• mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra...
    6 KB (794 words) - 15:22, 20 January 2024
  • Thumbnail for Lie algebra
    of g l ( n , F ) {\displaystyle {\mathfrak {gl}}(n,F)} . Cartan's criterion (by Élie Cartan) gives conditions for a finite-dimensional Lie algebra of...
    61 KB (10,459 words) - 23:14, 17 September 2024
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    did not make much use of the fact. A basic result that Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate...
    12 KB (1,837 words) - 13:51, 30 October 2024
  • criterion Cartan decomposition Cartan's equivalence method Cartan formalism (physics) Cartan involution Cartan's magic formula Cartan relations Cartan map Cartan...
    2 KB (201 words) - 05:40, 27 September 2024
  • Thumbnail for Solvable Lie algebra
    g ] {\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]} . This is Cartan's criterion for solvability. Lie's Theorem states that if V {\displaystyle V}...
    11 KB (1,606 words) - 19:14, 8 August 2024
  • Infinitesimal transformation Lie's third theorem Maurer–Cartan form Cartan's theorem Cartan's criterion Local Lie group Formal group law Hilbert's fifth problem...
    4 KB (360 words) - 19:55, 10 January 2024
  • Thumbnail for Real form (Lie theory)
    g0 is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable...
    6 KB (818 words) - 14:46, 20 June 2023
  • {\displaystyle \square } A typical application of the above criterion is the proof of Cartan's criterion for solvability of a Lie algebra. It says: if g ⊂ g l...
    41 KB (5,909 words) - 15:09, 15 September 2024
  • are equivalent. This conjecture is a vertex algebras analogue of Cartan's criterion for semisimplicity in the theory of Lie algebras because it relates...
    8 KB (1,177 words) - 05:18, 1 August 2024
  • abstract way—without using an explicit formula for the eigenvalue—using Cartan's criterion; see Sections 4.3 and 6.2 in the book of Humphreys. A Casimir element...
    20 KB (3,663 words) - 00:25, 22 September 2024