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Nonassociative rings and algebras Infinite-dimensional Lie algebras Object

Virasoro algebra

Builds on Lie algebraWitt algebraLie algebra extension

Description

The Virasoro algebra Vir\mathrm{Vir} is the complex Lie algebra with basis LnL_n for n∈ℤn \in \Z and CC, and bracket [Lm,Ln]=(m−n) Lm+n+C12 (m3−m) δm+n,0,[Lm,C]=0,[L_m, L_n] = (m - n)\, L_{m+n} + \frac{C}{12}\, (m^3 - m)\, \delta_{m+n, 0}, \qquad [L_m, C] = 0, for m,n∈ℤm, n \in \Z. Here δm+n,0\delta_{m+n, 0} is 11 if m+n=0m + n = 0 and 00 otherwise.

The element CC is central. The map Ln↦LnL_n \mapsto L_n, C↦0C \mapsto 0 is a surjective Lie algebra homomorphism onto the Witt algebra with kernel ℂC\C C. So Vir\mathrm{Vir} is a central extension of the Witt algebra by a one-dimensional abelian Lie algebra. The extension does not split, and up to isomorphism it is the only nontrivial one-dimensional central extension of the Witt algebra.

Check on a small case: for m∈{−1,0,1}m \in \set{-1, 0, 1} the factor m3−mm^3 - m vanishes. So L−1,L0,L1L_{-1}, L_0, L_1 have the same brackets as in the Witt algebra and span a copy of 𝔰𝔩2(ℂ)\mathfrak{sl}_2(\C). For m=2m = 2 one gets [L2,L−2]=4L0+C2[L_2, L_{-2}] = 4 L_0 + \frac{C}{2}.

Remarks

In a representation where CC acts by a scalar cc, the number cc is called the central charge. Highest weight representations of Vir\mathrm{Vir} are labelled by the central charge cc and the eigenvalue hh of L0L_0 on a highest weight vector. The Virasoro algebra acts on every vertex operator algebra, through the modes of the conformal vector.

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Associative algebra, concept, no evidence levelAssociative algebraconceptCommutator of a ring, concept, no evidence levelCommutator of a ringconceptEndomorphism, concept, no evidence levelEndomorphismconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectLaurent polynomial ring, concept, no evidence levelLaurent polynomial ri…conceptLie algebra, concept, no evidence levelLie algebraconceptLie algebra extension, concept, no evidence levelLie algebra extensionconceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptLie ideal, concept, no evidence levelLie idealconceptMatrix, definition, no evidence levelMatrixdefinitionMatrix, concept, no evidence levelMatrixconceptRing, concept, no evidence levelRingconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptVector space, definition, no evidence levelVector spacedefinitionJacobi identity, statement, no evidence levelJacobi identitystatementSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLie algebra of an associative algebra, example, no evidence levelLie algebra of an ass…exampleMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionGeneral linear Lie algebra, example, no evidence levelGeneral linear Lie al…exampleDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial linear Lie algebra, example, no evidence levelSpecial linear Lie al…exampleWitt algebra, object, no evidence levelWitt algebraobjectVirasoro algebra, object, no evidence levelVirasoro algebraobject
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  1. Associative algebra concept
  2. Commutator of a ring concept
  3. Endomorphism concept
  4. General linear group object
  5. Identity matrix object
  6. Laurent polynomial ring concept
  7. Lie algebra concept
  8. Lie algebra extension concept
  9. Lie bracket of vector fields concept
  10. Lie ideal concept
  11. Matrix definition
  12. Matrix concept
  13. Ring concept
  14. Trace of a matrix concept
  15. Vector space definition
  16. Jacobi identity statement
  17. Square matrix definition
  18. Linear map definition
  19. Lie algebra of an associative algebra example
  20. Multilinear map definition
  21. Alternating multilinear map definition
  22. General linear Lie algebra example
  23. Determinant definition
  24. Special linear group object
  25. Special linear Lie algebra example
  26. Witt algebra object
  27. Virasoro algebra object

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Further reading

  1. Web https://en.wikipedia.org/wiki/Virasoro_algebra
About this page
TagR9WFXME6GC
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Kindobject
WritingW0 stub
MSC17B68, 81R10, 17B65
Instance ofLie algebra
UsesLie algebra, Witt algebra, Lie algebra extension
Same as
Changed3 November 2026

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