Regular element of a Lie algebra

In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. In the specific case of nxn matrices over an algebraically closed field (such as the complex numbers), an element M is regular if and only if its Jordan normal form contains a single Jordan block for each eigenvalue. In that case, the centralizer is the set of polynomials of degree less than n evaluated at the matrix M, and therefore the centralizer has dimension n (but it is not necessarily an algebraic torus).

If the matrix M is diagonalisable, then it is regular if and only if there are n different eigenvalues. To see this, notice that M will commute with any matrix P that stabilises each of its eigenspaces. If there are n different eigenvalues, then this happens only if P is diagonalisable on a same basis as M is, and in fact P is a linear combination of the first n powers of M in this case, so that the centralizer is an algebraic torus of complex dimension n (and of dimension 2n as a real manifold); since this is the smallest possible dimension of a centralizer in this case, such a matrix M is regular. However if there are equal eigenvalues, then the centralizer, which is the product of the general linear groups of the eigenspaces of M, has strictly larger dimension, and M is not regular in this case.

For a connected compact Lie group G, and its Lie algebra g, the regular elements can also be described explicitly. In g they form an open and dense subset. In G, the regular elements form an open dense subset also; and if T is a maximal torus of G, the elements t of T that are regular in G determine the regular elements of G, which make up the union of the conjugacy classes in G of regular elements in T. The regular elements t are themselves explicitly given as the complement of a set in T, determined by the adjoint action of G, and making up a union of subtori.[1]

References

Notes

  1. ↑ Mark R. Sepanski, Compact Lie groups (2007), p. 156.