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Nonassociative rings and algebras Infinite-dimensional Lie algebras Object

Witt algebra

Builds on Lie algebraLaurent polynomial ringLie bracket of vector fieldsSpecial linear Lie algebra

Description

The Witt algebra WW is the complex Lie algebra with basis {Ln:n∈ℤ}\set{L_n : n \in \Z} and bracket [Lm,Ln]=(m−n) Lm+n,m,n∈ℤ.[L_m, L_n] = (m - n)\, L_{m+n}, \qquad m, n \in \Z. It is realized by the derivations of the Laurent polynomial ring ℂ[z,z−1]\C[z, z^{-1}], with Ln=−zn+1ddzL_n = -z^{n+1} \frac{d}{dz}.

Check of the bracket: for vector fields u ddzu\, \frac{d}{dz} and v ddzv\, \frac{d}{dz} the bracket is (uv′−vu′)ddz(u v' - v u') \frac{d}{dz}. With u=−zm+1u = -z^{m+1} and v=−zn+1v = -z^{n+1} this gives ((n+1)−(m+1))zm+n+1ddz=(m−n)(−zm+n+1ddz)=(m−n)Lm+n\big((n+1) - (m+1)\big) z^{m+n+1} \frac{d}{dz} = (m - n)\left(-z^{m+n+1} \frac{d}{dz}\right) = (m - n) L_{m+n}. Since the bracket of vector fields is antisymmetric and satisfies the Jacobi identity, so does the bracket of WW.

The elements L−1,L0,L1L_{-1}, L_0, L_1 span a subalgebra isomorphic to 𝔰𝔩2(ℂ)\mathfrak{sl}_2(\C): indeed [L1,L−1]=2L0[L_1, L_{-1}] = 2L_0 and [L0,L±1]=∓L±1[L_0, L_{\pm 1}] = \mp L_{\pm 1}. The element L0L_0 acts diagonally: [L0,Ln]=−nLn[L_0, L_n] = -n L_n, so WW is graded by ℤ\Z.

Remarks

Up to isomorphism, the Witt algebra has exactly one nontrivial one-dimensional central extension, the Virasoro algebra. The name also refers to a related finite-dimensional Lie algebra over a field of characteristic p>0p > 0: the Lie algebra of derivations of the truncated polynomial ring k[z]/(zp)k[z]/(z^p), of dimension pp.

Reading path

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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 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 algebraobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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 bracket of vector fields concept
  9. Lie ideal concept
  10. Matrix definition
  11. Matrix concept
  12. Ring concept
  13. Trace of a matrix concept
  14. Vector space definition
  15. Jacobi identity statement
  16. Square matrix definition
  17. Linear map definition
  18. Lie algebra of an associative algebra example
  19. Multilinear map definition
  20. Alternating multilinear map definition
  21. General linear Lie algebra example
  22. Determinant definition
  23. Special linear group object
  24. Special linear Lie algebra example
  25. Witt algebra object

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Witt_algebra
About this page
TagA7H11C2JMM
URIm2://A7H11C2JMM
Kindobject
WritingW0 stub
MSC17B68, 17B66, 17B65
Instance ofLie algebra
UsesLie algebra, Laurent polynomial ring, Lie bracket of vector fields, Special linear Lie algebra
Same as
Changed3 November 2026

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