Lie algebras MOC

Witt algebra

Let denote the algebra of Laurent polynomials over a field . The Witt algebra over is the derivation subalgebra of the Lie algebra . lie It is equivalently characterized as follows:¹ For each , define the derivation

Then is the Lie algebra of such derivations. The bracket may be expressed as

and a basis is given by

Proof of equivalence

Let denote the first characterization. Let , and set . Then

whence , furthermore

whence . Since these results hold for in place of , these two operators concur for all powers of .

In , the Witt algebra admits a unique nontrivial 1-dimensional central extension, the Virasoro algebra.

Properties

• For any , forms a Lie subalgebra isomorphic to [[Special linear Lie algebra|]], in particular

\begin{align*} \mathfrak{p} = \mathbb{K}d_{-1} + \mathbb{K}d_{0} + \mathbb{K}d_{1} \cong_{\cat{Lie}{\mathbb{K}}} \opn{\mathfrak{sl}}{2}\mathbb{K} \end{align*}

Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.9, pp. 31–32 ↩