Monoids, Embedding Functors and Quantum Groups

Abstract

We show that the left regular representation π_l of a discrete quantum group (A,Δ) has the absorbing property and forms a monoid (π_l,\tilde{m},\tildeη) in the representation category Rep(A,Δ). Next we show that an absorbing monoid in an abstract tensor *-category C gives rise to an embedding functor E:C->Vect_C, and we identify conditions on the monoid, satisfied by (π_l,\tilde{m},\tildeη), implying that E is *-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb} the generalized Tannaka theorem produces a discrete quantum group (A,Δ) such that C is equivalent to Rep_f(A,Δ). Thus, for a C^*-tensor category C with conjugates and irreducible unit the following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A,Δ), (2) C admits an absorbing monoid, (3) there exists a *-preserving embedding functor E: C->\mathrm{Hilb}.

arXiv commentFinal version, to appear in Int. Journ. Math. (Added some references and Subsection 1.2.) Latex2e, 21 pages