Symplectic Novikov Lie Algebra

TL;DR

This paper investigates symplectic Lie algebras with a Novikov left symmetric product, establishing conditions for associativity, classifying low-dimensional cases, and exploring the geometric properties of associated affine connections.

Contribution

It characterizes symplectic Novikov Lie algebras, proves their complete reducibility and two-step solvability, and provides classifications and geometric analyses for specific dimensions.

Findings

SNLA is completely reducible and two-step solvable

Classification results for 4-dimensional and nilpotent 6-dimensional cases

Methods for constructing examples of SNLA

Abstract

It is well known that a symplectic Lie algebra admit a left symmetric product. In this work, we study the case where this product is Novikov, we show that the left-symmetric product associated to the symplectic Lie algrbra is Novikov if and only if it is associative. In this case the symplectic Lie algebra is called symplectic Novikov Lie algebra(SNLA). We show that any SNLA is completely reducible two-step solvable. The classification of $4$-dimensional and nilpotent $6$-dimensional, also some methods for building large classes of examples are presented. Finally, we give a geometric study of the affine connection associated with an SNLA.