We introduce the notion of $lambda$-double Lie algebra, which coincides with usual double Lie algebra when $lambda = 0$. We state that every $lambda$-double Lie algebra for $lambda eq0$ provides the structure of modified double Poisson algebra on the
free associative algebra. In particular, it confirms the conjecture of S. Arthamonov (2017). We prove that there are no simple finite-dimensional $lambda$-double Lie algebras.
We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group
algebra of a group $G$ or $H=U(mathfrak{g})$ the universal enveloping algebra of a Lie algebra $mathfrak{g}$, then we prove that Rota-Baxter operators on $H$ are in one to one correspondence with corresponding Rota-Baxter operators on groups or Lie algebras.
We classify all Rota-Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.
We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for
a solution $r$ the element $r+tau(r)$ is $L$-invariant.
