Hall 123

Crossed Products of Hopf algebras, the Drinfeld Double construction and a Duality Theorem.

Sandipan De

IMSc

Throughout H will denote a finite dimensional Hopf algebra over
a field k. Suppose H acts on a k-algebra A outerly, we show that A
in A crossed H in A crossed H crossed H^* is "basic
construction tower" and use this to sketch the prove a uniqueness theorem
which says that if H, K are finite dimensional Hopf algebras acting
outerly on a k-algebra A such that the resulting algebras A crossed H and
A crossed K are isomorphic via an algebra isomorphism which maps A onto A,
then H and K are isomorphic as Hopf algebras. Further we show that for
certain unital inclusion of algebras A in B ( where B is infinite crossed
product of H and H^*'s), B is given by action of D(H)^cop on A where D(H)
is the Drinfeld Double of H. And also try to sketch the proof of a duality
theorem which states that if H acts an a k-algebra A, then A crossed H
crossed H^* and A tensor End(H) are isomorphic as algebras.