Symmetry Groups Algebras Commutators Conserved Quantities

[friend]

Symmetry, Groups, Algebras, Commutators, Conserved Quantities

OK, maybe this is asking too much, hopefully not.

I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given.

If I understand what I'm reading, there is a connection between finite groups and algebras, though I think I'm confused between what objects are involved in each and what each acts upon if anything. And I understand that there is a conserved quantity for every symmetry, but this is only for symmetries of Action integral, right? What I'm not sure about, though, is whether there is a connection between algebras and commutation relations. Any insight you could give in these areas would be appreciated. Thanks.

 

[friend]

OK, maybe this is asking too much, hopefully not.

I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given.

If I understand what I'm reading, there is a connection between finite groups and algebras, though I think I'm confused between what objects are involved in each and what each acts upon if anything. And I understand that there is a conserved quantity for every symmetry, but this is only for symmetries of Action integral, right? What I'm not sure about, though, is whether there is a connection between algebras and commutation relations. Any insight you could give in these areas would be appreciated. Thanks.

For example, as I understand it, conserved quantities only apply to continuous symmetries of the Action integral. Or are there conserved quantities for finite symmetries as well?

I thought I heard people say that the commutation relations define the algebra, is this right?

Moderator: Do you think this thread should be moved to Quantum Physics Forum? That is what is motivating these questions.