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In mathematics, a Chevalley basis for a
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complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
Lie algebra is a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
constructed by
Claude Chevalley Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foun ...
with the property that all
structure constants In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
are integers. Chevalley used these bases to construct analogues of Lie groups over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s, called
Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...
s. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators ''H'' and ''E'' indexed by simple
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
and their negatives \pm\alpha_i. The Cartan-Weyl basis may be written as : _i,H_j0 : _i,E_\alpha\alpha_i E_\alpha Defining the dual root or coroot of \alpha as :\alpha^\vee = \frac One may perform a change of basis to define :H_=(\alpha_i^\vee, H) The Cartan integers are :A_=(\alpha_i,\alpha_j^\vee) The resulting relations among the generators are the following: : _,H_0 : _,E_A_ E_ : _,E_= H_ : _,E_\pm(p+1)E_ where in the last relation p is the greatest positive integer such that \gamma -p\beta is a root and we consider E_ = 0 if \beta + \gamma is not a root. For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if \beta \prec \gamma then \beta + \alpha \prec \gamma + \alpha provided that all four are roots. We then call (\beta, \gamma) an extraspecial pair of roots if they are both positive and \beta is minimal among all \beta_0 that occur in pairs of positive roots (\beta_0, \gamma_0) satisfying \beta_0 + \gamma_0 = \beta + \gamma. The sign in the last relation can be chosen arbitrarily whenever (\beta, \gamma) is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.


References

* * * Lie groups Lie algebras {{algebra-stub