Journal of Lie Theory 34 (2024), No. 4, 829--862
Copyright Heldermann Verlag 2024
Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
Abdelmalek Abdesselam
Department of Mathematics, University of Virginia, Charlottesville, U.S.A.
malek@virginia.edu
Alexander Thomas
Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
athomas@math.univ-lyon1.fr
[Abstract-pdf]
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.
Keywords: Lie algebras, invariant theory, transvectants, 6j-symbols.
MSC: 17B05, 13A50.
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