Chapter

Cartan basis, simple roots and fundamental weights

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0016
Cartan basis, simple roots and fundamental weights

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Cartan basis, simple roots and fundamental weights are described. In the Cartan basis generators are grouped into two types. The generators called Hj are Hermitian and mutually commute. There are r of them, where r is the rank of the algebra. The remaining dr generators (where d is the dimension of the algebra) are not Hermitian but come in Hermitian conjugate pairs. They can be viewed as eigenvectors of the Hj to the eigenvalue αj. These eigenvalues are collected into an r-component vector α called a root. Simple roots are a certain subset of the roots, which form a basis in the r-dimensional space. Weyl reflections are defined under which roots are reflected into roots. When the generators are represented by matrices the Hj are represented by diagonal matrices, i.e. the states in the representation can be taken as eigenstates of the Hj to the eigenvalue μj and the r-component vector μ is called the weight. In a finite-dimensional representation there is necessarily a state of highest weight. The fundamental weights are a certain subset of highest weights that provide a basis for highest weights.

Keywords: Cartan basis; Cartan subalgebra; simple roots; fundamental weights; weyl reflections

Chapter.  3762 words. 

Subjects: Mathematical and Statistical Physics

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