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Math Help - bialgebra, coideal

  1. #1
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    Mar 2010
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    bialgebra, coideal

    • Hi! These are my problems, thanks:

      1)
      I can't figure out what bialgebra is. In my book (Montgomery "Hopf Algebras and Their Actions on Rings") it is said:
      "A k-space B is a bialgebra if (B, m, u) - is an algebra, (B, d, e) - is a coalgebra, and either of the following (equivalent) conditions holds:
      1) d and e are algebra morhisms,
      2) m and u are coalgebra morphisms.
      (m - multiplication, u - unit, d - comultiplication, e - counit)

      In Wikipedia: Bialgebra - Wikipedia, the free encyclopedia.

      So I can't understand why this is the same thing. Why is there a twist map in the first diagram?

      2)
      "A subspace I of coalgebra C is a coideal if d(I) lies in IxC+CxI (x - tensor product), and if e(I)=0.
      It is easy to check that if I is a coideal, then the k-space C/I is a coalgebra with a comultiplication induced from d..."

      I think i almost get it, but i can't write it down strictly. How to prove it correctly?
    Last edited by dsmmbrd; March 24th 2010 at 01:53 PM.
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