Bialgebras induced by special left Alia algebras

TL;DR

This paper explores the structure of Nijenhuis left Alia bialgebras, showing how they can be derived from Nijenhuis associative D-bialgebras, and provides methods to construct Nijenhuis operators on such algebras and coalgebras.

Contribution

It introduces the concept of Nijenhuis left Alia bialgebras and demonstrates their relation to Nijenhuis associative D-bialgebras, offering new insights and construction methods.

Findings

Nijenhuis left Alia bialgebras can be induced from Nijenhuis associative D-bialgebras.

A method for constructing Nijenhuis operators on left Alia algebras and coalgebras.

Establishment of the relationship between Nijenhuis structures and special left Alia bialgebras.

Abstract

Special left Alia algebras were introduced by Dzhumadil'daev in [J. Math. Sci. (N.Y.) 161(2009), 11-30] when studying the classification of algebras with skew-symmetric identity of degree 3. A special left Alia algebra (resp. coalgebra) $(A, [,]_{(f,g)})$ (resp. $(A, \Delta_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A, \cdot)$ (resp. $(A, \delta)$) together with two linear maps $f, g: A\longrightarrow A$ (resp. $F, G: A\longrightarrow A$). We find that if $((A, \cdot), f)$ (resp. $((A, \delta), F)$) is a Nijenhuis associative algebra (resp. coassociative coalgebra) such that $f\circ g=g\circ f$ (resp. $F\circ G=G\circ F$), then $((A, [,]_{(f,g)}), f)$ (resp. $((A, \Delta_{(F,G)}), F)$) is a Nijenhuis left Alia algebra (resp. coalgebra). A bialgebraic structure, named Nijenhuis associative D-bialgebra and denoted by $((A, \cdot, \delta), f, F)$, for $((A, \cdot), f)$ and $((A, \delta), F)$ was presented in [J. Algebra 639(2024), 150-186]. In this paper, we investigate the bialgebraic structure, named Nijenhuis left Alia bialgebra and denoted by $((A, [,], \Delta), N, S)$, for a Nijenhuis left Alia algebra $((A, [,]), N)$ and a Nijenhuis left Alia coalgebra $((A, \Delta), S)$, such that Nijenhuis special left Alia bialgebra $((A, [,]_{(f,g)}, \Delta_{(F,G)}), f, F)$ can be induced by Nijenhuis commutative cocommutative associative D-bialgebra $((A, \cdot, \delta), f, F)$. We also provide a method to construct Nijenhuis operators on a left Alia algebra (resp. coalgebra).