Virasoro algebra

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The Virasoro algebra, denoted Vir, is an infinite-dimensional Lie algebra, defined as central extension of the complexification of the Lie algebra of vector fields on the circle. One may think of it as a deformed version of the Lie algebra for the group of orientation-preserving diffeomorphisms of the circle. The representation theory of Virasoro algebra is rich, and has diverse applications in Mathematics and Physics.

Formal Definition[edit | edit source]

Vir is the Lie algebra over the field of complex numbers with the following generators:

  • ,with n running through every integer,

with the following relations:

  • ,
  • , with m and n each running through every integer

where is 1 when and is zero otherwise.

Representation Theory[edit | edit source]

Applications[edit | edit source]

See Also[edit | edit source]

Reference[edit | edit source]

  • Kac, V. G. and Raina, A. K.-- Highest Weight Representations of Infinite Dimensional Lie Algebras, ISBN 9971-50-396-4
  • Frenkel and ben-Zvi, Vertex algebras and algebraic curves, ISBN 0821828940, p.41(definition), p.326(geometric description)
  • Kac's article in Encyclopedia of Mathematics, Springer: [1]