Geodesic
Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
Journal of Lie theory, Tome 34 (2024) no. 4, pp. 829-862
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For a simple complex Lie algebra $\mathfrak g$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak g$ as vector space. For $\mathfrak{sl}_n({\mathbb C})$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed.
Classification : 17B05, 13A50
Mots-clés : Lie algebras, invariant theory, transvectants, 6j-symbols 
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     author = {A. Abdesselam and A. Thomas},
     title = {Structure {Constants} for {Simple} {Lie} {Algebras} from a {Principal} {sl\protect\textsubscript{2}-Triple}},
     journal = {Journal of Lie theory},
     pages = {829--862},
     year = {2024},
     volume = {34},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JLT_2024_34_4_JLT_2024_34_4_a4/}
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A. Abdesselam; A. Thomas. Structure Constants for Simple Lie Algebras from a Principal sl2-Triple. Journal of Lie theory, Tome 34 (2024) no. 4, pp. 829-862. http://geodesic.mathdoc.fr/item/JLT_2024_34_4_JLT_2024_34_4_a4/