Witt algebra

The Witt algebra is studied in mathematics , it is a special Lie algebra . It is used in mathematical physics , as in string theory and conformal field theory . It is named after the German mathematician Ernst Witt .

definition

Let be a basis of a vector space with as an integer index. The through the commutator relation ${\displaystyle L_{j}}$ ${\displaystyle j}$

${\displaystyle [L_{j},L_{k}]:=(j-k)\cdot L_{j+k}}$

defined Lie algebra is called Witt algebra. Such algebras are obtained as derivatives algebra over the ring of Laurent polynomials .

Realization through vector fields

In most applications, one looks at derivatives over . The Witt algebra can be implemented as follows using complex-valued vector fields: ${\displaystyle \mathbb {C} }$

${\displaystyle L_{n}:=-z^{n+1}{\frac {\partial }{\partial z}}}$

sl (2, K) as a sub-algebra

From the above commutator relations it immediately follows that for the Unter-Lie algebra generated by is the same . This three-dimensional Unter Lie algebra is isomorphic to sl (2, K) . ${\displaystyle n>0}$ ${\displaystyle L_{-n},L_{0},L_{n}}$ ${\displaystyle K\cdot L_{-n}+K\cdot L_{0}+K\cdot L_{n}}$

Central expansion

If you put Witt algebra through the Kozykel

${\displaystyle \omega (L_{m},L_{n}):={\frac {1}{12}}(n^{3}-n)\delta _{m+n,0}}$

extended centrally, one obtains the Virasoro algebra .

swell

Igor Frenkel , James Lepowsky , Arne Meurman : Vertex Operator Algebras and the Monster , Academic Press, New York (1988) ISBN 0-12-267065-5