Finite-dimensional representations for a class of generalized intersection matrix Lie algebras

TL;DR

This paper classifies finite-dimensional semi-simple quotients of generalized intersection matrix Lie algebras and establishes a correspondence between their irreducible modules.

Contribution

It provides a complete classification of finite-dimensional semi-simple quotients and irreducible modules for a class of generalized intersection matrix Lie algebras.

Findings

Finite-dimensional semi-simple quotients are of type M(n, a, c, d).

Irreducible modules of gIM(M_n) correspond to those of M(n, a, c, d).

Complete classification of finite-dimensional irreducible modules achieved.

Abstract

In this paper, we study a class of generalized intersection matrix Lie algebras $\gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{\bf a}, {\bf c},{\bf d})$. Particularly, any finite dimensional irreducible $\gim(M_n)$ module must be an irreducible module of $M(n,{\bf a}, {\bf c},{\bf d})$ and any finite dimensional irreducible $M(n,{\bf a}, {\bf c},{\bf d})$ module must be an irreducible module of $\gim(M_n)$.