Tridiagonal pairs of Krawtchouk type arising from finite-dimensional irreducible ๐๐โ-modules
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ID: 286883
2025
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Abstract
Let ๐ฝ denote an algebraically closed field with characteristic 0. One of the well-studied Lie algebras over ๐ฝ is the special linear algebra ๐๐2 which has a Chevalley basis {e,h,f}. Since the special orthogonal algebra ๐๐4 is the Lie algebra over ๐ฝ isomorphic to ๐๐2โจ๐๐2, ๐๐4 has a Chevalley basis {e1, h1, f1, e2, h2, f2}. In 2022, J. Morales found an automorphism *:๐๐4โ๐๐4 using some parameters p1,p2โ๐ฝ such that p1,p2โ{0,1}. By this automorphism, he found another Chevalley basis {e1*,h1*,f1*,e2*,h2*,f2*} of ๐๐4. He also described a simple construction of a finite-dimensional irreducible ๐๐4-module V in which ๐๐4 acts by derivation. In this paper, we construct four tridiagonal pairs on V via the actions of the Chevalley bases of ๐๐4. We show that these tridiagonal pairs are of Krawtchouk type and that the isomorphism of these tridiagonal pairs depends on the parameters p1 and p2. Consequently, we display four Lie algebra homomorphisms from the tetrahedron algebra โ to ๐๐4 and via these homomorphisms, we describe how the generators of โ act on V. Finally, we show that the irreducible ๐๐4-module V is isomorphic to a tensor product of two evaluation modules.
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| Authors | Pagaygay, Aaron |
| Journal | Malay Journal |
| Year | 2025 |
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| Keywords | Keywords not found |
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