- Cartan's criterion
Cartan's criterion is an important mathematical theorem in the foundations of
Lie algebratheory that gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear formon defined by the formula: where tr denotes the trace of a linear operator. The criterion is named after Élie Cartan.
Cartan's criterion states: : "A finite-dimensional Lie algebra over a field of
characteristic zerois semisimple if and only if the Killing form is nondegenerate. A Lie algebra is solvable if and only if "
More generally, a finite-dimensional Lie algebra is reductive if and only if it admits a nondegenerate
invariant bilinear form.
Jean-Pierre Serre, "Lie algebras and Lie groups." 1964 lectures given at Harvard University. Second edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 1992. viii+168 pp. ISBN 3-540-55008-9
Modular Lie algebra
Wikimedia Foundation. 2010.
См. также в других словарях:
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Trace (algèbre) — Pour les articles homonymes, voir Trace. En algèbre linéaire, la trace d une matrice carrée A est définie comme la somme de ses coefficients diagonaux et notée Tr(A). La trace peut être vue comme une forme linéaire sur l espace vectoriel des… … Wikipédia en Français
Killing form — In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. In an example of Stigler s law of eponymy, the Killing form was actually invented… … Wikipedia
Real form (Lie theory) — Lie groups … Wikipedia
List of Lie groups topics — This is a list of Lie group topics, by Wikipedia page. Contents 1 Examples 2 Lie algebras 3 Foundational results 4 Semisimple theory … Wikipedia
Lie algebra — In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term… … Wikipedia
Solvable Lie algebra — In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. That is, writing for the derived Lie algebra of g, generated by the set of values [x,y] for x and y in g, the derived series … Wikipedia
Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… … Wikipedia
Tychonoff's theorem — For other theorems named after Tychonoff, see Tychonoff s theorem (disambiguation). In mathematics, Tychonoff s theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey… … Wikipedia
Holonomy — Parallel transport on a sphere depends on the path. Transporting from A → N → B → A yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection. In differential… … Wikipedia