Great link, thanks! I alway enjoy reading summaries of other fields written by/for mathematicians, since the assumption of mathematical background allows a lot of fluff to be cut out (compared to, e.g. reading a textbook about physics intended for physics students).
For background reading, the notes by Teleman 2005, "Representation Theory" [1] are a good intro to the topic.
Learning about representation theory helped me understand the power of thinking about mathematical objects in terms of their action on other objects. Just as representation theory studies group actions on vector spaces, the theory of modules is best described as the study of ring actions on commutative groups. Many things happen to be rings (e.g. endomorphism rings of functions, where addition=pointwise addition and multiplication=composition), and modules allow us to apply (almost all of) vector space theory to better understand ring-like objects.
For background reading, the notes by Teleman 2005, "Representation Theory" [1] are a good intro to the topic.
Learning about representation theory helped me understand the power of thinking about mathematical objects in terms of their action on other objects. Just as representation theory studies group actions on vector spaces, the theory of modules is best described as the study of ring actions on commutative groups. Many things happen to be rings (e.g. endomorphism rings of functions, where addition=pointwise addition and multiplication=composition), and modules allow us to apply (almost all of) vector space theory to better understand ring-like objects.
[1] https://math.berkeley.edu/~teleman/math/RepThry.pdf
[2] Another favorite from John Baez: https://groups.google.com/d/msg/sci.physics.research/aiMUJrO...