Question

In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've seen sources that say that the SU(N) group has N²−1 generators, but actually these are generators for the Lie algebra aren't they?

Is this also true for representations? When we say a field is in the adjoint rep, does this typically mean the adjoint rep of the algebra rather than of the gauge group?

Answer 1

When discussing generators in Physics, it generally refers to the generators of the Lie algebra rather than the Lie group. This distinction is also true for representations. The Lie algebra captures the structure of the group and the generators help understand the underlying symmetries in a physical theory.

Step-by-step explanation:

In Physics papers, when discussing generators, it is typically referring to the generators of the Lie algebra rather than the generators of the Lie group. For example, the SU(N) group has N²−1 generators, but these are generators for the Lie algebra. Similarly, when referring to a field being in the adjoint representation, it usually means the adjoint representation of the algebra rather than the gauge group.

The Lie algebra of a Lie group is a vector space that captures the structure of the group, while the Lie group itself represents the actual transformations or symmetries of a physical system. So, by studying the generators of the Lie algebra, we can understand the underlying symmetries of a physical theory.

Understanding the Lie algebra and its generators is crucial in modern theoretical physics, particularly in the study of gauge theories and symmetries.