A characterization of the unitary highest weight modules by Euclidean Jordan algebras
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2012
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| _version_ | 1866911998310088704 |
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| author | Bai, Zhanqiang |
| author_facet | Bai, Zhanqiang |
| contents | Let $\mathfrak{co}(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\mathfrak{co}(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\mathfrak{co}(J)_{\mathbb{C}})$. In particular, we find an quadratic element in $U(\mathfrak{co}(J)_{\mathbb{C}})$. A prime ideal in $U(\mathfrak{co}(J)_{\mathbb{C}})$ equals the Joseph ideal if and only if it contains this quadratic element. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1203_6434 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | A characterization of the unitary highest weight modules by Euclidean Jordan algebras Bai, Zhanqiang Representation Theory Let $\mathfrak{co}(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\mathfrak{co}(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\mathfrak{co}(J)_{\mathbb{C}})$. In particular, we find an quadratic element in $U(\mathfrak{co}(J)_{\mathbb{C}})$. A prime ideal in $U(\mathfrak{co}(J)_{\mathbb{C}})$ equals the Joseph ideal if and only if it contains this quadratic element. |
| title | A characterization of the unitary highest weight modules by Euclidean Jordan algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/1203.6434 |