A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY

Abstract

The notion of non-abelian phase space of a Lie algebra was first formulated and then discussed by Kuperschmidt. In this paper, we further study the non-abelian phase spaces in terms of left-symmetric algebras. We interpret the natural appearance of left-symmetric algebras from the intrinsic algebraic properties and the close relations with the classical Yang–Baxter equation. Furthermore, using the theory of left-symmetric algebras, we study some interesting geometric structures related to phase spaces. Moreover, we also discuss the generalized phase spaces with certain non-trivial algebraic structures on the dual spaces.