Near-derivations And Their Applications to Lie Algebras

Abstract

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 ) .