The notion of a double Poisson algebra on a given associative algebra was introduced by M. Van den Bergh in 2008 as a noncommutative analog of Poisson algebra. The goal behind this notion was to develop noncommutative Poisson geometry. The notion of double Lie algebra (of weight zero) naturally arose directly from the definition of double Poisson algebra.

In this talk, I will introduce the notion of a double Lie algebra of a nonzero weight lambda, which coincides with usual double Lie algebra when the weight lambda is equal to zero. I will focus on the algebraic ideas that lead us to the definition and on  connections between structures of double Lie algebras and Rota-Baxter operators of special type on the corresponding matrix algebra. As a motivation, I will speak on the connection between lambda-double Lie algebra and structures of modified double Poisson algebra on the free associative algebra. Also, some properties of finite-dimensional lambda-double Lie algebras will be discussed.