Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l(b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class alpha of the constant functor with value k(x) on the orbit category (F) over bar (c) of F-centric subgroups Q of P of b which "glues together" the second cohomology classes alpha(Q) of Aut((F) over bar)(Q) with values in k(x) in Kulshammer-Puig [Invent. Math. 102 (1990) 17-71]. We show that if a exists, there is a canonical quasi-hereditary k-algebra (F) over bar (b) such that Alperin's weight conjecture becomes equivalent to the equality 1(b) = 1((F) over bar (b)). By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779-856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of F-c by the center functor Z. If both invariants alpha C exist, we show that there is an O-algebra C(b) associated with b having (F) over bar (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b) = l(L(b)); furthermore, if b has an abelian defect group, C(b) is isomorphic to a source algebra of the Brauer correspondent of b. (C) 2004 Elsevier Inc. All rights reserved.