1. Classification of the reducible Verma modules over the Jacobi algebra G2
- Author
-
V. K. Dobrev, S. Doi, and Naruhiko Aizawa
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Verma module ,Diagram (category theory) ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Algebra ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Modeling and Simulation ,Lie algebra ,FOS: Mathematics ,Embedding ,Representation Theory (math.RT) ,Algebra over a field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematical Physics ,Mathematics - Abstract
In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram., 41pages, many figures
- Published
- 2021