#### 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.

Done