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$k$-Para-K\"ahler Lie algebras are a generalization of para-K\"ahler Liealgebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. Inthis paper, we show that the characterization of para-K\"ahler Lie algebras asleft symmetric bialgebras can be generalized to $k$-para-K\"ahler Lie algebrasleading to the introduction of two new structures which are different but bothgeneralize the notion of left symmetric algebra. This permits also theintroduction of generalized $S$-matrices.  We determine then all the $k$-symplectic Lie algebras of dimension $(k+1)$and all the six dimensional 2-para-K\"ahler Lie algebras.

On $k$-para-K\"ahler Lie algebras a subclass of $k$-symplectic Lie algebras