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Irreducible cuspidal modules of simple $n$-Lie algebras

Bakhrom Omirov, Gulkhayo Solijanova

Abstract

This work devoted to the description of irreducible cuspidal modules over simple $n$-Lie algebras. Since the description of irreducible modules over $n$-Lie algebra $O^n$ are already well understood, we focus here on the irreducible cuspidal modules over $n$-Lie algebras of Wronskians and Jacobians. First, for a given $n$-Lie algebra $\mathcal{L}$, we analyze the possible Lie and Leibniz structures on $\wedge^{n-1} \mathcal{L}$ and $\otimes^{n-1} \mathcal{L}$ by thoroughly examining existing structures. Next, we classify the irreducible cuspidal modules over the $n$-Lie algebra of Wronskians defined on Laurent polynomials with degree-preserving derivations. Furthermore, we prove that these modules remain irreducible over the $n$-Lie algebra of Jacobians.

Irreducible cuspidal modules of simple $n$-Lie algebras

Abstract

This work devoted to the description of irreducible cuspidal modules over simple -Lie algebras. Since the description of irreducible modules over -Lie algebra are already well understood, we focus here on the irreducible cuspidal modules over -Lie algebras of Wronskians and Jacobians. First, for a given -Lie algebra , we analyze the possible Lie and Leibniz structures on and by thoroughly examining existing structures. Next, we classify the irreducible cuspidal modules over the -Lie algebra of Wronskians defined on Laurent polynomials with degree-preserving derivations. Furthermore, we prove that these modules remain irreducible over the -Lie algebra of Jacobians.
Paper Structure (19 sections, 17 theorems, 115 equations)

This paper contains 19 sections, 17 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie and Leibniz structures on $\wedge^{n-1}\mathcal{L}$ and $\otimes^{n-1}\mathcal{L}$.
  4. Structures related to Leibniz $n$-algebras
  5. Structures related to $n$-Lie algebras
  6. Modules over $n$-Lie algebras
  7. Irreducible cuspidal $W_{n-1}$ and $S_n$-modules.
  8. Irreducible cuspidal $W_{n-1}$-modules
  9. Irreducible cuspidal $S_{n}$-modules
  10. Simplification of the Balanced Equality for $W_{n-1}$ and $S_n$-modules.
  11. Simplification of the Balanced Equality for $W_{n-1}$-modules.
  12. Simplification of the Balanced Equality for $S_{n}$-modules.
  13. Irreducible cuspidal modules over $n$-Lie algebra $W^{n}$.
  14. The case of $T(U, \alpha)$ with $dim U=1$.
  15. The case of $T(U, \alpha)$ with $dim U > 1$.
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $(\mathcal{L}, [ -, \dots, - ])$ be a Leibniz $n$-algebra. Then the equality holds.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 27 more