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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.07348 |
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| _version_ | 1866915822295842816 |
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| author | Lau, Michael Mathieu, Olivier |
| author_facet | Lau, Michael Mathieu, Olivier |
| contents | Let $J$ be a unital Jordan algebra, and let $\widehat{\mathfrak{sl}}_2(J)$ be the universal central extension of its Tits-Kantor-Koecher Lie algebra.
In Part A, we study the category of $(\widehat{\mathfrak{sl}}_2(J), SL_2(K))$-modules. We characterize the dominant $J$-spaces, which are analogous to the dominant highest weights appearing in classical settings. A family of universal envelopes $\mathcal{U}_n(J)$ associated to such modules is introduced and studied. We also prove some finiteness theorems.
In Part C, we define the notion of smooth $\widehat{\mathfrak{sl}}_2(J)$-modules for augmented Jordan algebras $J$, and investigate the category of smooth modules in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules of this category are finite dimensional when $J$ is finitely generated.
The free unital Jordan algebra $J(D)$ over $D$ variables is an elusive object, but finiteness and Ext-vanishing properties suggest that the smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues might form a generalized highest weight category. However, we prove that such an assertion would contradict recently obtained information about the growth of free Jordan algebras. See [24] and [13] for more details. It then follows that the category of smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues is not a generalized highest weight category when $D\geq 2$.
Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free Jordan Algebras and Representations of $\widehat{\mathfrak{sl}}_2(J)$ Lau, Michael Mathieu, Olivier Representation Theory 17B10, 17B60, 17C05, 17C50 Let $J$ be a unital Jordan algebra, and let $\widehat{\mathfrak{sl}}_2(J)$ be the universal central extension of its Tits-Kantor-Koecher Lie algebra. In Part A, we study the category of $(\widehat{\mathfrak{sl}}_2(J), SL_2(K))$-modules. We characterize the dominant $J$-spaces, which are analogous to the dominant highest weights appearing in classical settings. A family of universal envelopes $\mathcal{U}_n(J)$ associated to such modules is introduced and studied. We also prove some finiteness theorems. In Part C, we define the notion of smooth $\widehat{\mathfrak{sl}}_2(J)$-modules for augmented Jordan algebras $J$, and investigate the category of smooth modules in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules of this category are finite dimensional when $J$ is finitely generated. The free unital Jordan algebra $J(D)$ over $D$ variables is an elusive object, but finiteness and Ext-vanishing properties suggest that the smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues might form a generalized highest weight category. However, we prove that such an assertion would contradict recently obtained information about the growth of free Jordan algebras. See [24] and [13] for more details. It then follows that the category of smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues is not a generalized highest weight category when $D\geq 2$. Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov. |
| title | Free Jordan Algebras and Representations of $\widehat{\mathfrak{sl}}_2(J)$ |
| topic | Representation Theory 17B10, 17B60, 17C05, 17C50 |
| url | https://arxiv.org/abs/2502.07348 |