Linearly recursive sequences, Hopf algebras, and ``ring''

Paddy McCrudden (3/9/97)

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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Steve Lack

Last modified: Tue May 19 15:06:25 EST 1998