Chevalley basis

In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups.

The generators of a Lie group are split into the generators H and E such that:

$[H_{\alpha_i},H_{\alpha_j}]=0$

$[H_{\alpha_i},E_{\alpha_j}]=A_{ij}E_{\alpha_j}$

$[E_{\alpha_i},E_{\alpha_j}]=H_{\alpha_j}$

$[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}$

where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.

References

• Simple Groups of Lie Type by Roger W. Carter, ISBN 0-471-50683-4

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Lie groups Lie algebras Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Gumball (band) Steady Brook, Newfoundland and Labrador Look at other dictionaries: Chevalley–Warning theorem — In algebra, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1936) and a slightly weaker form of the theorem, known as… …   Wikipedia Chevalley–Shephard–Todd theorem — In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by… …   Wikipedia Jordan–Chevalley decomposition — In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley (also known as Dunford decomposition, named after Nelson Dunford, as well as SN decomposition), expresses a linear operator as the sum of its… …   Wikipedia 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 Group of Lie type — In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… …   Wikipedia Groupe de type de Lie — En mathématiques, un groupe de type de Lie G(k) est un groupe (non nécessairement fini) de points rationnels d un groupe algébrique linéaire réductif (en) G à valeur dans le corps commutatif k. La classification des groupes simples finis… …   Wikipédia en Français Hecke algebra — is the common name of several related types of associative rings in algebra and representation theory. The most familiar of these is the Hecke algebra of a Coxeter group , also known as Iwahori Hecke algebra, which is a one parameter deformation… …   Wikipedia Spinor — In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the… …   Wikipedia Jordan normal form — In linear algebra, a Jordan normal form (often called Jordan canonical form)[1] of a linear operator on a finite dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some… …   Wikipedia Jordan'sche Normalform — Die jordansche Normalform ist ein Begriff aus dem mathematischen Teilgebiet der linearen Algebra. Sie ist ein einfacher Vertreter der Äquivalenzklasse der zu einer trigonalisierbaren Matrix (trigonalisierbaren linearen Abbildung) ähnlichen… …   Deutsch Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”