On $π$-systems of symmetrizable Kac-Moody algebras
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| Format: | Preprint |
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2026
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| _version_ | 1866917452828377088 |
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| author | Raghavan, K. N. Roy, Krishanu Viswanath, S. |
| author_facet | Raghavan, K. N. Roy, Krishanu Viswanath, S. |
| contents | Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $π$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of $\mathfrack{g}$, and were originally studied by Dynkin, Morita and Naito. We show that the binary relation introduced by Morita defines a partial order on the set of $\mathfrack{g}$ of finite, untwisted affine or hyperbolic type. We also formulate general principles for constructing $π$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $π$-systems of a given $\mathfrack{g}$. Among other applications, we use this to determine the set of maximal hyperbolic Dynkin diagrams in ranks $3$-$10$ relative to the Morita partial order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_00469 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $π$-systems of symmetrizable Kac-Moody algebras Raghavan, K. N. Roy, Krishanu Viswanath, S. Rings and Algebras 17B22 (17B67) Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $π$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of $\mathfrack{g}$, and were originally studied by Dynkin, Morita and Naito. We show that the binary relation introduced by Morita defines a partial order on the set of $\mathfrack{g}$ of finite, untwisted affine or hyperbolic type. We also formulate general principles for constructing $π$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $π$-systems of a given $\mathfrack{g}$. Among other applications, we use this to determine the set of maximal hyperbolic Dynkin diagrams in ranks $3$-$10$ relative to the Morita partial order. |
| title | On $π$-systems of symmetrizable Kac-Moody algebras |
| topic | Rings and Algebras 17B22 (17B67) |
| url | https://arxiv.org/abs/2605.00469 |