In gauge theory, we understand matter and gauge fields by transformation under Lie algebras. We have seen that string theory naturally realizes such algebra. In heterotic string, momentum-winding quantum numbers provide representation of a state. Considering massive states and twisted states, this algebra is generalized it to the affine Lie algebra. It is generalized to twisted Lie algebra that describes twisted sector spectrum, which is a special feature in orbifold theory. Thus, it is crucial to understand the structure of the algebra.