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- Regular_element_of_a_Lie_algebra abstract "In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in the complex general linear group of nxn matrices, a diagonalisable matrix M will commute with any matrix P that stabilises its 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. Non-diagonalisable matrices can be regular even though they contain eigenvalues with multiplicity: it is necessary and sufficient that their Jordan normal form contains a single Jordan block for each eigenvalue (in which case the centralizer is again the set of polynomials of degree less than n in M, and therefore of dimension n, but it is not an algebraic torus 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.".
- Regular_element_of_a_Lie_algebra wikiPageID "31861096".
- Regular_element_of_a_Lie_algebra wikiPageRevisionID "536000394".
- Regular_element_of_a_Lie_algebra hasPhotoCollection Regular_element_of_a_Lie_algebra.
- Regular_element_of_a_Lie_algebra subject Category:Lie_algebras.
- Regular_element_of_a_Lie_algebra subject Category:Lie_groups.
- Regular_element_of_a_Lie_algebra type Abstraction100002137.
- Regular_element_of_a_Lie_algebra type Algebra106012726.
- Regular_element_of_a_Lie_algebra type Cognition100023271.
- Regular_element_of_a_Lie_algebra type Content105809192.
- Regular_element_of_a_Lie_algebra type Discipline105996646.
- Regular_element_of_a_Lie_algebra type Group100031264.
- Regular_element_of_a_Lie_algebra type KnowledgeDomain105999266.
- Regular_element_of_a_Lie_algebra type LieAlgebras.
- Regular_element_of_a_Lie_algebra type LieGroups.
- Regular_element_of_a_Lie_algebra type Mathematics106000644.
- Regular_element_of_a_Lie_algebra type PsychologicalFeature100023100.
- Regular_element_of_a_Lie_algebra type PureMathematics106003682.
- Regular_element_of_a_Lie_algebra type Science105999797.
- Regular_element_of_a_Lie_algebra comment "In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in the complex general linear group of nxn matrices, a diagonalisable matrix M will commute with any matrix P that stabilises its each of its eigenspaces.".
- Regular_element_of_a_Lie_algebra label "Regular element of a Lie algebra".
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- Regular_element_of_a_Lie_algebra sameAs Q7309598.
- Regular_element_of_a_Lie_algebra sameAs Regular_element_of_a_Lie_algebra.
- Regular_element_of_a_Lie_algebra wasDerivedFrom Regular_element_of_a_Lie_algebra?oldid=536000394.
- Regular_element_of_a_Lie_algebra isPrimaryTopicOf Regular_element_of_a_Lie_algebra.