Math 2.0

Nonassociative rings and algebras Lie algebras: basic notions Example

Heisenberg algebra

Builds on General linear Lie algebraCenter of a Lie algebraHeisenberg group

Example

Let kk be a field. The three-dimensional Heisenberg algebra π”₯3\mathfrak h_3 over kk has a basis x,y,zx, y, z with [x,y]=z,[x,z]=0,[y,z]=0.[x, y] = z, \quad [x, z] = 0, \quad [y, z] = 0. A concrete model is the space of strictly upper triangular 3Γ—33 \times 3 matrices in the general linear Lie algebra 𝔀𝔩3(k)\mathfrak{gl}_3(k), with x=E12,y=E23,z=E13.x = E_{12}, \quad y = E_{23}, \quad z = E_{13}. Check: E12E23=E13E_{12} E_{23} = E_{13} and E23E12=0E_{23} E_{12} = 0, so [x,y]=z[x, y] = z. Also E13E_{13} has zero product with E12E_{12} and E23E_{23} in both orders, so [x,z]=[y,z]=0[x, z] = [y, z] = 0. Since the model is a subspace of 𝔀𝔩3(k)\mathfrak{gl}_3(k) closed under brackets, π”₯3\mathfrak h_3 is a Lie algebra.

Remarks

The center of π”₯3\mathfrak h_3 is the line kzk z, and [π”₯3,π”₯3]=kz[\mathfrak h_3, \mathfrak h_3] = k z. So π”₯3\mathfrak h_3 is nilpotent but not abelian. More generally, π”₯2n+1\mathfrak h_{2n+1} has basis x1,…,xn,y1,…,yn,zx_1, \dots, x_n, y_1, \dots, y_n, z with [xi,yj]=Ξ΄ijz[x_i, y_j] = \delta_{ij} z and all other brackets of basis elements zero. For k=ℝk = \R it is the Lie algebra of the Heisenberg group. The canonical commutation relation [q^,p^]=iℏ[\hat q, \hat p] = i\hbar of quantum mechanics has the form [x,y]=z[x, y] = z, with zz acting as the scalar iℏi\hbar.

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  1. Associative algebra concept
  2. Center of a Lie algebra concept
  3. Centre of a group concept
  4. Commutator of a ring concept
  5. Endomorphism concept
  6. General linear group object
  7. Lie group concept
  8. Matrix concept
  9. Nilpotent group concept
  10. Ring concept
  11. Heisenberg group example
  12. Jacobi identity statement
  13. Lie algebra of an associative algebra example
  14. General linear Lie algebra example
  15. Heisenberg algebra example

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Heisenberg_group
About this page
TagHBVZGRKCWQ
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WritingW0 stub
MSC17B30, 17B81
Instance ofLie algebra
UsesGeneral linear Lie algebra, Center of a Lie algebra, Heisenberg group
Changed3 November 2026

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