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arXiv · 2605.00469

On ${\pi}$-systems of symmetrizable Kac-Moody algebras

K. N. Raghavan·Krishanu Roy·S. Viswanath
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Abstract

Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-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 $\pi$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $\pi$-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.

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BibTeXRIS

K. N. Raghavan, Krishanu Roy, S. Viswanath. 2026-05-01. On ${\pi}$-systems of symmetrizable Kac-Moody algebras. https://arxiv.org/abs/2605.00469

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