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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 *H _{j}* are Hermitian and mutually commute. There are

*r*of them, where

*r*is the rank of the algebra. The remaining

*d*–

*r*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

*H*to the eigenvalue

_{j}*α*. These eigenvalues are collected into an

_{j}*r*-component vector

**called a root. Simple roots are a certain subset of the roots, which form a basis in the**

*α**r*-dimensional space. Weyl reflections are defined under which roots are reflected into roots. When the generators are represented by matrices the

*H*are represented by diagonal matrices, i.e. the states in the representation can be taken as eigenstates of the

_{j}*H*to the eigenvalue

_{j}*μ*and the

_{j}*r*-component vector

**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.**

*μ*

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

*Chapter.*
*3762 words.*

*Subjects: *
Mathematical and Statistical Physics

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