On Lie algebras twisted by fractional differential operators

Abstract

In this paper, we explore the possibility of having fractional differential operator-twisted Lie algebras on function spaces over ℝ. It is known that under the bracket [·,·]_dⁿ/dxⁿ , C^∞(ℝ) forms a twisted Lie algebra only when n = 1, for non-negative integer values of n. We explore the possibility of having a twisted Lie algebra structure on C^∞(ℝ) when n ∈ (0,1]. Results show that using both the Riemann-Liouville and Caputo definition for fractional derivatives do not yield a twisted Lie algebra structure. On the other hand, using the Khalil definition gives rise to such structures. We then further explore twisted Lie algebras on time scales using the Khalil definition, showing that the desired structures only arise on closed subsets of ℝ, in which, every point is right dense. We then show that such subsets are equivalent to closed connected subsets of ℝ.