Witt algebra

A basis for the Witt algebra is given by the vector fields ${\displaystyle L_{n}=-z^{n+1}{\frac {\partial }{\partial z}}}$, for n in ${\displaystyle \mathbb {Z} }$.

The Lie bracket of two basis vector fields is given by

${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}.}$

This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal field theory and string theory.

Note that by restricting n to 1,0,-1, one gets a subalgebra. Taken over the field of complex numbers, this is just the Lie algebra ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ of the Lorentz group ${\displaystyle \mathrm {SO} (3,1)}$. Over the reals, it is the algebra sl(2,R) = su(1,1). Conversely, su(1,1) suffices to reconstruct the original algebra in a presentation.[1]

Over a field k of characteristic p>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring

k[z]/z^p

The Witt algebra is spanned by L_m for −1≤ m ≤ p−2.

• Virasoro algebra
• Heisenberg algebra

• Élie Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909).
• "Witt algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]