Bialgebras induced by special left Alia algebras

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. $(A,{\mathrm{\Delta }}_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A,\cdot )$ (resp. $(A,\delta )$) together with two linear maps $f,g:A⟶A$ (resp. $F,G:A⟶A$). 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. $((A,{\mathrm{\Delta }}_{(F,G)}),F)$) 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 $((A,[,],\mathrm{\Delta }),N,S)$, for a Nijenhuis left Alia algebra $((A,[,]),N)$ and a Nijenhuis left Alia coalgebra $((A,\mathrm{\Delta }),S)$, such that Nijenhuis special left Alia bialgebra $((A,[,{]}_{(f,g)},{\mathrm{\Delta }}_{(F,G)}),f,F)$ 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).