hw 1 bonus problems

What about the bonus problems? I hope you are thinking about those too, they’re very interesting and instructive.

On number 6: Clearly it is easy to find a finitely generated algebra B. Even inside a pretty simple B (which ones have you tried?) you can find subalgebras which are *not* finitely generated. But how do you prove that something is not finitely generated?

One tricky thing about this question is that there is so much room to experiment with! I imagine different people will find rather different examples. But if you have no idea how to get started, think about this: what is a finitely generated algebra B that is reasonably easy for you to think about? Inside there, pick an “easy” infinite set of generators for A. Do you need them all? If so, good! If not, try again. 🙂

On number 7: How do you get started? Any good ideas? Presumably you know about or have looked up Catalan numbers, to see that they arise in  *many, many* combinatorial contexts. Can you connect any of those with the algebra we are studying here?

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2 comments

  1. Brian Cruz

    For number 6: For it to be not finitely generated does it mean that we have an infinite list of generators of which a finite subset does not generate the algebra? Or does it mean there exists no finite set in the world that generates the algebra?

    For number 7: I thought I had proved my basis was a basis, but I’m having a difficult time showing that no two elements in my set are equivalent to any of the others via the relations. The good thing is that I know my basis is the right one because I’ve found a bijection with a combinatorial interpretation of the Catalan numbers (the so-called dyck paths). Any thoughts?

  2. 6. It means that there exists no finite set in the world that generates the algebra.

    7. My comments here might help:
    https://hopfcombinatorics.wordpress.com/2012/02/05/constructing-an-algebra
    Here the relations all tell you that one monomial is equal to another monomial, so you know a priori that you have a basis for A_n given by monomials in the e_i. The relations give you rules to transform one monomial to another, so you’ll need to show that no two of your C_n monomials can be obtained from each other by a sequence of those rules.

    I hope that helps.

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