- 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?