Skew Symmetric Extended Affine Lie algebras

TL;DR

The paper introduces Skew Symmetric Extended Affine Lie Algebras (SSEALA) from a skew-symmetric matrix $B$, unifying Hamiltonian and contact-type EALAs and positioning SSEALA as a broad class containing HEALA and KEALA. It develops a structural framework for studying representations by reducing irreducible integrable modules with finite weight spaces to jet-module analysis over the derivation-algebra $\widetilde{H_B} \ltimes \mathcal A$, and proves that irreducible jet modules are of the form $V_0 \otimes \mathcal A$ with $V_0$ tied to $\mathfrak{sp}_{2m}$. In the level-zero case (non-degenerate $B$), irreducible integrable modules with finite weight spaces decompose as $V(\mu) \otimes P \otimes \mathcal A$ for finite-dimensional $\mathfrak g$-modules $V(\mu)$ and $\mathfrak{sp}_{2m}$-modules $P$, with explicit action described and automorphism-twist flexibility. For non-zero level KEALA (degenerate $B=\bar J$, $N=2m+1$), irreducibles arise as quotients $L(P' \otimes \mathcal A_{2m})$ of induced modules, with $P'$ a finite-dimensional $\mathfrak{sp}_{2m}$-module. These results extend the understanding of representations for toroidal and full toroidal algebras within the EALA framework, linking finite-dimensional pieces to Laurent polynomial actions and providing constructions via induced modules.

Abstract

For any skew symmetric matrix over complex numbers, we introduce an EALA and it is called Skew Symmetric Extended Affine Lie Algebra (SSEALA). This way we get a large class of EALAs and most often they are non-isomorphic. In this paper we study irreducible integrable modules for SSEALA with finite dimensional weight spaces. We classify all such modules in the level zero case with non-degenerate skew symmetric matrix.

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.1
• Definition 2.2
• Example 2.1
• Example 2.2
• Example 2.3
• Proposition 3.1