Uniformly bounded weight modules for Map extended Special and Map extended Hamiltonian Lie algebras

TL;DR

The paper tackles the classification of irreducible uniformly bounded Harish-Chandra modules for map extended Special and map extended Hamiltonian Lie algebras under a nontrivial action of $t^{ extbf{r}} ens 1$ and $t^{ extbf{0}} ens 1$. It develops a quasi-associative framework for the action of $A_{N} ens B$, proves that $t^{ extbf{r}}$ acts injectively on weight spaces and that $t^{ extbf{r}} ens b$ corresponds to a single scalar $oldsymbol{\\psi}(b)$-twist, while $D(u, extbf{r})$ with $ extbf{r} eq extbf{0}$ scales by $oldsymbol{\\psi}(b)$ on weight spaces. Consequently, weight modules decompose as $V=igoplus_{ extbf{r}y extbf{0}} V_{oldsymbol{\\Lambda}+ extbf{r}}$, with $V_{oldsymbol{\\Lambda}+ extbf{r}}$ carrying a Jet-module structure for the underlying extended algebras $rak{S}_{N}$ and $ ilde{rak{H}}_{N}$, respectively. The results connect to and extend prior classifications of map-extended and Cartan-type Harish-Chandra modules by showing that irreducibles arise from quasi-associative actions and jet-module realizations, thereby enriching the landscape of Harish-Chandra representations for map-extended Cartan-type algebras. The findings have potential implications for understanding representation theories of infinite-dimensional Lie algebras in mathematical physics, especially in contexts where toroidal and loop-extended structures appear.

Abstract

This paper explores the irreducible uniformly bounded weight modules of map extended Special Lie algebras and map extended Hamiltonian Lie algebras under some condition on the action of the Laurent polynomial ring A_{N}.

Theorems & Definitions (61)

• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Lemma 2.1: [17]
• Definition 2.2.1
• Definition 2.2.2
• Definition 2.2.3
• Theorem 2.2
• Theorem 2.3
• Definition 2.4.1