First Exploration of a Novel Generalization of Lie and Malcev Algebras Leading to the Emergence of Complex Structures

Abstract

Alejandro Jesus Bermejo Valdes

We introduce a novel algebraic structure, termed algebra B, which generalizes Lie algebras through the definition of a non-associative and unital algebra. We demonstrate that algebra B retains essential properties such as bilinearity and antisymmetry, and satisfies both the Jacobi and Malcev identities; this occurs when the product operations of Lie and Malcev algebras are derived from the product operation defined in our algebra. By examining the connections between algebra B, Malcev algebras, and Lie algebras, we establish that algebra B effectively generalizes these structures. Furthermore, we show that algebra B can generate complex entities with a novel structure distinct from those currently known. Our findings lay the groundwork for future investigations into the practical applications and further theoretical development of this new algebraic framework.

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