Journal of Lie Theory

no. 4
Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
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 
@article{JOLT_2024_34_4_a4,
     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},
     doi = {10.5802/jolt.1363},
     zbl = {1567.17013},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1363/}
}
TY  - JOUR
AU  - A. Abdesselam
AU  - A. Thomas
TI  - Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
JO  - Journal of Lie Theory
PY  - 2024
SP  - 829
EP  - 862
VL  - 34
IS  - 4
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1363/
DO  - 10.5802/jolt.1363
ID  - JOLT_2024_34_4_a4
ER  - 
%0 Journal Article
%A A. Abdesselam
%A A. Thomas
%T Structure Constants for Simple Lie Algebras from a Principal sl2-Triple
%J Journal of Lie Theory
%D 2024
%P 829-862
%V 34
%N 4
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1363/
%R 10.5802/jolt.1363
%F JOLT_2024_34_4_a4
A. Abdesselam; A. 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

Cited by Sources: