## Linearly recursive sequences, braided hopf algebras and ring

### Paddy McCrudden – 3 September 1997

Let L be the k-vector space of all linearly recursive sequences over k. In this semianar we define two multiplications, convolution and quantum convolution on L, and show that this makes L into an algebra. The results were shown in [1].
We prove the results by constructing the right adjoints to the functors [-,k] : k-Coalg ----> (k-Alg)^op [-,k] : Comon(Mod(H)) ----> (Mon(Mod(H)))^op
Where H is a Braided Hopf k-Algebra with invertible antipode. Usually these are called "Ring" and denoted by a superscripted circle. The new reserch in this seminar is the following: Let V be a closed braided monoidal category. Then there exists a monoidal 2-functor [-,I] : V^oprev ----> V thus there exists an induced 2-functor [-,I] : Comon(V^rev) --> Mon(V)^op We say that V has Algebra-Coalgebra Duality if there exists a right 2-adjoint to this functor. An examples of such a V is Modules over a commutative ring. Then Main result is that if V has Algebra-Coalgebra Duality, then so does Mod(H), where H is a braided hopf Monoid in V with invertible antipode. It is also claimed (but not yet proven!) that these results extend appropriately with Mon(V) and Comon(V) replaced by V-Cat and V^op-Cat. References: [1] Ng and Taft, Quantum Convolution of Linearly Recursive Sequences, Preprint, to appear in J.Alg.

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