Journal of Lie Theory

no. 3
A Characterization of the Unitary Highest Weight Modules by Euclidean Jordan Algebras
Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 747-778
\def\c{{\frak c}} \def\o{{\frak o}} \def\C{{\Bbb C}} Let $\c\o(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\c\o(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\c\o(J)_\C)$. In particular, we find an quadratic element in $U(\c\o(J)_\C)$. A prime ideal in $U(\c\o(J)_\C)$ equals the Joseph ideal if and only if it contains this quadratic element.
DOI: 10.5802/jolt.748
Classification: 22E47, 17B10, 17C99
Keywords: Euclidean Jordan algebras, unitary highest weight module, quadratic relation, Joseph Ideal 
@article{JOLT_2013_23_3_a8,
     author = {Z. Bai},
     title = {A {Characterization} of the {Unitary} {Highest} {Weight} {Modules} by {Euclidean} {Jordan} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {747--778},
     year = {2013},
     volume = {23},
     number = {3},
     doi = {10.5802/jolt.748},
     zbl = {1279.22020},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.748/}
}
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Z. Bai. A Characterization of the Unitary Highest Weight Modules by Euclidean Jordan Algebras. Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 747-778. doi: 10.5802/jolt.748

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