ON REPRESENTATIONS OF AFFINE TEMPERLEY–LIEB ALGEBRAS, II K. Erdmann and R.M. Green Mathematical Institute Oxford University 24–29 St. Giles’ Oxford OX1 3LB England E-mail:
[email protected] Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England E-mail:
[email protected] Abstract. We study some non-semisimple representations of affine Temperley–Lieb algebras and related cellular algebras. In particular, we classify extensions between simple standard modules. Moreover, we construct a completion which is an infinite dimensional cellular algebra. To appear in the Pacific Journal of Mathematics 1. Introduction arXiv:math/9811017v1 [math.RT] 4 Nov 1998 The affine Temperley–Lieb algebra, TL(An−1), is an infinite dimensional algebra which occurs as a quotient of the Hecke algebrab associated to a Coxeter system of type An−1. It occurs naturally in the context of statistical mechanics [10, 11], and may beb thought of as an algebra of diagrams (see [3, §4]). In [6], the second author used the diagram calculus and the theory of cellular algebras as described by Gra- ham and Lehrer [4] to classify and characterise most finite dimensional irreducible modules for TL(An−1) and for a larger algebra of diagrams, Dn. In fact, all simple b 1991 Mathematics Subject Classification. 16D70, 16W80. Part of this work was done at the Newton Institute, Cambridge, and supported by the special semester on Representations of Algebraic Groups, 1997. The second author was supported in part by an E.P.S.R.C. postdoctoral research assistantship. Typeset by AMS-TEX 1 2 K.