Twisted Lie brackets and the Witt algebra

Abstract

In this paper, we explore twisted Lie brackets on a general commutative associative algebra A. Given a linear map ψ: A → A and for any a, b ∈ A, we define the bilinear form [,]_ψ by [a, b]_ψ := a⋅ ψ(b) - b⋅ ψ(a). We show that for any linear map ψ, skew-symmetry is always satisfied, however, the Jacobi identity is not. We identify necessary and sufficient conditions on ψ so that [,]_ψ satisfies the Jacobi identity, making it a twisted Lie bracket on A. We also show that when ψ is a linear derivation on A, [,]_ψ will be Lie bracket. We then give examples on C∞(ℝⁿ). Finally, we show that the Witt algebra is in fact a twisted Lie algebra by proving that it is isomorphic to C∞_poly(S¹) with the twisted Lie bracket [,]_d/dθ.