Virasoro Algebra

The Virasoro algebra is an infinitely dimensional Lie algebra and thus belongs to the field of mathematics . It is used in mathematical physics , especially in string theory and in conformal field theory . There it is treated as an algebra over the complex numbers, but instead of the complex numbers any fields with the characteristic 0 can also be used. It was introduced in 1970 by Miguel Virasoro as part of string theory. In mathematics, it plays an important role in the construction of the group of monsters .

construction

The starting point is the Witt algebra over a field of characteristic 0 (for example ), which is generated by elements with the commutator relations . A Virasoro algebra is defined as the central extension of this Witt algebra. That is, there is a short exact sequence of Lie algebras ${\displaystyle W}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle d_{n},n\in \mathbb {Z} }$ ${\displaystyle [d_{m},d_{n}]\,=\,(m-n)d_{m+n}}$ ${\displaystyle V}$

${\displaystyle 0\rightarrow K\cdot c\rightarrow V\rightarrow W\rightarrow 0}$.

${\displaystyle K\cdot c}$is a one-dimensional vector space that can be thought of as being contained in. It should lie in the center of , which is sometimes referred to as the “central charge” of Virasoro algebra. The Virasoro algebra is then generated by and elements that are archetypes of . Certain options are available for the commutator relations . A common choice is ${\displaystyle V}$ ${\displaystyle c}$ ${\displaystyle V}$ ${\displaystyle c}$ ${\displaystyle V}$ ${\displaystyle c}$ ${\displaystyle L_{n}}$ ${\displaystyle d_{n}}$

• ${\displaystyle [v,c]\,=\,0}$for all , because is in the center of , ${\displaystyle v\in V}$ ${\displaystyle c}$ ${\displaystyle V}$
• ${\displaystyle [L_{m},L_{n}]:=(m-n)\cdot L_{m+n}+{\frac {c}{12}}(m^{3}-m)\delta _{m+n,0}}$for everyone . ${\displaystyle m,n\in \mathbb {Z} }$

It stands for the Kronecker Delta . The central part of the commutator relation is called; In the most general case, this proportion can be chosen as with . The present choice is motivated by the fact that for vanishes and is therefore mapped isomorphically to in the above sequence , the latter being a Lie algebra that is isomorphic to sl (2, K) . The factor is explained by the fact that there are certain representations of Virasoro algebra in which this factor then disappears; this is just a convenient convention. ${\displaystyle \delta }$ ${\displaystyle {\frac {c}{12}}(m^{3}-m)\delta _{m+n,0}}$ ${\displaystyle \alpha m^{3}+\beta m}$ ${\displaystyle \alpha ,\beta \in K}$ ${\displaystyle m^{3}-m}$ ${\displaystyle m=-1,0,1}$ ${\displaystyle K\cdot L_{-1}+K\cdot L_{0}+K\cdot L_{1}\subset V}$ ${\displaystyle K\cdot d_{-1}+K\cdot d_{0}+K\cdot d_{1}\subset W}$ ${\displaystyle 1/12}$

An alternative convention

An alternative choice of commutator relations is obtained if one passes from to . A short calculation shows ${\displaystyle L_{n}}$ ${\displaystyle L_{n}':=L_{n}-{\frac {c}{24}}\delta _{n,0}}$

• ${\displaystyle [L_{m}',L_{n}']:=(m-n)\cdot L_{m+n}'+{\frac {cm^{3}}{12}}\delta _{m+n,0}}$,

that is, one can make the linear term of the central part of the commutator relations disappear.

Equivalences

Two central extensions of Witt algebra and are called equivalent if there is a Lie algebra isomorphism with and is. ${\displaystyle 0\rightarrow K\cdot c\,{\stackrel {i_{1}}{\rightarrow }}\,V_{1}\,{\stackrel {p_{1}}{\rightarrow }}\,W\rightarrow 0}$ ${\displaystyle 0\rightarrow K\cdot c\,{\stackrel {i_{2}}{\rightarrow }}\,V_{2}\,{\stackrel {p_{2}}{\rightarrow }}\,W\rightarrow 0}$ ${\displaystyle \phi \colon V_{1}\to V_{2}}$ ${\displaystyle i_{2}=\phi \circ i_{1}}$ ${\displaystyle p_{1}=p_{2}\circ \phi }$

One can show that, apart from equivalence, there is only one central extension that is not equivalent to a semidirect sum , namely the Virasoro algebra introduced above. ${\displaystyle 0\rightarrow K\cdot c\rightarrow V\rightarrow W\rightarrow 0}$ ${\displaystyle K\cdot c\oplus W}$

swell

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