Near-derivations And Their Applications to Lie Algebras

Abstract E.B. Vinberg’s concept 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 $$({\\mathcal {S}}({\\mathfrak q}),\\{\\ ,\\,\\})$$ ( S ( q ) , { , } ) of a Lie algebra $${\\mathfrak q}$$ 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 $$({\\mathcal {S}}({\\mathfrak q}),\\{\\ ,\\,\\})$$ ( S ( q ) , { , } ) yields a pencil of compatible Poisson brackets on $${\\mathfrak q}^*$$ q ∗ and (2) using D one may naturally construct a Poisson-commutative subalgebra of $${\\mathcal {S}}({\\mathfrak q})$$ S ( q ) . A special attention is given to near-derivations of $$({\\mathcal {S}}({\\mathfrak q}),\\{\\ ,\\,\\})$$ ( S ( q ) , { , } ) induced from near-derivations of $${\\mathfrak q}$$ q . This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $${\\mathfrak q}$$ q and Nijenhuis operators in $${\\mathfrak {gl}}({\\mathfrak q})$$ gl ( q ) .