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Chevalley-Serre Relations

The Chevalley-Serre relations, sometimes called simply the "Serre relations," present a finite-dimensional semisimple Lie algebra over the complex numbers for each Cartan matrix. That is, if (A_(ij)) is a k×k Cartan matrix then, up to isomorphism, there exists a unique finite-dimensional semisimple Lie algebra g over the complex numbers (whose Cartan matrix is equivalent to (A_(ij))) such that g is defined by a set of 3k generators {e_i,f_i,h_i}_(i=1)^k subject to the following Chevalley-Serre relations:

1. [h_i,h_j]=0

2. [e_i,f_i]=h_i and [e_i,f_j]=0 if i!=j

3. [h_i,e_j]=A_(ij)e_j

4. [h_i,f_j]=-A_(ij)f_j

5. ad(e_i)^(1-A_(ij))(e_j)=0 if i!=j

6. ad(f_i)^(1-A_(ij))(f_j)=0 if i!=j.

Moreover, g has rank k and the h_i's generate a Cartan subalgebra. For proof, see Serre (1987).

See also

Cartan Matrix

Portions of this entry contributed by Shawn Westmoreland

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References

Fuchs, J. Affine Lie Algebras and Quantum Groups, An Introduction with Applications in Conformal Field Theory. Cambridge, England: Cambridge University Press, pp. 38-39, 1992.Samelson, H. Notes on Lie Algebras. New York: Springer-Verlag, p. 73, 1990.Serre, J. Complex Semisimple Lie Algebras. New York: Springer-Verlag, pp. 48-49 and 52-55, 1987.

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Chevalley-Serre Relations

Cite this as:

Weisstein, Eric W., with contributions by Shawn Westmoreland. "Chevalley-Serre Relations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Chevalley-SerreRelations.html

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