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Flat degenerations of flag supermanifolds for basic Lie superalgebras

Ibrahim Ahmad

TL;DR

The paper advances the degeneration program for flag varieties to the realm of basic Lie superalgebras by constructing and analyzing the super-analogue of favourable modules. It defines a coordinate ring $R(\lambda)$ from finite-dimensional quotients, equips it with a multiplication via a Cartan map, and proves the existence of a flat one-parameter family whose general fibre recovers $R(\lambda)$ while the special fibre is governed by essential monomials, yielding a toric-type degeneration in favorable cases. The work develops a straightening law for the superalgebra, introduces a favourable-module framework, and constructs an explicit $\mathbb{C}[t]$-deformation whose associated graded algebra is monomial and toric in nature, connecting to supergeometry and toric supervarieties in the sense of Jankowski. It thus extends the classical degeneration program to the super setting, with concrete corollaries for Kus–Fourier bases and typified basic Lie superalgebras. This provides a bridge between representation-theoretic bases, PBW-type filtrations, and toric geometry in a supersymmetric context, with potential applications to the study of flag supermanifolds and their coordinate rings.

Abstract

Motivated by bases of representations compatible with the PBW filtration for basic Lie superalgebras by Kus and Fourier, we generalise the construction of degenerations of flag varieties via favourable modules to the super setup. In the classical setup, this method of degenerating flag varieties by Feigin, Fourier and, Littelmann relies on constructing bases of representations of Lie algebras such that in the coordinate ring of an embedded flag varietiy their multiplication can be identified with an affine semigroup modulo terms of higher degree. By killing off said terms of higher degree via a filtration construction, one gets a toric variety the embedded flag variety degenerates into. By adapting these techniques we provide a similar construction and discuss when one can get a degeneration into a toric supervariety, as defined by Jankowski

Flat degenerations of flag supermanifolds for basic Lie superalgebras

TL;DR

The paper advances the degeneration program for flag varieties to the realm of basic Lie superalgebras by constructing and analyzing the super-analogue of favourable modules. It defines a coordinate ring from finite-dimensional quotients, equips it with a multiplication via a Cartan map, and proves the existence of a flat one-parameter family whose general fibre recovers while the special fibre is governed by essential monomials, yielding a toric-type degeneration in favorable cases. The work develops a straightening law for the superalgebra, introduces a favourable-module framework, and constructs an explicit -deformation whose associated graded algebra is monomial and toric in nature, connecting to supergeometry and toric supervarieties in the sense of Jankowski. It thus extends the classical degeneration program to the super setting, with concrete corollaries for Kus–Fourier bases and typified basic Lie superalgebras. This provides a bridge between representation-theoretic bases, PBW-type filtrations, and toric geometry in a supersymmetric context, with potential applications to the study of flag supermanifolds and their coordinate rings.

Abstract

Motivated by bases of representations compatible with the PBW filtration for basic Lie superalgebras by Kus and Fourier, we generalise the construction of degenerations of flag varieties via favourable modules to the super setup. In the classical setup, this method of degenerating flag varieties by Feigin, Fourier and, Littelmann relies on constructing bases of representations of Lie algebras such that in the coordinate ring of an embedded flag varietiy their multiplication can be identified with an affine semigroup modulo terms of higher degree. By killing off said terms of higher degree via a filtration construction, one gets a toric variety the embedded flag variety degenerates into. By adapting these techniques we provide a similar construction and discuss when one can get a degeneration into a toric supervariety, as defined by Jankowski
Paper Structure (9 sections, 18 theorems, 94 equations)

This paper contains 9 sections, 18 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Essential monomials and multiplications
  4. A basis and straightening laws for $R(\lambda)$
  5. Favourable modules and and a vanishing ideal of $R(\lambda)$
  6. Flat degeneration
  7. The ideal of the family
  8. Constructing a $\mathbb{C}[t]$-basis
  9. The supergeometric interpretation

Key Result

Theorem 1

(Corollary result1) There exists a morphism of superschemes $\kappa:\mathrm{Spec}\mathcal{R}\rightarrow\mathbb{A}^{1|0}$ that is flat and the fibers are

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Theorem 2.1: Kac_Lie
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1: Cartan mapping
  • proof
  • Remark 3.2
  • Theorem 3.3: Cheng-Wang
  • Definition 3.4: Gröbner_Super
  • ...and 40 more