Irreducible Generalized Restricted Representations of g∆2

Ibraev, Sherali S.; Kainbayeva, Larissa; Yensebayeva, Gulzat M.; Ibrayeva, Anar A.; Parmenova, Manat Z.; Yeshmurat, Gulnur K.; Abuova, Nabat A.

doi:10.3390/sym17122072

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Irreducible Generalized Restricted Representations of $g (∆_{2})$

by

Sherali S. Ibraev

,

Larissa Kainbayeva

^*,

Gulzat M. Yensebayeva

,

Anar A. Ibrayeva

,

Manat Z. Parmenova

,

Gulnur K. Yeshmurat

and

Nabat A. Abuova

Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda University, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan

^*

Author to whom correspondence should be addressed.

Symmetry 2025, 17(12), 2072; https://doi.org/10.3390/sym17122072

Submission received: 29 October 2025 / Revised: 26 November 2025 / Accepted: 30 November 2025 / Published: 3 December 2025

(This article belongs to the Special Issue Symmetries in Algebraic Combinatorics and Their Applications)

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Abstract

Generalized restricted representations were introduced to facilitate the study of representations of modular Lie algebras that lack a restricted structure. At present, generalized restricted representations are defined for Lie algebras of the Cartan type and for contragredient Lie algebras over fields of positive characteristic. Classifications of irreducible generalized restricted representations have been obtained for both classes of Lie algebras. However, irreducible generalized restricted representations with non-restricted highest weights, as well as their dimensions, remain largely unexplored. For contragredient Lie algebras, the dimensions of such representations are known only in the case of a three-dimensional simple algebra over a field of characteristic 2. In this paper, we study irreducible generalized restricted representations of the ten-dimensional simple contragredient Lie algebra

g (∆_{2})

with a Cartan matrix of

∆_{2} = (\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix})

over an algebraically closed field of characteristic

p = 2 .

We provide a complete classification of the irreducible generalized restricted

g (∆_{2})

modules and their dimensions (Theorem 1). These modules are parameterized by the fundamental weights

ω_{1}

and

ω_{2}

and by elements of the finite field

F_{2^{s}}

.

Keywords: contragredient Lie algebra; simple Lie algebra; generalized restricted representation; irreducible representation

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MDPI and ACS Style

Ibraev, S.S.; Kainbayeva, L.; Yensebayeva, G.M.; Ibrayeva, A.A.; Parmenova, M.Z.; Yeshmurat, G.K.; Abuova, N.A. Irreducible Generalized Restricted Representations of $g (∆_{2})$ . Symmetry 2025, 17, 2072. https://doi.org/10.3390/sym17122072

AMA Style

Ibraev SS, Kainbayeva L, Yensebayeva GM, Ibrayeva AA, Parmenova MZ, Yeshmurat GK, Abuova NA. Irreducible Generalized Restricted Representations of $g (∆_{2})$ . Symmetry. 2025; 17(12):2072. https://doi.org/10.3390/sym17122072

Chicago/Turabian Style

Ibraev, Sherali S., Larissa Kainbayeva, Gulzat M. Yensebayeva, Anar A. Ibrayeva, Manat Z. Parmenova, Gulnur K. Yeshmurat, and Nabat A. Abuova. 2025. "Irreducible Generalized Restricted Representations of $g (∆_{2})$ " Symmetry 17, no. 12: 2072. https://doi.org/10.3390/sym17122072

APA Style

Ibraev, S. S., Kainbayeva, L., Yensebayeva, G. M., Ibrayeva, A. A., Parmenova, M. Z., Yeshmurat, G. K., & Abuova, N. A. (2025). Irreducible Generalized Restricted Representations of $g (∆_{2})$ . Symmetry, 17(12), 2072. https://doi.org/10.3390/sym17122072

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Irreducible Generalized Restricted Representations of $g (∆_{2})$

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