Lie algebra and Dynkin index

Abstract

The Lie algebra ${\mathfrak{e}}_{7\frac{1}{2}}$ comes from P. Deligne’s work on the exceptional series of Lie groups. Using the triality algebra, J. Landsberg and L. Manivel construct this Lie algebra in 2006. In this paper, we study the structure of ${\mathfrak{e}}_{7\frac{1}{2}}$ following B. Gross and N. Wallach’s work on the highest root and Heisenberg parabolic subalgebra. The process of removing the lowest root from the extended Dynkin diagram of ${\mathcal{E}}_8$ will contribute to the components of ${\mathfrak{e}}_{7\frac{1}{2}}$. We also study the branching rule of ${\mathfrak{e}}_{7\frac{1}{2}}$ to $\mathfrak{s}\mathfrak{l}(3,\mathbb{C})$ and ${\mathfrak{g}}_2$. We use the computer algebra system SageMath to carry out the branching rule calculations. Then we calculate the Dynkin indices for ${\mathfrak{e}}_{7\frac{1}{2}}$. We find that the number 24 behaves like the ‘dual Coxeter number’ of ${\mathfrak{e}}_{7\frac{1}{2}}$.

2020 Mathematics Subject Classification:

• 17B25
• 17B10
• 17B60
• 17-04

Keywords:

• Dual Coxeter number
• Dynkin index
• exceptional series of Lie groups
• sextonions

Data availability statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Disclosure statement

The author declares that he/she has no conflict of interest.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

This work is part of the author’s PhD thesis [35]. The author appreciates the supervision of Professor Jing-Song Huang during the study at The Hong Kong University of Science and Technology. The author thanks Professor Eric Marberg’s helpful comments for the improvement of my thesis. The author thanks Professor Jan Draisma for pointing out an error of my manuscript. The author thanks Professor Daniel Bump for answering my questions about the maximal subgroups of type ${\mathcal{A}}_2$ in ${\mathcal{E}}_7$ and SageMath. Last but not least, the author appreciates the annonymous referees’ helpful suggestions and comments.

Additional information

Funding

This work is supported by the Shandong Provincial Natural Science Foundation under grant ZR2022QA056.