In this paper, we consider the existence and asymptotic behavior normalized solutions for the following Choquard equation involving Sobolev critical exponent

under the prescribed L²-norm

{\int }_{{\mathbb{R}}^3}{u}^2=c^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\int_{\mathbb R^{3}} \ u^{2}=c^{2}$$\end{document}

with c > 0, where I_α denotes the Riesz potential. Let

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${5 \over 3} < q < 3$$\end{document}

. When α > 0 small enough, we obtain the existence of the positive ground state solutions, which converge to a least energy solution of the limiting critical local problem as α → 0⁺.

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

-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.