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Using face algebras (i.e. algebras of L-operators of IRF models), weconstruct modular tensor categories with positive definite inner product, whosefusion rules and S-matrices are the same as (or slightly different from) thoseobtained by $U_q (\frak{sl}_{N})$ at roots of unity. Also we obtain state-sumsof ABF models on framed links which give quantum SU(2)-invariants ofcorresponding 3-manifolds.Comment: AMS-LaTeX, 36 pages, to appear in Commun. Math. Phy

Face Algebras and Unitarity of SU(N) L -TQFT