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Zero-level integrable modules over twisted affine Lie superalgebras

Hajar Kiamehr, Senapathi Eswara Rao, Malihe Yousofzadeh

TL;DR

This work addresses the problem of classifying zero-level integrable finite-weight modules over twisted affine Lie superalgebras. The authors reduce the problem to parabolic induction from modules over a simpler subalgebra $\mathscr{L}$, which is either a ${\mathbb Z}$-graded abelian Lie algebra or a direct sum of such an algebra with the quadratic Lie superalgebra $\mathcal{Q}$, and they characterize finite-dimensional simple $\mathcal{Q}$-modules via Clifford-algebra methods. A detailed analysis of ${\mathbb Z}$-graded simple $\mathcal{Q}$-modules, including loop-module structures and bounded weight behavior, is carried out, especially for the exceptional type $A(2m,2n)^{(4)}$ where the quadratic component arises. The main result asserts that any zero-level integrable finite-weight module is parabolically induced from a module over $\mathscr{L}$, and provides a complete description of the relevant base modules, linking the centerless core representations to the full twisted affine setting. Overall, the paper delivers a complete zero-level classification by decomposing into abelian and quadratic components and detailing the Clifford-algebraic underpinnings.

Abstract

The main result of this paper is the characterization of zero-level integrable finite weight modules, over twisted affine Lie superalgebras. We prove that such a module is parabolically induced from a module which is obtained, in a prescribed way, from a module over a Lie algebra $\mathscr{L}$ which is either a $\bbbz$-graded abelian Lie algebra or a direct sum of a $\bbbz$-graded abelian Lie algebra and the so-called quadratic Lie superalgebra $\mathcal{Q}$. We give also a complete characterization of both finite dimensional simple $\mathcal{Q}$-modules as well as bounded finite weight $\bbbz$-graded simple $\mathcal{Q}$-modules.

Zero-level integrable modules over twisted affine Lie superalgebras

TL;DR

This work addresses the problem of classifying zero-level integrable finite-weight modules over twisted affine Lie superalgebras. The authors reduce the problem to parabolic induction from modules over a simpler subalgebra , which is either a -graded abelian Lie algebra or a direct sum of such an algebra with the quadratic Lie superalgebra , and they characterize finite-dimensional simple -modules via Clifford-algebra methods. A detailed analysis of -graded simple -modules, including loop-module structures and bounded weight behavior, is carried out, especially for the exceptional type where the quadratic component arises. The main result asserts that any zero-level integrable finite-weight module is parabolically induced from a module over , and provides a complete description of the relevant base modules, linking the centerless core representations to the full twisted affine setting. Overall, the paper delivers a complete zero-level classification by decomposing into abelian and quadratic components and detailing the Clifford-algebraic underpinnings.

Abstract

The main result of this paper is the characterization of zero-level integrable finite weight modules, over twisted affine Lie superalgebras. We prove that such a module is parabolically induced from a module which is obtained, in a prescribed way, from a module over a Lie algebra which is either a -graded abelian Lie algebra or a direct sum of a -graded abelian Lie algebra and the so-called quadratic Lie superalgebra . We give also a complete characterization of both finite dimensional simple -modules as well as bounded finite weight -graded simple -modules.
Paper Structure (13 sections, 15 theorems, 141 equations, 3 tables)

This paper contains 13 sections, 15 theorems, 141 equations, 3 tables.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Conventions
  4. Graded associative algebras
  5. Clifford superalgebras
  6. Finite weight ${\mathbb Z}$-graded simple $\mathcal{Q}$-modules
  7. ${\mathbb Z}$-graded simple modules over split extensions of $\mathcal{Q}$
  8. Bounded finite weight ${\mathbb Z}$-graded simple $\mathfrak{g}$-modules
  9. Affine Lie superalgebras
  10. Parabolically induced modules
  11. ${A(2m,2n)^{(4)}}$
  12. Zero-level finite weight modules over ${\bigoplus_{k\in{\mathbb Z}}\mathcal{L}^{k\delta}}$
  13. Zero-level integrable finite weight modules

Key Result

Lemma 3.1

Suppose that $V$ is a ${\mathbb Z}$-graded $\mathcal{Q}$-module.

Theorems & Definitions (19)

  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4: Finite dimensional simple $\mathfrak{k}$-modules
  • Proposition 3.5: Finite dimensional simple $\mathfrak{g}$-modules
  • Proposition 3.6
  • Proposition 3.7
  • Lemma 3.8
  • Proposition 3.9
  • Theorem 3.10: Characterization of bounded finite weight ${\mathbb Z}$-graded simple $\mathfrak{g}$-module
  • ...and 9 more