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In mathematics, a '''Chevalley basis''' for a [[Simple Lie group|simple]] [[complex number|complex]] [[Lie algebra]] is
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a [[Basis (linear algebra)|basis]] constructed by [[Claude Chevalley]] with the property that all [[Algebra_over_a_field#Structure_coefficients|structure constants]] are integers. Chevalley used these bases to construct analogues of [[Lie group]]s over [[finite field]]s, called [[Chevalley group]]s.
 
The generators of a Lie group are split into the generators ''H'' and ''E'' indexed by simple [[Root_system|roots]] and their negatives <math>\pm\alpha_i</math>. The relations among the generators are the following:
 
:<math>[H_{\alpha_i},H_{\alpha_j}]=0</math>
:<math>[H_{\alpha_i},E_{\alpha_j}]=\alpha_j(H_{\alpha_i}) E_{\alpha_j}</math>
:<math>[E_{-\alpha_i},E_{\alpha_i}] = H_{\alpha_i} </math>
:<math>[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}</math>
where the last relation is imposed only if <math>\beta + \gamma</math>  is a root and '''p''' is the greatest positive integer such that {{nowrap|γ &minus; ''p''β}} is a root. The sign in the relation can be fixed arbitrarily.
 
==References==
*''Simple Groups of Lie Type'' by Roger W. Carter, ISBN 0-471-50683-4
 
[[Category:Lie groups]]
[[Category:Lie algebras]]
 
 
{{algebra-stub}}

Latest revision as of 11:36, 8 December 2014

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