Quiver Poisson algebras

• DOI: 10.1016/j.jalgebra.2007.03.034
• Publication Date: 2007-04-09T13:03:36Z
• Authors: Yuan Yao; Yu Ye; Pu Zhang
• Keywords: Indecomposable module; Quiver; Poisson algebra; Poisson manifold

AbstractInner Poisson algebras on a given associative algebra are introduced and characterized, which gives a way of constructing non-commutative Poisson structures. Applying these to the finite-dimensional path algebras kQ→, together with the decomposition into indecomposable Lie ideals of the standard Poisson structure on kQ→, we classify all the inner Poisson structures on kQ→, which turn out to be the piecewise standard Poisson algebras. We also determine all the finite quivers Q→ without oriented cycles such that kQ→ admits outer Poisson structures: these are exactly the finite quivers without oriented cycles such that there exist two non-trivial paths α and β lying in a reduced closed walk, which cannot be connected by a sequence of non-trivial paths.