Admissible hom-Mock Lie Algebras: Dual Representations, Matched Pairs, Manin Triples and Bialgebras

Gbêvèwou Houndedji, Bakary Kourouma and Cyrille Haliya

This work addresses the admissible representations of hom-Mock Lie algebras by studying their dual representations. The analysis establishes the conditions for admissibility, under which admissible hom-Mock Lie algebras are defined via their adjoint representations. These conditions allow for a systematic study of the structural properties and internal symmetries of such algebras. Matched pairs of admissible hom-Mock Lie algebras are constructed, providing a framework to understand the interaction between two compatible algebraic structures. In particular, the properties of adjoint and coadjoint representations of regular admissible hom-Mock Lie algebras are examined, revealing key insights into their representation theory. Furthermore, admissible hom-Mock Lie bialgebras are developed based on the compatibility of dual structures. Their equivalence to matched pairs and Manin triples is demonstrated using a standard symmetric bilinear form defined on the direct sum of a hom-Mock Lie algebra and its dual. These findings contribute to the ongoing development of hom-type algebras and extend classical Lie theoretic ideas to more generalized algebraic systems.

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