Dynkin Indices, Casimir Elements and Branching Rules

Abstract

The Dynkin index is introduced by E. B. Dynkin in his famous work on the classification of semisimple subalgebras of semisimple Lie algebras in 1952. Dynkin index offers a way to study the different embeddings of a simple subalgebra into a complex simple Lie algebra, and the Dynkin index is also used in the Wess–Zumino–Witten (WZW) model of the conformal field theory. In this paper, we work on the Dynkin indices of representations of

\mathcal{A}\mathcal{D}\mathcal{E}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal A}{\cal D}{\cal E}$$\end{document}

-type complex simple Lie algebras, as well as some non-

\mathcal{A}\mathcal{D}\mathcal{E}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal A}{\cal D}{\cal E}$$\end{document}

-type Lie algebras. As an application of computational Lie theory, we work on the branching rules from the complex simple exceptional Lie algebras to

\mathfrak{s}\mathfrak{l}(3,\mathbb{C})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak s}{\mathfrak l}(3, \ {\mathbb C})$$\end{document}

and

{\mathfrak{g}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak g}_{2}$$\end{document}

. As a result, we get the Dynkin indices of

\mathfrak{s}\mathfrak{l}(3,\mathbb{C})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak s}{\mathfrak l}(3, \ {\mathbb C})$$\end{document}

and

{\mathfrak{g}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak g}_{2}$$\end{document}

in the exceptional Lie algebras. In this process, we find a new Dynkin index of

{\mathfrak{g}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak g}_{2}$$\end{document}

in

{\mathfrak{e}}_8

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak e}_{8}$$\end{document}

, i.e., 4. This number is not listed in Dynkin’s paper of 1952.

Keywords

Nilpotent orbit / Dynkin index / representation

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