Journal of Lie Theory

no. 4
Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
Abdelmalek Abdesselam1; Alexander Thomas2
1Department of Mathematics, University of Virginia, Charlottesville, U.S.A.
2Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 829-862
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.
DOI: 10.5802/jolt.1363
Classification: 17B05, 13A50
Keywords: Lie algebras, invariant theory, transvectants, 6j-symbols

Abdelmalek Abdesselam  1 ; Alexander Thomas  2

1 Department of Mathematics, University of Virginia, Charlottesville, U.S.A.
2 Institut Camille Jordan, Université Lyon 1, Villeurbanne, France 
Abdelmalek Abdesselam; Alexander Thomas. Structure Constants for Simple Lie Algebras from a Principal sl2-Triple. Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 829-862. doi: 10.5802/jolt.1363
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     title = {Structure {Constants} for {Simple} {Lie} {Algebras} from a {Principal} {sl\protect\textsubscript{2}-Triple}},
     journal = {Journal of Lie Theory},
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