On Lie-algebraic solutions of the type IIB matrix model

TL;DR

This paper systematically classifies Lie algebra solutions to the type IIB matrix model up to dimension six, revealing their structure, representations, and associated non-commutative spaces, advancing understanding of algebraic solutions in string theory.

Contribution

It provides a comprehensive classification of Lie algebra solutions to the type IIB matrix model up to dimension six, including their matrix representations and non-commutative geometries.

Findings

Existence of Lie algebra solutions corresponding to nilpotent and solvable algebras.

Explicit matrix representations and star-product formulations.

Construction of non-commutative spaces and nilmanifolds from these algebras.

Abstract

A systematic search for Lie algebra solutions of the type IIB matrix model is performed. Our survey is based on the classification of all Lie algebras for dimensions up to five and of all nilpotent Lie algebras of dimension six. It is shown that Lie-type solutions of the equations of motion of the type IIB matrix model exist and they correspond to certain nilpotent and solvable Lie algebras. Their representation in terms of Hermitian matrices is discussed in detail. These algebras give rise to certain non-commutative spaces for which the corresponding star-products are provided. Finally the issue of constructing quantized compact nilmanifolds and solvmanifolds based on the above algebras is addressed.