dpa said:
Hi Everyone,
I am kind of looking some online text to understand Lie Algebra, Group Theory and so forth.
I usually need application (everyday/science context how it is used) and intuition more than mere mathematical definition to understand topics. So I need some text that gives very deep understanding of the text.
And I am not sure whether that topic falls under Linear and Abstract Algebra or under some other thread.
Thank You
Sincerely
DPA
I don't know about textbooks, but I think I can offer one way to see how group theory is useful.
Consider a chess game. In chess we have rules for every chess piece in terms of moving a piece whether its just a movement or whether it involves moving your piece to steal an opponents piece.
Now consider the fact that you can always 'undo' every action that you carried out: in other words, if you gave me the list of every move you and your opponent gave for the entire game as well as the final state of the game: I could reverse the whole process all the way back to the original setup of the board.
If you think in terms of a symmetry, you can see that with a group, you can understand a particular kind of process symmetry in this chess example: one important property of groups is that they are reversible and thus have a form of symmetry in this context.
Now consider that instead of just chess pieces, we want to take this idea and consider all kinds of transformations. The transformation may involve rotating a shape or something along the lines of solving a rubix cube. It might also have to do with number theory, which is what has happened with crytography: because public-key cryptography can be seen in a group-theoretic context (you must be able to get back the original message from the encrypted message for cryptography to be useful), then by finding things out about groups, you also indirectly learn something about cryptographic algorithms.
This idea of reversibility, or inversibility, or symmetry or whatever you want to call it is an important way to study systems. Also if something really is a group, then it means that because of this symmetry you can create a process and undo it: I know I've said this many times but realize that many things in mathematics do not have this property, but if something is a group, then it must which means we can look at the consequences of something having this property.
The key word is consequence: mathematics (a large part of it at least) is concerned with taking an idea that seems useful and to exploring what the consequences of that idea has on particular attributes: in other words, we are given this object that has all these properties (inverses, identities, associativity and closure) and then we say "well what does this mean for these kinds of structures in the context of _____"
