Face algebras and unitarity of SU(N)_L-TQFT

Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by $U_q (\frak{sl}_{N})$ at roots of unity. Also we obtain state-sums of ABF models on framed links which give quantum SU(2)-invariants of corresponding 3-manifolds.