Problem 1
The tangent space at the identity carries a bracket from left-invariant vector fields and describes infinitesimal symmetry.
Faculty solution guide
Lie Groups, Lie Algebras and Exponential Maps
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Lie Groups, Lie Algebras and Exponential Maps.
Problem 4
The argument must address connecting global symmetry with infinitesimal brackets with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in gauge theory, particle physics, control, robotics, geometric phases, and characteristic classes, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.