# 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}.

*Proceedings of the Samahang Pisika ng Pilipinas*

**37**, SPP-2019-PB-01 (2019). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2019-PB-01.