Witt algebra

Let denote the algebra of Laurent polynomials over a field . The Witt algebra over is the derivation subalgebra of the Lie algebra . #m/def/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

#state/tidy | #lang/en | #SemBr