Near-derivations and their applications to Lie algebras

E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra $(\mathcal S(\mathfrak q),\{\ ,\,\})$ of a Lie algebra $\mathfrak q$. We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation $D$ of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ yields a pencil of compatible Poisson brackets on $\mathfrak q^*$ and (2) using $D$ one may naturally construct a Poisson-commutative subalgebra of $\mathcal S(\mathfrak q)$. A special attention is given to near-derivations of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ induced from near-derivations of $\mathfrak q$. This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $\mathfrak q$ and Nijenhuis operators in $\mathfrak{gl}(\mathfrak q)$.