772eaf71eab45cbabda9809959fb754,

title = "Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex",

abstract = "We introduce the quantum Manin (m|n)-superspace Aqm|n equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra Ωq(m|n) equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra Ωq and its Manin dual Ωq! are made into Uq(gl(m|n))-module superalgebras, either for q generic, or for q root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple Uq(gl(m|n))-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.",

keywords = "Quantum (dual) Grassmann superalgebra with quantum divided power superstructure, Quantum Manin superspace, Quantum differential operator, module (super-)algebra, quantum Deligne-Morgan-Manin de Rham super complex",

author = "Ge Feng and Naihong Hu and Marc Rosso",

note = "Publisher Copyright: {\textcopyright} 2025 Elsevier Inc.",

year = "2026",

month = feb,

day = "1",

doi = "10.1016/j.jalgebra.2025.09.009",

language = "英语",

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journal = "Journal of Algebra",

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T1 - Quantum super-symmetries (I)

T2 - Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex

AU - Feng, Ge

AU - Hu, Naihong

AU - Rosso, Marc

PY - 2026/2/1

Y1 - 2026/2/1

N2 - We introduce the quantum Manin (m|n)-superspace Aqm|n equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra Ωq(m|n) equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra Ωq and its Manin dual Ωq! are made into Uq(gl(m|n))-module superalgebras, either for q generic, or for q root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple Uq(gl(m|n))-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.

AB - We introduce the quantum Manin (m|n)-superspace Aqm|n equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra Ωq(m|n) equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra Ωq and its Manin dual Ωq! are made into Uq(gl(m|n))-module superalgebras, either for q generic, or for q root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple Uq(gl(m|n))-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.

KW - Quantum (dual) Grassmann superalgebra with quantum divided power superstructure

KW - Quantum Manin superspace

KW - Quantum differential operator, module (super-)algebra, quantum Deligne-Morgan-Manin de Rham super complex

UR - https://www.scopus.com/pages/publications/105016792144

U2 - 10.1016/j.jalgebra.2025.09.009

DO - 10.1016/j.jalgebra.2025.09.009

M3 - 文章

AN - SCOPUS:105016792144

SN - 0021-8693

VL - 687

SP - 492

EP - 531

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex

Abstract

Keywords

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