CELLULAR ALGEBRAS

A class of associative algebras (''cellular'') is defined by means of multiplicative properties of a basis. They are shown to have cell repr esentations whose structure depends on certain invariant bilinear form s. One thus obtains a general description of their irreducible represe ntations and block theory as well as criteria for semisimplicity. Thes e concepts are used to discuss the Brauer centraliser algebras, whose irreducibles are described in full generality, the Ariki-Koike algebra s, which include the Hecke algebras of type A and B and (a generalisat ion of) the Temperley-Lieb and Jones' recently defined ''annular'' alg ebras. In particular the latter are shown to be non-semisimple when th e defining parameter delta satisfies gamma(g(n))(-delta/2)=1, where ga mma(n), is the n-th Tchebychev polynomial and g(n) is a quadratic poly nomial.