# Hopf algebras [Lecture notes] by Stefaan Caenepeel and J. Vercruysse

By Stefaan Caenepeel and J. Vercruysse

Syllabus 106 bij WE-DWIS-12762 "Hopf algebras en quantum groepen - Hopf algebras and quantum teams"

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Then any element m ∈ M is contained in a finite dimensional subcomodule of M . Proof. Let {ci | i ∈ I} be a basis of C, and write m i ⊗ ci , ρ(m) = i∈I where only finitely many of the mi are different from 0. The subspace N of M spanned by the mi is finite dimensional. We can write aijk cj ⊗ ck , ∆(ci ) = j,k and then mi ⊗ aijk cj ⊗ ck , ρ(mi ) ⊗ ci = i i,j,k hence mi ⊗ aijk cj ∈ N ⊗ C, ρ(mk ) = i,j so N is a subcomodule of M . 5 Let C be a coalgebra. Then the categories MC and C cop M are isomorphic.

If x ∈ G(C) is grouplike, then the associated C-coaction on A is given by ρ(a) = xa. If x ∈ G(C), then we call (C, x) a coring with a fixed grouplike element. For M ∈ MC , we call M coC = {m ∈ M | ρ(m) = m ⊗A x} the submodule of coinvariants of M ; note that this definition depends on the choice of the grouplike element. Also observe that AcoC = {b ∈ A | bx = xb} is a subring of A. An adjoint pair of functors Let i : B → A be a ring morphism. i factorizes through AcoC if and only if x ∈ G(C)B = {x ∈ G(C) | xb = bx, for all b ∈ B}.

2 Subcoalgebras and coideals Let C = (C, ∆, ε) be a coalgebra, and D a k-submodule of C. D is called a subcoalgebra of C if the comultiplication ∆ restricts and corestricts to ∆|D : D → D ⊗ D in this case, D = (D, ∆|D , ε|D ) is itself a coalgebra. 3 Let (Ci )i∈I be a family of subcoalgebras of C. Show that subcoalgebra of C. i∈I Ci is a again a A k-submodule I of C is called - a left coideal if ∆(I) ⊂ C ⊗ I; - a right coideal if ∆(I) ⊂ I ⊗ C; - a coideal if ∆(I) ⊂ I ⊗ C + C ⊗ I and ε(I) = 0. 4 Let k be a field, I be a left and right coideal of the coalgebra C; show that I is a subcoalgebra.