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Nonassociative rings and algebras Lie algebras: basic notions Example

Special linear Lie algebra

Builds on General linear Lie algebraTrace of a matrixLie idealSpecial linear group

Example

Let kk be a field and n≥2n \ge 2. The special linear Lie algebra is 𝔰𝔩n(k)={X∈𝔤𝔩n(k):tr⁡X=0},\mathfrak{sl}_n(k) = \set{X \in \mathfrak{gl}_n(k) : \tr X = 0}, the set of n×nn \times n matrices of trace zero, with the bracket [X,Y]=XY−YX[X, Y] = XY - YX of the general linear Lie algebra.

Check that it is a Lie algebra: it is a linear subspace, and tr⁡[X,Y]=tr⁡(XY)−tr⁡(YX)=0\tr[X, Y] = \tr(XY) - \tr(YX) = 0 for all X,YX, Y. So 𝔰𝔩n(k)\mathfrak{sl}_n(k) is closed under brackets, and it is even an ideal of 𝔤𝔩n(k)\mathfrak{gl}_n(k). Its dimension is n2−1n^2 - 1.

For n=2n = 2 the standard basis is e=(0100),h=(100−1),f=(0010),e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, with [h,e]=2e[h, e] = 2e, [h,f]=−2f[h, f] = -2f and [e,f]=h[e, f] = h.

Remarks

For k=ℝk = \R or ℂ\C, 𝔰𝔩n(k)\mathfrak{sl}_n(k) is the Lie algebra of the special linear group SLn(k)SL_n(k), because det⁡(etX)=ettr⁡X\det(e^{tX}) = e^{t \tr X}. Over a field of characteristic 00, 𝔰𝔩n(k)\mathfrak{sl}_n(k) is simple for n≥2n \ge 2. Over ℂ\C, 𝔰𝔩n(ℂ)\mathfrak{sl}_{n}(\C) is the simple Lie algebra of type An−1A_{n-1}.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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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 matrixobjectLie 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…example
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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. Lie ideal concept
  7. Matrix definition
  8. Matrix concept
  9. Ring concept
  10. Trace of a matrix concept
  11. Vector space definition
  12. Jacobi identity statement
  13. Square matrix definition
  14. Linear map definition
  15. Lie algebra of an associative algebra example
  16. Multilinear map definition
  17. Alternating multilinear map definition
  18. General linear Lie algebra example
  19. Determinant definition
  20. Special linear group object
  21. Special linear Lie algebra example

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Further reading

  1. Web https://en.wikipedia.org/wiki/Special_linear_Lie_algebra
About this page
Tag1ZAT1Z967A
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Kindexample
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MSC17B20, 17B45, 22E60
Instance ofLie algebra
UsesGeneral linear Lie algebra, Trace of a matrix, Lie ideal, Special linear group
Same as
Changed3 November 2026

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